On Synthesis and Optimization of Cooling Water ... - ACS Publications

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On Synthesis and Optimization of Cooling Water Systems with Multiple Cooling Towers Khunedi Vincent Gololo†,‡ and Thokozani Majozi*,† † ‡

Department of Chemical Engineering, University of Pretoria, Lynnwood Road, Pretoria, 0002, South Africa Modelling and Digital Science, Council for Scientific and Industrial Research (CSIR), Pretoria, South Africa ABSTRACT: Cooling water systems are generally designed with a set of heat exchangers arranged in parallel. This arrangement results in higher cooling water flow rate and low cooling water return temperature, thus reducing cooling tower efficiency. Previous research on cooling water systems has focused mainly on heat exchanger network thus excluding the interaction between heat exchanger network and the cooling towers. This paper presents a technique for grassroot design of cooling water system for wastewater minimization, which incorporates the performances of the cooling towers involved. The study focuses mainly on cooling systems consisting of multiple cooling towers that supply a common set of heat exchangers. The heat exchanger network is synthesized using the mathematical optimization technique. This technique is based on superstructure in which all opportunities for cooling water reuse are explored. The cooling tower model is used to predict the thermal performance of the cooling towers. Two case studies are presented to illustrate the proposed technique. The first case resulted in nonlinear programming (NLP) formulation and the second case yield mixed integer nonlinear programming (MINLP). The nonlinearity in both cases is because of the bilinear terms present in the energy balance constraints. In both cases, the cooling towers operating capacity were debottlenecked without compromising the heat duties.

1. INTRODUCTION Industrial development and other economic activities have led to an increase in fresh water consumption and contamination of freshwater resources. One of the major water using operations in industries is the cooling water systems. Cooling water systems use equipments such as cooling towers to remove waste heat from the process to the atmosphere. These systems also generate wastewater through the blowdown mechanisms. Escalating costs of waste treatment, stricter environmental regulations on industrial effluent and scarce water resources have led to studies which concern various means of minimizing water usage and waste generation. Previous research on cooling water systems has focused mainly on debottlenecking the cooling towers through synthesis and optimization of the cooling water networks. The common technique used in this regard was based on graphical analysis approach.1-3 This technique was derived from the principles of pinch analysis developed for heat exchanger networks synthesis by Linnhoff and co-workers.4-7 The principles of pinch analysis were also adapted for mass exchange network synthesis8 and later applied for targeting and synthesis of wastewater minimization problems.9 Few authors used the superstructural modeling techniques to optimize the cooling water systems.10-12 Besides the application of this technique for cooling water systems optimization, it was also used by Takama et al.,13 Gunaratnma et al.,14 and Alva-Argaez et al.15 for optimization of various water using systems. The synthesis of cooling water systems which takes into consideration the interaction between cooling water networks and the cooling towers has not been fully explored. The following section gives a brief overview of developments in this regard. r 2011 American Chemical Society

1.1. Cooling Water System Design: Pinch Analysis Based Techniques. Heat exchanger networks in a cooling system are

generally designed with a set of heat exchangers arranged in parallel. This design means that all heat exchangers receive cooling water at the same supply temperature and the outlet streams are mixed before being returned to the cooling tower. In cases where some of the cooling duties do not require cooling water at the cooling water supply temperature, this arrangement results in higher cooling water flow rate and low cooling water return temperature thus reducing the cooling tower efficiency.16 In most cases not all heat exchanger inlet temperatures should be at the cooling source supply temperature. Higher inlet temperatures can be tolerated as long as the ΔTmin approach is satisfied. If the outlet temperature of the cooling water from one heat exchanger is at least ΔTmin lower than the process temperature in any heat exchanger, it could be reused to supply those heat exchangers thereby reducing the freshwater consumption. Mixing two or more cooling water outlet streams from various heat exchangers can result in a stream with temperature at least ΔTmin lower than the process temperature in other heat exchangers thus giving an opportunity for cooling water reuse. The recent study on cooling water system design was conducted by Kim and Smith.1 They developed a methodology for grassroot design of a cooling water system with one cooling Special Issue: Water Network Synthesis Received: June 30, 2010 Accepted: December 2, 2010 Revised: November 26, 2010 Published: January 13, 2011 3775

dx.doi.org/10.1021/ie101395v | Ind. Eng. Chem. Res. 2011, 50, 3775–3787

Industrial & Engineering Chemistry Research source taking into account the cooling tower performance. The cooling tower model was developed to predict the outlet conditions of water from the cooling tower for a given inlet water conditions. The design for the overall cooling water system was carried out by investigating the interaction between cooling water network and the cooling tower performance. The authors further used the principles of pinch analysis for targeting and design of cooling water network. The water mains method by Kuo and Smith17 was readily used for cases were maximum reuse was allowed, where the heat exchanger outlet temperature was allowed to go as high as possible. In most practical situations, the return water temperature is constrained. The cooling water supply line for these situations does not make a pinch with the composite curve, which implies that the water mains method cannot be readily applied. The concept of pinch migration and temperature shift was introduced to handle problems were process pinch does not exist. Although the authors designed the water network by considering cooling tower performance, the research was only limited to one cooling source which is not the case in most practical situations. Kim and Smith18 further presented a systematic method for the design of cooling water system to reduce makeup water. The makeup water reduction was achieved by reuse of water from wastewater generation processes. The authors assumed that wastewater was available at allowable contaminant concentration and with unlimited quantities. To reduce fresh makeup water usage, Kim and Smith18 added wastewater before or after the cooling tower depending on its temperature. If the wastewater temperature was higher than the inlet cooling system temperature, it was added before the cooling tower. This ensured that the cooling water did not gain heat before entering the cooling water using operations. Thus wastewater with lower temperature could be added after the cooling tower. The cooling water system network was designed using the procedure developed by Kim and Smith.1 Majozi and Nyathi19 also developed a methodology for a cooling water system design by combining graphical approach and mathematical programming. The graphical approach was used for targeting minimum cooling water flow rate. The mathematical optimization technique was then used for water using operations network design. Only cooling systems with at least two cooling towers were considered. This work did not include the interaction of the heat exchanger network with the cooling tower performance. Panjeshahi et al.3 also developed a similar methodology using advanced pinch design (APD). The authors used an optimization approach for targeting. The objective was to find the solution that minimizes capital and operating cost at maximum water reuse and minimum cooling water flow rate. The advanced pinch design method was developed which allowed the interaction between the cooling tower performance and heat exchanger network. APD was divided into three stages. In the first stage the feasible region was defined, the region was further explored in the second stage to target the supply line and the third stage was designing cooling water network for target conditions with pinch migration concept using water mains method. The cooling tower model derived by Kr€oger20 was used to predict the cooling tower outlet water conditions. This research was also limited to one cooling source. 1.2. Cooling Water System Design: Mathematical Optimization Techniques. Takama et al.13 were first to use mathematical

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Figure 1. Superstructure for water using network.

programming for targeting and designing water using networks in the refinery. This technique involved superstructure in which all possible network features were explored. The possible features included recycles and reuse as shown in Figure 1. The superstructure was optimized subject to material balance at each node and across each unit. The strength of this technique lies in its ability to handle many practical constraints, for example, forced or forbidden matches, capital cost functions, control, and safety constraints. Kim and Smith,11 Majozi and Moodley,10 and Castro et al.21 solved the problems of cooling water using networks through mathematical optimization techniques. The formulated problems were generally nonconvex NLP10,21 or MINLP.11,10 Optimization of nonlinear problem generally yields a local optimum solution depending on the starting point. Thus it is important to best initialize the problem. Kim and Smith11 extended their work1 by incorporating pressure drop into their design. Targeting was done by applying the method of Kim and Smith.1 The mathematical optimization technique was used to synthesize the heat exchanger network with the least pressure drop. Their formulation was MINLP model which was solved by first linearizing the problem through setting the outlet temperature of each heat exchanger to maximum value and using linear estimation of pressure drop correlations. The solution of the linearized problem was then used as a starting point for MINLP model. Majozi and Moodley10 developed a mathematical optimization technique for synthesis of cooling water systems comprising at least two cooling towers. The technique involves a superstructure, in which all possible water reuse and recycle opportunities are exploited. The cooling water systems consisting of multiple cooling towers supplying water at different temperatures were debottlenecked by simultaneously determining the minimum amount of the overall circulating cooling water and the corresponding cooling water networks. The following four cases were considered: Case I

Unbounded cooling water return temperature to the cooling tower without a dedicated source or sink for any cooling water using operation. The formulation entails bilinear terms which are nonconvex thereby rendering the model NLP. Case II Unbounded cooling water return temperature to the cooling tower with a dedicated source or sink for any cooling water using operation. The formulation also consists of bilinear terms and binary variables thus rendering the model MINLP. Case III Specified maximum cooling water return temperature to the cooling tower without a dedicated source or sink for any cooling water using operation. The formulation entails bilinear terms which are nonconvex thereby rendering the model NLP. 3776

dx.doi.org/10.1021/ie101395v |Ind. Eng. Chem. Res. 2011, 50, 3775–3787

Industrial & Engineering Chemistry Research Case IV Specified maximum cooling water return temperature to the cooling tower with a dedicated source or sink for any cooling water using operation. The formulation also consists of bilinear terms and binary variables thus rendering the model MINLP. Cases I and II were linearized using the technique by Savelski and Bagajewicz.22 The technique was designed for water utilization systems but equally applies in cooling water systems design. Savelski and Bagajewicz22 stated that the condition for optimality exists when the outlet concentration is at its maximum allowable level. If this condition is satisfied, the water flow rate will be at its minimum. In the case of heat exchanger networks, Majozi and Moodley10 demonstrated that the optimal solution exists if the outlet temperature is at its maximum allowable level. The reformulation linearization technique by Quesada and Grossmann23 was used to linearize Cases III and IV. A relaxed model was solved to obtain the global optimum solution for LP problem. This solution was used as an initial starting point for the exact NLP (Case III) and MINLP (Case IV) models. 1.3. Cooling Tower Model. The prediction of cooling tower thermal performance dates back to 1925 by Merkel.24 Bernier16 and Kr€oger20 applied Merkel’s theory to develop the cooling tower model. Bernier16 evaluated the cooling tower thermal performance by deriving a one-dimensional model based on the thermal behavior of water droplet in a spray type cooling tower. The model was able to predict the cooling tower outlet temperature and change in air humidity. The major assumptions for this model were as follows: • Lewis factor is unity • No packing inside the towerThe author further used Merkel’s theory to predict the coefficient of performance [(KaV)/mw] as shown in eq 1. This equation was derived from energy balance between the surrounding air and water droplet. The following assumptions were used: • Water evaporation inside the tower is negligible • The resistance surrounding water droplets is negligible • Constant heat capacity of water • Transfer coefficients are independent of temperature Z Tw, out KaV dTw ¼ cpw ð1Þ mw Tw, in Hw - Ha Bernier16 further used Merkel’s theory to express [(KaV)/ mw] as a function of air and water flow rate as shown in eq 2. This implies that the effect of inlet wet bulb temperature and the inlet water temperature on the coefficient of performance is negligible. y KaV mw ¼ x ð2Þ mw ma The values of x and y parameters can be determined experimentally for a given cooling tower packing. The experimental work completed by the author showed a good approximation of [(KaV)/mw]. The correlation coefficient for the regression was in magnitude of 0.99. Kr€oger20 suggested similar correlation for counter flow fills as shown below. Kafi ad mw dda ¼ ATDbdb ð3Þ mw Afr ma where ad, dda, ATD, and bdb are system parameters.

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Fisenko, et al.25 also derived a one-dimensional mathematical model for a counter flow mechanical draft cooling tower by solving heat transfer, mass transfer, and dynamic equations of a falling water droplet. Equation 4 was used to evaluate the cooling tower efficiency. Tw, out - Tw, in ð4Þ η ¼ Tw, out - Twb Kim and Smith1 also developed the cooling tower model which predicts the thermal performance a cooling tower. The authors derived a cooling tower model with the following major assumptions. • Adiabatic operation in the cooling tower • Water and dry air flow rate are constant • Drift and leakage losses neglectedThe performance of the cooling tower was assessed by changing the inlet conditions of water and air. The cooling tower performance was then measured by calculating the effectiveness, which is described as the ratio of actual energy transfer to maximum possible energy transfer. Khan and Zubair26 also developed a model which incorporates the evaporation and drift losses. Khan et al.27 extended the work done by Khan and Zubair26 to investigate fouling on the thermal performance of a cooling tower. Qureshi and Zubair28 developed a cooling tower model which accounts for heat transfer in the spray zone, packing, and rain zone. They further developed a fouling model to predict fouling on packing. The mass transfer coefficient was calculated from the same correlation used by Bernier.16 Equation 5 was used to evaluate the cooling tower effectiveness. This should not be confused with the cooling tower efficiency by Fisenko et al.25 Ha, out - Ha, in ε ¼ ð5Þ Hs, w - Ha, in This paper presents a technique for grassroot design of cooling water system which incorporates the performances of the cooling towers involved. The study focuses mainly on cooling systems consisting of multiple cooling towers that supply a common set of heat exchangers. The heat exchanger network is synthesized using the mathematical optimization technique. This technique is based on superstructure in which all opportunities for cooling water reuse are explored. The cooling tower model is used to predict the thermal performance of the cooling towers while taking the thermal conditions of the associated heat exchanger network into account.

2. COOLING WATER SYSTEMS MODEL DEVELOPMENT The cooling water system consists of cooling towers and heat exchanger network. Therefore the mathematical model for designing cooling system entails the heat exchanger network model and the cooling tower model. The heat exchanger model entails a superstructure in which all possible cooling water reuse are explored. The optimum heat exchanger network design is found by minimizing the cooling tower inlet flow rates. The interaction between the heat exchanger network and the cooling towers is investigated using the cooling tower model derived by Kr€oger20 by considering a control volume as shown in Figure 2. The following assumptions were made: • Interface water temperature is the same as the bulk temperature 3777


Industrial & Engineering Chemistry Research

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Figure 2. Control volume.

• Air and water properties are the same at any horizontal cross section • Heat and mass transfer area is identicalThe governing equations that predict the thermal performance of a cooling tower are given by eqs 6-8. Equations 6 and 7 define the mass and energy balance for the control volume, respectively. Equation 8 defines the air enthalpy change for the control volume. dmw dw ð6Þ ¼ ma dz dz ! dTw ma 1 dHa dw ¼ - Tw ð7Þ dz dz cpw mw cpw dz dHa Kafi Afi ¼ ðLef ðHas - Ha Þ þ ð1 - Lef ÞHv ðws - wÞÞ dz ma ð8Þ In eqs 6-8, afi is the wetted area divided by the corresponding volume of the fill and Afr is a frontal area. The Lewis factor, Lef, appearing in eq 8 is the relationship between the heat-transfer coefficient and the mass-transfer coefficient, that is, h/(Kcpma) = Lef. Lewis factor appears in many governing heat and mass transfer equations. A number of authors like Bernier16 assumed the Lewis factor to be unity. Klopper and Kr€oger29 used expression given in eq 9 to predict the value of Lewis factor. The authors studied the influence of Lewis factor on the performance prediction of wet cooling tower. Their findings were that the influence of Lewis factor diminishes when the inlet ambient air is relatively hot and humid. ws þ 0:622 0:667 ws þ 0:622 - 1 =ln Lef ¼ 0:866 w þ 0:622 w þ 0:622 ð9Þ They further elaborated that increasing Lewis factor increases heat rejection, decreases water outlet temperature and decreases water evaporation rate. The heat exchanger network model is based on the following two possible practical cases. Case I. Specified maximum cooling water return temperature to the cooling tower without a dedicated source or sink for any cooling water using operation. This situation arises when packing material inside the cooling tower is sensitive to temperature and any

cooling tower can supply any water using operation while the water using operation can return to any cooling tower. Case II. Specified maximum cooling water return temperature to the cooling tower with a dedicated source or sink for any cooling water using operation. This is similar to Case I except that the geographic constrains are taken into account. A particular cooling tower can only supply a particular set of heat exchangers and these heat exchangers can only return water to the same supplier. Mathematical Formulation. The mathematical formulation entails the sets, parameters, continuous variables, and constraints described in the Glossary. The mathematical optimization formulation was developed from the superstructure given in Figure 3 by considering energy and mass balance equations across each cooling water using operation and at each node. Two cases that were considered are given in the following sections. 2.1. Case I. In this case, there is no dedicated source or sink for any cooling water using operation. The water using operation can be supplied by one or more cooling towers. The maximum cooling water return temperatures to the cooling towers are also specified. This situation arises when packing material inside the cooling tower is sensitive to temperature and any cooling tower can supply any water using operation and the water using operation can return to any cooling tower. The model was developed by considering mass and energy balance constraints across each equipment and at each node. The mass and energy balance constraints are given by constraints 10-15 and constraints 16-20, respectively. The design constraints are given by constraints 21-23. Mass Balance Constraints. Constraint 10 stipulates that the total cooling water is made up of circulating water from all cooling towers: X OSðnÞ ð10Þ CW ¼ n∈N

Constraints 11 and 12 ensure that the inlet and outlet cooling water flow rates for any cooling tower are equal. These are the mass balance constraints across any cooling tower. X OSðnÞ ¼ CSðn, iÞ - MðnÞ "n ∈ N ð11Þ n∈N

OSðnÞ ¼

X

CRði, nÞ - BðnÞ - EðnÞ

"n ∈ N

ð12Þ

i∈I

The water using operation inlet flow rate is defined by constraint 13. It stipulates that the total water flow rate to water using operation i is made up of cooling water from cooling towers and reuse cooling water from other operations. X X CSðn, iÞ þ FRði0 , iÞ "i ∈ I ð13Þ Fin ðiÞ ¼ n∈N

i0 ∈ I

The outlet flow rate for any water using operation is given by constraint 14. The constraint states that the total water flow rate from water using operation i is made up of reuse cooling water to other operations and cooling water recycling back to the cooling towers. X X Fout ðiÞ ¼ CRði, nÞ þ FRði, i0 Þ "i ∈ I ð14Þ i∈I

3778

i0 ∈ I



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Figure 3. Superstructure for a cooling system.

Constraint 15 ensures that water is conserved through each cooling water using operation. Fin ðiÞ ¼ Fout ðiÞ "i ∈ I

ð15Þ

Energy Balance Constraints. All energy balance constraints were derived considering the energy balance at a node where streams are mixing. This was done to calculate the resultant temperature for mixing streams. The energy balance for all streams supplying any operation i yields constraint 16, which is the definition of inlet temperature into operation i. P P FRði, i0 ÞTout ði0 Þ þ CSðn, iÞTðnÞ 0 n∈N i ∈I "i ∈ I ð16Þ Tin ðiÞ ¼ Fin ðiÞ The addition of make up water results in a change in cooling water temperature. This change is catered for by the energy balance constraint 17. Ts ðnÞ ¼

MðnÞTamb þ OSðnÞTctout ðnÞ "n ∈ N ðCSðnÞÞ

ð17Þ

The return temperature to any cooling tower is calculated from the energy balance constraint 18. This constraint was derived by considering the energy balance for all streams supplying a cooling tower P CRði, nÞTout ðiÞ i∈I P Tret ðnÞ ¼ "n ∈ N ð18Þ CRði, nÞ i∈I

Energy balance across water using operation i is given by

constraint 19 ðTout ðiÞ - Tin ðiÞÞFin ðiÞcp ¼ Q ðiÞ "i ∈ I

ð19Þ

By substituting constraint 16 into constraint 19, the bilinear term Fin(i)Tin(i) can be eliminated and constraint 16 and constraint 19 will be replaced by constraint 20. X X Q ðiÞ þ cp CSðn, iÞTðnÞ þ cp FRði0 , iÞTout ði0 Þ n∈N

i0 ∈ I

¼ Fin ðiÞcp Tout ðiÞ

"i ∈ I

ð20Þ

Design Constraints. The equipments within the cooling water system have the maximum allowable flow rates and temperatures. The design constraints ensure that all the equipments are operated within their specified design limits. Constraints 21 and 22 ensure that the cooling towers are operated below their maximum throughputs and the maximum allowable temperatures respectively. OSðnÞ e OSu ðnÞ "n ∈ N

ð21Þ

u Tret ðnÞ e Tret ðnÞ "n ∈ N

ð22Þ

Constraint 23 ensures that the flow rate through water using operations does not exceed its maximum design flow rate. Fin ðiÞ e Finu ðiÞ "i ∈ I

ð23Þ

The formulation for Case I entails constraints 10-15, 17-18, and 20-23. The objective function of this model is to minimize the total cooling water as given in constraint 10. Constraints 17, 18, 3779



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Figure 4. Flowchart for cooling water system model.

and 20 consist of bilinear terms which are nonconvex thus rendering the model NLP. This model is difficult to initialize because the starting point might be infeasible or the solution might be locally optimum. To overcome these difficulties the technique proposed by Quesada and Grossmann23 was used to linearize the bilinear terms. This technique uses the upper and the lower bounds to create a convex space for the bilinear terms as shown in the next section. Reformulation Linearization Technique. The technique starts by assigning a variable to all bilinear terms as shown in below. Let crtði, nÞ ¼ CRði, nÞTout ðiÞ "n ∈ N, i ∈ I frtði0 , iÞ ¼ FRði0 , iÞTout ðiÞ fntðiÞ ¼ Fin ðiÞTout ðiÞ tcsðn, iÞ ¼ CSðn, iÞTs ðnÞ

The upper and the lower bound for variables in each bilinear term were defined as follows: The lower bound for the flow rates is zero and the upper bound was given a value. The lower bound for the temperatures is the wet bulb temperature and the upper bound was also assigned a value. The bilinear term CR(i,n)Tout(i) can now be defined by constraints 24-27. u u crtði, nÞ g OSu ðnÞTout ðiÞ þ CRði, nÞTout ðiÞ - OSu ðnÞTout ðiÞ "n ∈ N, "n ∈ N

ð24Þ L crtði, nÞ g CRði, nÞTout ðiÞ "n ∈ N, "i ∈ I L crtði, nÞ e OSu ðnÞTout ðiÞ þ CRði, nÞTout ðiÞ u ðiÞ "n ∈ N, "i ∈ I - OSu ðnÞTout

i ∈ I, i0 ∈ I "i ∈ I "n ∈ N, i ∈ I

ð25Þ

u ðiÞ "n ∈ N, "i ∈ I crtði, nÞ e CRði, nÞTout

ð26Þ ð27Þ

Similarly, the bilinear term FR(i0 ,i)Tout(i) is defined by constraints 28-31, the bilinear term Fin(i)Tout(i) by constraints 3780



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Introduction of linearization variables require constraint 18, constraint 20, and constraint 17 to be modified as shown in constraints 40-42, respectively. P crtði, nÞ i∈I "n ∈ N ð40Þ Tret ðnÞ ¼ P CRði, nÞ Q ðiÞ þ cp

X

i∈I

tcsðn, iÞ þ cp

n∈N

X

X

frtði0 , iÞ

i0 ∈ I

¼ fntðiÞcp "i ∈ I tcsðn, iÞ ¼ MðnÞTamb þ OSðnÞTctout ðnÞ

ð41Þ "n ∈ N ð42Þ

i∈I

The relaxed LP model for Case I consists of Constraints 10-15, 21-23, and 24-42. To get the solution for Case 1, the relaxed model is first solved by minimizing the total cooling water. The solution of the relaxed model is then used as a starting point for solving the exact model. 2.2. Case II. In this case there are dedicated source and sink for any cooling water using operation. This implies that no premixing or postsplitting of cooling water return is allowed. A set of heat exchanger can only be supplied by one cooling tower. Furthermore, the return cooling water from cooling water using operation must supply the source cooling tower. However, reuse of water within the network is still allowed. All the constraints in Case I are still applicable. Few constraints needed to be added to control the source and the sink. Constraints 43 and 44 prevent premixing. Constraint 43 ensures that the supply flow rate from any cooling tower to operation i cannot exceed the maximum flow rate. Constraint 44 ensures that cooling water using operation i can only be supplied by a maximum of one cooling tower.

Figure 5. Base case.10

32-35 and the bilinear term CS(n,i)Ts(n) by constraints 36-39. u ðiÞ frtði0 , iÞ g Finu ðiÞTout ði0 Þ þ FRði0 , iÞTout u - Finu ðiÞTout ðiÞ 0

0

frtði , iÞ g FRði

i ∈ I, i0 ∈ I

ð28Þ

0

ð29Þ

L , iÞTout ðiÞ

i ∈ I, i ∈ I

L ðiÞ frtði0 , iÞ e Finu ðiÞTout ði0 Þ þ FRði0 , iÞTout

i ∈ I, i0 ∈ I

L - Finu ðiÞTout ðiÞ

u frtði0 , iÞ e FRði0 , iÞTout ðiÞ i ∈ I, i0 ∈ I

ð30Þ

CSðn, iÞ e Finu ðiÞysðn, iÞ "i ∈ I "n ∈ N X ysðn, iÞ e 1"n ∈ N

Postsplitting can also be prevented by constraints 45 and 46. Constraint 45 ensures that the return flow rate from operation i to any cooling tower cannot exceed the maximum flow rate into that operation. Constraint 46 ensures that cooling water using operation i can supply a maximum of one cooling tower. CRði, nÞ e Finu ðiÞyrði, nÞ "i ∈ I "n ∈ N X yrði, nÞ e 1 "n ∈ N

ð31Þ ð33Þ

L L fntðiÞ e Finu ðiÞTout ðiÞ þ Fin ðiÞTout ðiÞ - Finu ðiÞTout ðiÞ "i ∈ I ð34Þ

u ðiÞ "i ∈ I fntðiÞ e Fin ðiÞTout

ð35Þ

tcsðn, iÞ g OS ðnÞTs ðnÞ þ CSðn, iÞTsu ðnÞ - OSu ðnÞTsu ðnÞ "n ∈ N, i ∈ I

ð36Þ

tcsðn, iÞ g CSðn, iÞTsL ðnÞ

ð37Þ

u

"n ∈ N, i ∈ I

tcsðn, iÞ g OSu ðnÞTs ðnÞ þ CSðn, iÞTsL ðnÞ - OS

u

ðnÞTsL ðnÞ

"n ∈ N, i ∈ I

tcsðn, iÞ g CSðn, iÞTsu ðnÞ

"n ∈ N, i ∈ I

ð45Þ ð46Þ

n∈N

Constraints 47 and 48 ensure that the source and the sink cooling water supply is the same for a particular cooling water using operation. yrði, nÞ e ysðn, iÞ ! X X þ 2ysðn, iÞ yrðn, iÞ "i ∈ I "n ∈ N ð47Þ n∈N

n∈N

yrði, nÞ e ysðn, iÞ ! X X ysðn, iÞ yrðn, iÞ "i ∈ I - 2n∈N

ð38Þ ð39Þ

ð44Þ

n∈N

u u fntðiÞ g Finu ðiÞTout ðiÞ þ Fin ðiÞTout ðiÞ - Finu ðiÞTout ðiÞ "i ∈ I ð32Þ

L fntðiÞ g Fin ðiÞTout ðiÞ "i ∈ I

ð43Þ

"n ∈ N

ð48Þ

n∈N

The formulation for Case II entails constraints 10-15, 17-18, 20-23, and 43-48. Constraints 43-48 consist of binary variables, 3781



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while constraints 17, 18, and 23 consist of bilinear terms which are nonconvex. This renders the model MINLP. Similar to Case I, the model was linearized using the linearization relaxation procedure by Quesada and Grossmann23 as shown in constraints 24-42. The relaxed model thus entails constraints 10-15, 21-22, and 24-48. Table 1. Cooling Tower Design Information cooling towers

Tretu ()

OSu

CT01 CT02

50 50

9.6 16

CT03

55

20

Table 2. Limiting Cooling Water Data operations

Tinu (C)

Toutu (C)

Fin (kg/s)

Q(i) (kW)

OP01 OP02

30 40

45 60

9.52 3.57

600 300

OP03

25

50

7.62

800

OP04

45

60

7.14

600

OP05

40

55

4.76

300

OP06

30

45

11.1

700

The solution procedure starts by first solving the relaxed model by minimizing the total cooling water. The solution of the relaxed model is then used as a starting point for solving the exact model. To synthesize the overall cooling water system, both the cooling tower model and the heat exchanger network model should be solved simultaneously. The algorithm for synthesizing the overall cooling water system is given in the following section. Solution Algorithm. The solution procedure can be applied for both cases considered. The first step is to optimize the heat exchanger network model without the cooling towers. The results from the first iteration, which are cooling water return (CWR) temperatures and flow rates, become the input to the cooling tower models. Each cooling tower model then predicts the outlet water temperatures and flow rates. This is done by first assuming the outlet water temperature of a cooling tower. The assumption is done by subtracting 0.5 C from the given cooling tower inlet temperature. The three governing mass and heat transfer equations, that is, equation 6-8 are then solved numerically using fourth-order Runge-Kutta method starting from the bottom of the cooling tower moving upward at stepsize Δz. When the maximum height is reached, the temperature at this point will be compared with the CWR temperature. If the two agree within a specified tolerance, the cooling tower model will

Figure 6. Final design of the cooling water system. 3782



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Figure 7. Final design of the cooling water system.

stop and the outlet temperature will be given as the assumed temperature, else the inlet temperature will be adjusted until the CWR temperature agrees with the calculated temperature. The predicted outlet cooling tower temperatures and flow rates then become the input to the heat exchanger network model. If the outlet temperature of the cooling tower model agrees with the previous inlet temperature to the heat exchanger network model, the algorithm stops which implies that final results have been obtained. Otherwise the iteration continues. The solution algorithm flowchart is shown in Figure 4.

3. CASE STUDIES The application of the proposed technique is demonstrated by considering one example for Cases I and II. This example was extracted from the paper by Majozi and Moodley.10 3.1. Base Case. Cooling water system in Figure 5 shows a set of heat exchanger networks which are supplied by a set of cooling

towers. Each cooling water using operation is supplied by fresh water from the cooling tower and return back to the cooling tower. The implication of these arrangements results in higher return cooling water flow rate and low return cooling water temperature thus reducing cooling tower efficiency.16 The heat duties, temperature limits and design information are shown in Tables 1 and 2. Turet is the maximum allowable temperature for packing inside the cooling towers while OSu is the maximum flow rate of the cooling tower before flooding. Tuin and Tuout are the thermodynamic temperature limits for the inlet and outlet temperature of the cooling water using operation respectively. 3.2. Case I. In this case each cooling tower can supply any cooling water using operation. The return streams from any cooling water using operation can go to any cooling tower. The return temperature to any cooling tower is however specified. Figure 6 shows the heat exchanger network after applying the methodology described above. 3783



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Table 3. Results Summary stream

base case (kg/s)

results (kg/s)

make up

2.52

2.33

blowdown

0.50

0.47

31.94

24.80

base case (kg/s)

results (kg/s)

makeup

2.52

2.41

blowdown

0.50

0.48

31.94

25.69

circulating water

Table 4. Results Summary stream

circulating water

Figure A1. Control volume.

By exploiting the opportunity for cooling water reuse, the overall circulating water decreased by 22% and one cooling tower was eliminated. The cooling tower inlet temperatures are at their maximum values. These results show the opportunity to increase the heat duties, through expansions, without investing on a new cooling tower. The only additional investment required is on piping for reuse streams. For this case study the makeup and the blowdown was also decreased by 7%. However the decrease in makeup and blowdown cannot be guaranteed for all practical case studies. The results summary for Case I are shown in Table 3. 3.3. Case II. In this case, a cooling tower can only supply a dedicated set of heat exchangers. This implies that each operation can only be supplied by one cooling tower. The return streams from any cooling water using operation can only go to its supplier cooling tower. The return temperatures to the cooling towers are also specified. Figure 7 shows the heat exchanger network after applying the methodology described above. By allowing for the cooling water reuse, the overall circulating water decreased by 20%. This will decrease the pumping power requirement for the circulating pump thus reducing the pumping cost. The cooling towers spare capacity is also increased giving opportunities for increased heat load without investing in a new cooling tower. To satisfy the required heat duties with the reduced flow rate, the return temperature to the cooling towers is increased to the maximum value. The makeup and the blowdown are also decreased by 4%. As above-mentioned, the decrease in makeup and blowdown cannot be guaranteed for all practical case studies. The results summary for Case II are shown in Table 4.

’ APPENDIX: COOLING TOWER MODEL DERIVATION This section presents a derivation for the cooling tower model by Kr€oger.20 The model was derived by considering a control volume as shown Figure A1. The following assumptions were made: • Interface water temperature is the same as the bulk temperature • Air and water properties are the same at any horizontal cross section • Heat and mass transfer area is identicalThe governing equations that predict the thermal performance of a cooling tower were given by eqs A1, A4, and A15. The mass and energy balance equations for the control volume are given in eq A1 and A3, respectively. dmw dz ma ð1 þ wÞ þ mw þ dz dw ¼ ma 1 þ w þ dz þ mw ðA1Þ dz

4. CONCLUSIONS The mathematical technique for cooling water system synthesis with multiple cooling towers has been presented. This technique is more holistic because it caters for the effect of cooling tower performance on heat exchanger network. The cooling tower thermal performance is predicted using the mathematical model. The results obtained using this technique are more practical, since all components of the cooling water system are included in the analysis. The proposed technique has the advantage of debottlenecking the cooling towers, which implies that a given set of cooling towers can manage an increased heat load. Furthermore, the overall circulation water is also decreased with an added benefit of decreasing the overall power consumption of the circulating pumps. There is also a potential for the reduction of makeup and blowdown water flow rate. The proposed technique shows a potential for capital saving in grassroots and retrofit designs.

ðA3Þ

The above equation can be further simplified dmw dw ¼ ma dz dz dmw dTw ma Ha þ mw þ dz cpw T þ w dz dz dz dHa ¼ ma H a þ dz þ mw cpw Tw dz

ðA2Þ

Ignoring the second-order terms, the equation can further be simplified ! dTw ma 1 gHa dw - Tw ¼ ðA4Þ dz dz cpw mw cpw dz The enthalpy transfer between the air and water interface is given in eq A5 ðA5Þ dQ ¼ dQm þ dQc where dQm ¼ HV Kðws - wÞdA dQc ¼ hðTw - Ta ÞdA 3784

ðmass transfer enthalpyÞ ðA6Þ ðconvective heat transferÞ

ðA7Þ



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Figure A2. Cooling tower model flowchart.

Therefore, dQ ¼ HV Kðws - wÞdA þ hðTw - Ta ÞdA

ðA8Þ

The enthalpy of unsaturated air and saturated air are given as Ha ¼ cpa Ta þ wðHf gwo þ cpv Ta Þ

The difference of the above equations

½ðHas - Ha Þ - ðws - wÞHv cpma

where

# ! h h dQ ¼ h ðHas - Ha Þ þ 1 þ Hv ðws - wÞ dA ðA13Þ Kcpma Kcpma

The enthalpy change must be equal to the enthalpy change of air stream dHa 1 dQ Kafi Afi ¼ ¼ ðLef ðHas - Ha Þ þ ð1 - Lef ÞHv ðws - wÞÞ ma dz dz ma

ðA14Þ

ðA11Þ

or Tw - Ta ¼

"

ðA9Þ

Has ¼ cpa Tw þ ws ðHfgwo þ cpv Tw Þ ¼ cpa Tw þ ws HV ðA10Þ

Has - Ha ¼ ðcpa þ wcpv ÞðTw - Ta Þ þ ðws - wÞHv

Substituting the expression of Tw - Ta into the enthalpy transfer between the air and water interface equation.

ðA12Þ

where dA = afiAfrdz and h/(Kcpma) = Le. afi is the wetted area divided by the corresponding volume of the fill and Afr is a frontal area. h/(Kcpma) = Le is a Lewis factor. Lewis Factor. The expression for Lewis factor used by Klopper and Kr€oger29 is given by eq A15.

cpma ¼ cpa þ wcpv

Le ¼ 0:8660:667

the enthalpy transfer between the air and water interface 3785

ws þ 0:622 ws þ 0:622 - 1 =ln ðA15Þ w þ 0:622 w þ 0:622



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Coefficient of Performance. Kr€ oger20 suggested a correla-

tion for counter flow fills as shown eq A16 Kafi ad mw dda ¼ ATDbdb mw Afr ma

ðA16Þ

where ad, dda, ATD, and bdb are system parameters. Makeup and Blow Down. Equation A17 and A18 were used to calculate the makeup and blowdown flow rates. CC ðA17Þ FM ¼ FE CC - 1 where CC is cycle of concentration. FM ¼ FB þ FE

ðA18Þ

The governing equations (eq A2, A4, and A14) were solved numerically using fourth -order Runge-Kutta method. Figure A2 shows the flowchart for solving the cooling tower model.

’ AUTHOR INFORMATION Corresponding Author

*Tel: (þ27) 12 420 4130. Fax: (þ27) 12 420 5048. E-mail: [email protected].

’ ACKNOWLEDGMENT The authors thank the Council for Scientific and Industrial Research (CSIR) for funding this research. ’ GLOSSARY Sets

i = i|i is a cooling water using operation n = n |n is a cooling tower 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) TU out(i) = limiting outlet temperature of cooling water using operation i (C) TU in(i) = limiting inlet temperature of cooling water using operation i (C) (i) = maximum inlet flow rate of cooling water using operation FU in i (kg/s) TU ret(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 4.2 (kJ/kg C). Tamb = ambient temperature (C) Continuous Variables

OS(n) = operating capacity of cooling tower n (kg/s) CW = 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(i,n) = return cooling water to cooling tower n from cooling water using operating i (kg/s)

FR(i0 ,i) = reuse cooling water to cooling water using operating i0 from cooling water using operating i (kg/s) Fin(i) = total cooling water into cooling water using operating i (kg/s) Fout(i) = total cooling water from cooling water using operating i (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) Tst(n) = cooling water supply temperature form cooling tower n after adding make up (C) crt(i,n) = linearization variable for relaxation technique frt(i0 ,i) = linearization variable for relaxation technique fnt(i) = linearization variable for relaxation technique tcs(n,i) = linearization variable for relaxation technique

’ NOMENCLATURE a, afi = surface area per unit volume Afr = frontal area cp = specific heat capacity CW = cooling water CWR = cooling water return F = Flow rate H = Enthalpy h = heat transfer coefficient K = mass transfer coefficient Le = Lewis factor m = flow rate T = temperature V = volume W = humidity z = cooling tower height F = density η = efficiency ε = effectiveness ad, dda, ATD, and bdb = cooling towers fill parameters Subscripts

a = air c = cold h = hot ma = moist air min = minimum s = saturation w = water wb = wet bulb

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