Article pubs.acs.org/IECR
Complex Cooling Water Systems Optimization with Pressure Drop Consideration Khunedi V. Gololo†,‡ and Thokozani Majozi*,† †
Department of Chemical Engineering, University of Pretoria, Lynnwood Road, Pretoria, South Africa Modelling and Digital Science, Council for Scientific and Industrial Research (CSIR), Pretoria, South Africa
‡
ABSTRACT: Pressure drop consideration has shown to be an essential requirement for the synthesis of a 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 while the superstructure allows for reuse of the cooling water. The proposed technique offers the opportunity to debottleneck the cooling water systems with multiple cooling towers while maintaining a minimum pressure drop. This technique, which was previously used in a cooling water network with a single source, has been adapted in a cooling water network with multiple sources. The mathematical formulations exhibit 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. The case studies showed that the circulating cooling water flow rate can be reduced by up to 26% at a minimum cooling water network pressure drop.
1. INTRODUCTION Cooling water systems are used in many industries to remove waste heat from the process to the environment. This utility system consists of a cooling water network, which is used to exchange heat from the process into the cooling medium (i.e., water), and a cooling tower, which discharges heat from the cooling water to the environment. Research in this area has focused on individual components of the cooling water system. Bernier1 developed a onedimensional mathematical model that predicts the behavior of a cooling tower. The model was based on the thermal behavior of the water droplet in a spray-type cooling tower. The author assumed a Lewis factor of 1 and that the model was based on a tower with no packing. Klopper and Kröger2 later showed the influence of the Lewis factor on the performance prediction of a wet cooling tower. Kim and Smith3 also developed a cooling tower model by assuming a constant water flow rate through the tower. Kröger4 developed a comprehensive counter-current cooling tower model with a differential water flow rate by assuming that the interface temperature of water was the same as the bulk temperature. A more rigorous approach was taken by Qureshi and Zubair,5 who developed a mathematical model that incorporated all regions of a cooling tower (i.e., the spray, fill, and rain zones). The authors also studied the importance of including the fouling model. Castro et al.6 and Cortinovis et al.7,8 attempted a holistic approach for the synthesis of cooling water systems by considering the system as a whole. The authors developed mathematical models for optimizing the heat exchanger network with parallel configuration. The authors minimized the operating cost by changing the cooling tower fan speed and hot blowdown flow rate. These contributions were limited to one cooling source and did not explore any reuse/recycle opportunities in the cooling water network. Several authors used the holistic approach to optimize and synthesize the cooling water systems in which the opportunities for recycle and reuse are explored. Kim and Smith3 used the graphical © 2012 American Chemical Society
technique to debottleneck a cooling water system with a single source. Ponce-Ortega et al.9 presented a mathematical model for the synthesis of cooling water networks that was based on a stagewise superstructural approach. Their formulation was a mixed-integer nonlinear programming (MINLP) structure. This work included the cooling tower model, and the pressure drop for each cooler was considered. Panjeshahi and Ataei10 extended the work of Kim and Smith3 on cooling water system design by incorporating a comprehensive cooling tower model and an ozone treatment for the circulating cooling water. A different approach was taken by Majozi and Moodley,11 who developed a mathematical model for optimizing cooling water systems with multiple cooling towers. The authors considered four scenarios to develop the mathematical formulations, which yielded linear programming (LP), mixed-integer linear programming (MILP), nonlinear programming (NLP), and MINLP structures. This work was later improved by Gololo and Majozi,12 by incorporating the cooling tower model to study the interaction of the cooling water network and the cooling towers. Although the reuse/recycle philosophy offers a good debottlenecking opportunity, the topology of the associated cooling water network is more complex and, therefore, is prone to higher pressure drop than the conventional parallel design. Kim and Smith13 presented a paper on retrofit design of cooling water systems in which pressure drop was taken into consideration. The authors used a graphical technique to target the minimum circulating water flow rate and mathematical technique to synthesize a cooling water network. Their formulation was a MINLP structure. This work was limited to one cooling source. Special Issue: PSE-2012 Received: Revised: Accepted: Published: 7056
September 15, 2012 December 5, 2012 December 12, 2012 December 12, 2012 dx.doi.org/10.1021/ie302498j | Ind. Eng. Chem. Res. 2013, 52, 7056−7065
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Figure 1. Superstructure for a cooling system.12
water using operations, the cooling water using operations with their limiting temperatures and heat duties, the limiting temperature for each cooling tower fill, the dimensions for each cooling tower, and the coefficient of performance correlation for each cooling tower determine the minimum cooling water network pressure drop for a cooling water system with multiple cooling towers while maintaining a minimum circulating cooling water flow rate.
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 while the superstructure allows for reuse of the cooling water. This technique was previously used by Kim and Smith13 to synthesize a cooling water network with a single source. However, in this paper, the CPA is adapted for a cooling water network with multiple sources. Furthermore, the detailed cooling tower model is also incorporated.
4. 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 flow rate and in the second step the CPA is incorporated to synthesize the cooling water network with a minimum pressure drop. The cooling tower model developed by Kröger4 is used to predict the outlet conditions of the cooling towers and the overall effectiveness of the cooling towers. Using the superstructure in Figure 1, the mathematical formulation is developed considering two cases. The first case involves a cooling water system with no dedicated cooling water sources and sinks. This implies that a set of heat exchangers can be supplied by any cooling tower and return the cooling water to any cooling tower. The second case involves a cooling water system with dedicated cooling water sources and sinks. This implies that a set of heat exchangers 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.11 The cooling water network model is adapted from the work of Gololo and Majozi.12
2. MOTIVATION Majozi and Moodley11 and Gololo and Majozi12 conducted studies to debottleneck the cooling water systems with multiple cooling sources. In both papers, the topology of the debottlenecked cooling water network was more complex, because of the additional reuse/ recycle streams required. This has a potential to increase the pressure drop of the cooling water network, thus increasing the pumping cost. Therefore, it is imperative to consider the cooling water network pressure drop when synthesizing and optimizing cooling water systems. This paper aims to synthesis the cooling water network with a minimum pressure drop while maintaining a minimum circulating water flow rate. In the previous contributions, the complex cooling water system was debottlenecked; however, in this work, the adapted CPA is incorporated in the model to select the cooling water network topology with a minimum pressure drop. 3. PROBLEM STATEMENT The problem addressed in this paper can be stated as follows. Given a set of cooling towers with their dedicated set of cooling 7057
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4.1. Case I. In this case, there are no dedicated source and sink for any operation using cooling water. This implies that a set of heat exchanger scan be supplied by any cooling tower. The mathematical model consists of the following sets, parameters, and variables.
z(n,i) binary variable indicating the existence of a stream from cooling water using operation i to cooling tower n The following constraints are considered to develop the mathematical model for a cooling water network. Mass Balance Constraints. Constraint (1) sets the bound for circulating cooling water flow rate in cooling tower n. S is a slack variable used to relax the target.
Sets
i { i|i is a cooling water using operation} n { n|n is a cooling tower}
OS(n) = T (n) + S(n)
Parameters
Q(i) duty of cooling water using operation i (kW) Tctout(n) cooling water supply temperature from cooling tower n (°C) OSn(n) maximum design capacity of cooling tower n (kg/s) TUout(i) limiting outlet temperature of cooling water using operation i (°C) TUin(i) limiting inlet temperature of cooling water using operation i (°C) FUin(i) maximum inlet flow rate of cooling water using operation i (kg/s) TUret(n) limiting inlet temperature of cooling water using operation n (°C) B(n) blowdown flow rate for cooling tower n (kg/s) M(n) makeup flow rate for cooling tower n (kg/s) E(n) blowdown flow rate for cooling tower n (kg/s) cp specific heat capacity of water; cp = 4.2 kJ/(kg °C) Tamb ambient temperature (°C) T target (kg/s) di heat-exchanger tube inner diameter (m) do heat-exchanger outer tube diameter (m) A heat exchanger area (m)2 ρ density (kg/m3) μ viscosity (Ns/m2) Nt number of tubes in the heat exchanger ntp number of heat exchanger tube passes
(1)
Constraint (2) stipulates that the total cooling water is the sum of all cooling water from cooling tower n.
∑ OS(n)
CW =
(2)
n∈N
Constraints (3) and (4) ensure that the inlet and outlet cooling water flow rates of cooling tower n are equal.
∑ CS(n , i) − M(n)
OS(n) =
∀n∈N (3)
i∈I
∑ CR(i , n) − B(n) − E(n)
OS(n) =
∀n∈N
i∈I
(4)
The total water flow rate to water using operation i is the sum of all reused cooling water from operation i′ and the sum of cooling water flow rates from cooling tower n, as given in constraint (5).
∑ CS(n , i) + ∑ FR(i′, i)
Fin(i) =
n∈N
∀i∈I (5)
i ′∈ I
The total water flow rate from water using operation i is the sum of all reused cooling water to operation i′ and the sum of cooling water flow rates to cooling tower n, as given in constraint (6). Fout (i) =
Continuous Variables
∑ CR(i , n) + ∑ FR(i , i′) i∈I
Operating capacity of cooling tower n (kg/s) overall cooling water supply from all cooling tower (kg/s) CS(n,i) cooling water supply from cooling tower n to cooling water using operating i (kg/s) CR(n,i) return cooling water to cooling tower n from cooling water using operating i (kg/s) FR(i′,i) reuse cooling water to cooling water using operating i′ from cooling water using operating i (kg/s) Fin(i) total cooling water into cooling water using operation i (kg/s) Fout(i) total cooling water from cooling water using operation i (kg/s) fj flow rate through the heat exchanger (kg/s) Fp flow rate through piping (kg/s) Tin(i) inlet temperature of cooling water to cooling water using operation i (°C) Tout(i) outlet temperature of cooling water to cooling water using operation i (°C) Ts(n) cooling water supply temperature from cooling tower n after adding makeup (°C) ΔP pressure drop (kPa) S slack variable (kg/s) OS(n) CW
∀i∈I (6)
i ′∈ I
Constraint (7) ensures that water is conserved through each cooling water using operation i: Fin(i) = Fout(i)
∀i∈I
(7)
Energy Balance Constraints. Constraint (8) is the definition of inlet temperature into operation i Fin(i)Tin(i) =
∑ FR(i′, i)Tout(i′) + ∑ CS(n , i)Ts(n) i ′∈ I
n∈N
(8)
∀i∈I
Constraint (9) is the definition of circuit inlet temperature from cooling tower n after adding makeup. Ts(n)(OS(n) + M(n)) = M(n)Tamb + OS(n)Tct out(n) (9)
∀n∈N
Constraint (10) is the definition of the return temperature to cooling tower n: Tret(n) ∑ CR(i , n) = i∈I
∑ CR(i , n)Tout(i)
∀n∈N
i∈I
(10)
Binary Variables
The energy balance across water using operation i is given by constraint (11):
x(n,i) binary variable indicating the existence of a stream from cooling tower n to cooling water using operation i y(i′,i) binary variable indicating the existence of a stream from cooling water using operation i′ to cooling water using operation i
(Tout(i) − Tin(i))Fin(i)cp = Q (i) 7058
∀i∈I
(11)
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Figure 2. Cooling water system superstructure: (a) single source (b) and multiple sources.
Figure 3. Multiple-source cooling water system superstructure.
CR(i , n) ≤ FinL(i)z(i , n)
By substituting constraint (8) into constraint (11), the bilinear term Fin(i)Tin(i) can be eliminated and constraint (8) and constraint (11) can be replaced by constraint (12). Q (i) + cp
∑ xn,i + ∑ yi′ ,i = K
i ′∈ I
= Fin(i)cpTout(i)
∀i∈I
n
(12)
Tret(n) ≤
u Tret
∀n∈N
∀n∈N
(13) (14)
∀i∈I
CS(n , i) ≤
Nt1 =
(15)
i)
∀ i ∈ I, ∀ n ∈ N
(16)
CS(n , i) ≤ FinL(i)x(n , i)
∀ i ∈ I, ∀ n ∈ N
(17)
Nt 2 =
∀i∈I
(18)
FR(i′, i) ≤ FinL(i)y(i′, i)
∀i∈I
(19)
∀ i ∈ I, ∀ n ∈ N
π 2.8ρNt 2.8dodi 4.8
(24)
20ntp3ρ π 2Nt 2di 4
(25)
⎛ ⎞ 1 ΔP = Np⎜⎜ 0.36 ⎟⎟ ⎝ Fp ⎠
13
(26)
where Fp is the flow rate of any stream (CS(n, i), F(i′, i), CR(i, n))), and Np =
Constraints (20) and (21) assign a binary variable and set bounds for any stream from operation i to cooling source n. CR(i , n) ≤ Finu (i)z(i , n)
1.115567μ0.2 ntp 2.8A
The line pressure drop is calculated from constraint (26).
Constraints (18) and (19) assign a binary variable and set the bounds for any reuse stream from operation i′ to operation i. FR(i′, i) ≤ FinU (i)y(i′, i)
(23)
where
Constraints (16) and (17) assign a binary variable and set the bounds for any stream from cooling source n to operation i. FinU (i)x(n ,
(22)
ΔP(i) = Nt1Fin(i)1.8 + Nt 2Fin(i)2
Constraint (15) sets the upper limit of the inlet flow rate for operation i. Fin(i) ≤ Finu (i)
∀ i ∈ I, ∀ n ∈ N
i′
where K is a natural integer value. Pressure Drop Constraints. The cooling water network model by Gololo and Majozi12 is improved by incorporating the modified heat exchanger and pipe pressure drop correlations of Nie and Zhu,14 shown in constraints (23) and (26), respectively. In this paper, the correlation of Nie and Zhu14 is expressed in terms of mass flow rate.
Design Constraints. The cooling tower design constraints are given in constraints (13) and (14). Constraint (13) sets the upper limit of the flow rate for cooling tower n. Constraint (14) sets the upper limit of the return water temperature for cooling tower n. OS(n) ≤ OSu(n)
(21)
Constraint (22) is used to limit the number of inlet streams to any operation using water.
∑ CS(n , i)T(n) + cp ∑ FR(i′, i)Tout(i′) n∈N
∀ i ∈ I, ∀ n ∈ N
188.318ρ0.536 μ0.2 L π 1.8
(27) 13
The CPA from the paper of Kim and Smith is adapted to select the cooling water network with a minimum pressure drop. The authors used the superstructure shown in Figure 2a.
(20) 7059
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nonlinear programming (NLP). The model is solved by minimizing circulating cooling water flow rate (CW) given in constraint (2). The debottlenecking model with pressure drop consists of constraints (1)−(11) and (13)−(35). This formulation consists of binary terms and bilinear terms thus rendering the model MINLP. The objective function for this model is given in constraint (35). 4.2. Case II. In this case, there are dedicated source and sink for any cooling water using operation i. This implies that no premixing or post-splitting of cooling water return is allowed. A set of heat exchanger can only be supplied by one cooling tower. The mass and energy balance constraint in case I are included in case II. To control sources and sinks in each cooling water using operation i, constraints (16)−(21) and (36) and (37) are incorporated. Constraints (16) and (36) prevent pre-mixing. Constraint (36) ensures that cooling water using operation i can only be supplied by a maximum of one cooling tower n.
The superstructure is based on a single-source cooling water network. By modifying the superstructure for a single source cooling water systems, a multiple sources superstructure is shown in Figure 2b. The CPA used by Kim and Smith13 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. Constraint (28) is used to identify the maximum pressure drop path between the source and the sink. Pm − Pn ≥ ΔPmn
(28)
To cater for multiple sources and sinks, the superstructure in Figure 2b is modified by using a single imaginary source and sink, as shown in Figure 3. Constraint (29) is then used to define the pressure of source node n from the imaginary source node. PS , img − PS , n = ΔP
(29)
CS(n , i) ≤ FinU (i)x(n , i)
Constraints (30)−(33) represent the CPA adapted from Kim and Smith.13 Constraint (34) 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 a minimum pressure drop. PS , n − Pin , i + LV(1 − x(n , i)) ≥ ΔPn , i
(30)
Pout, i ′ − Pin, i + LV(1 − y(i′ , i)) ≥ ΔPi ′ , i
(31)
∑ x(n , i) ≤ 1
∀ i ∈ I, ∀ n ∈ N
∀n∈N (36)
n∈N
Post-splitting is prevented by constraints (20) and (37). Constraint (37) ensures that cooling water using operation i can supply a maximum of one cooling tower n. CR(i , n) ≤ FinU (i)z(i , n)
∑ z(i , n) ≤ 1
∀ i ∈ I, ∀ n ∈ N
(32)
Pout, i − PE , n + LV(1 − z(i , n)) ≥ ΔPi , n
(33)
(37)
Constraints (38) and (39) ensure that the source and the sink cooling water supply is the same for a particular cooling water using operation i.
x, y, and z are binary variables indicating the existence of a stream from any source n/operation i′ to operation i/sink n. LV is a large value. PE , n − PE , img ≥ ΔPn , img
x(i , n) ≤ z(n , i) + (2 −
n∈N
(38)
(34)
z(i , n) ≥ z(n , i) + (2 −
∑ x(n , i) − ∑ z(i , n)) n∈N
n∈N
(39)
The targeting model consists of constraints (2)−(11), (13)−(21), and (36)−(39). The formulation has bilinear terms and binary variables, thus rendering the model MINLP. The model is solved by minimizing circulating water flow rate (CW) given in constraint (2). The debottlenecking model with pressure drop consists of constraints (1)−(11) and (13)−(39). The formulation also consists of binary terms and bilinear terms, thus rendering the model MINLP. The objective function for this model is given in constraint (35). The MINLP models exhibit multiple local optimum solutions and they are also generally difficult to solve, because the starting point might yield suboptimal or infeasible results. Thus, it is important to obtain a good starting point before solving the exact MINLP problem. The solution procedure outlined in section 5.1 is used to address this problem. 4.3. Cooling Tower Model and the Overall Effectiveness of the Cooling Towers. Cooling Tower Model. The cooling tower model used in this paper was developed by Kröger.4 The model predicts the evaporation and the outlet water conditions. The author made the following assumptions: • Interface water temperature is the same as the bulk temperature. • Air and water properties are the same at any horizontal cross section.
⎛ CW ⎞ ⎟(ΔP )(1 h) OB = (cost power)⎜ ⎝ 1000 ⎠ (35)
where cost power :
∑ x(n , i) − ∑ z(i , n)) n∈N
The network topology with a minimum pressure drop is then synthesized by minimizing the objective function shown in constraint (35). The expression in constraint (35) also minimizes the slack variable, which is used to relax the targeted circulating water flow rate (CW). Other parameters in constraint (35) are used to make the expression dimensionally consistent.
+ (cost power)(CW)(3600 s)
(20)
∀ i ∈ I, ∀ n ∈ N
n∈N
Pin, i − Pout, i = ΔPi
(16)
c.u. kWh
CW m3 : 1000 s ΔP: kPa c.u. cost power : L kg CW: s
The targeting model consists of constraints (2)−(11) and (13)− (15). This formulation has bilinear terms, thus rendering the model a 7060
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• Heat- and mass-transfer areas are identical. The mathematical model that is shown in the Appendix is solved in the MATLAB platform. The Overall Effectiveness of the Cooling Towers. The effectiveness is defined as the ratio of actual heat transferred over the maximum theoretical amount of heat that can be transferred shown in constraint (40).15 In a circuit consisting of multiple cooling towers, the overall effectiveness for the cooling towers is evaluated based on the procedure by Gololo.16 The author used the thermodynamic definition of the effectiveness for one cooling tower to derive the expression for the overall effectiveness of cooling towers as shown in constraint (41). ε=
Q act Q max
=
mwcpw(Tw ,in − Tw ,out) mcap(Has ,out − Ha ,in)
(40)
where mw is the cooling tower water flow rate, cpw is the specific heat capacity of water, (Tw,in − Tw,out) is the cooling tower range, mcap is the water capacity rate, Has,out is the enthalpy of saturated air at the inlet water temperature, and Ha,in is the enthalpy of air at the inlet air temperature.
Figure 4. Illustration of the piecewise linearization technique.13
5.2. Overall Cooling Water System. Optimization of the overall cooling water system requires simultaneous solution of both the cooling water network and the cooling tower model. The procedure starts by targeting the overall circulating cooling water flow rate with no inclusion of pressure drop constraints. The results from the targeting model are the flow rate and inlet temperature for each cooling tower. The cooling tower model is then used to calculate the outlet temperature, flow rate, and evaporation for each cooling tower. These conditions are then used as the inputs to the cooling water network model with pressure drop constraints. Using the target, the model with pressure drop constraints is solved by minimizing the overall pressure drop. There is an iterative process between the cooling water network model and the cooling tower model, as shown in Figure 5. If the difference between the outlet temperature of the cooling tower model and the previous inlet temperature to the cooling water network is
