- Email: [email protected]

Contents lists available at SciVerse ScienceDirect

Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Synthesis of cooling water systems with multiple cooling towers Eusiel Rubio-Castro a, b, Medardo Serna-González b, *, José María Ponce-Ortega b, Mahmoud M. El-Halwagi c, d a

Chemical and Biological Sciences Department, Universidad Autónoma de Sinaloa, Culiacán 80000, Sinaloa, México Chemical Engineering Department, Universidad Michoacana de San Nicolás de Hidalgo, Morelia 58060, Michoacán, México c Chemical Engineering Department, Texas A&M University, College Station, TX 77843, USA d Adjunct Faculty at the Chemical and Materials Engineering Department, King Abdulaziz University, Jeddah, Saudi Arabia b

h i g h l i g h t s < An MINLP model is proposed to design cooling systems with multiple cooling sources. < It can account for practical feasibility constraints in cooling towers. < It considers that the hot streams require cooling water at different temperatures. < Signiﬁcant cost reduction can be realized using the proposed model.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 March 2012 Accepted 8 June 2012 Available online 21 June 2012

This paper presents a systematic approach for the synthesis of re-circulating cooling water systems consisting of a cooler network and a cooling tower network (i.e. a general arrangement of multiple interconnected cooling towers with different supply temperatures). An overall superstructure is developed, which contains all the different alternatives for the cooler and cooling tower networks, as well as all their potential interconnections for ﬁxed data of hot process streams. The synthesis problem is formulated as a mixed integer nonlinear programming problem. The continuous variables include the ﬂow rates and temperatures of the circulating cooling water in the superstructure while the integer variables describe the existence of the coolers and the cooling towers. The objective function is to minimize the total annual cost, which includes the investment costs of cooler units and cooling towers, as well as the operating costs due to make-up water and power consumption of the circulating-water pumps and cooling tower fans. The solution of the proposed MINLP formulation provides simultaneously both the optimal conﬁguration and operating conditions of the cooler and cooling tower network to comply with given process cooling demand at minimum total annual cost. One of the most important design variables is the number of cooling water sources (i.e. cooling water streams at different temperatures originated from the outlets of cooling towers) for removing heat from several hot process streams at different temperature ranges. Results show that cooling systems consisting of multiple cooling towers with different supply temperature yield signiﬁcant better results than traditional systems with a cooler network in parallel arrangement that is supplied with cooling water from a single cooling tower. 2012 Elsevier Ltd. All rights reserved.

Keywords: MINLP model Optimization Cooling water systems Cooling tower network Cooler network Multiple cooling water sources

1. Introduction Because of its non-harmful chemical composition, easy handling under the process conditions, and suitable thermal properties, water is widely used as a coolant to remove waste heat in industrial petrochemical and chemical facilities, electric power generating stations, refrigeration and air conditioning plants, pulp and paper * Corresponding author. Tel.: þ52 443 3273584; fax: þ52 443 3273581. E-mail address: [email protected] (M. Serna-González). 1359-4311/$ e see front matter 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2012.06.015

mills, and steel mills. In industry, the use of water as a heat-transfer medium for cooling operations is commonly practiced in open re-circulating cooling systems due to water scarcity and environmental considerations. As shown in Fig. 1, a traditional re-circulating cooling water system can be regarded as an integrated system that consists of three main components: the cooler network, the cooling tower and the circulating-water pumps. Cold water passes from the cooling tower to the heat exchange units in the cooler network, and the heated water from the cooler network returns to the top of the tower and ﬂows downward over a packing

958

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974 Evaporated water

Outlet air

Hot water THIN

THOUT

Cooling tower

Inlet air Tcout

Inlet air

Hot process stream

Makeup water

Tcin Cool water

Cooler network

Blowdown water

Pump

Fig. 1. Components of a typical re-circulating cooling water system.

or ﬁll, while air can ﬂow upward (counter ﬂow) or horizontally (cross-ﬂow). As a result of the direct contact between the water and the air in the packing region, part of the water is vaporized and the water temperature is reduced while the air humidity is increased. The moist air leaves at the top of the tower. The blowdown water from the recirculation loop and the water lost by evaporation and drift are replaced by supplying fresh or make-up water to the system. Since cooling water is reused after heat is dissipated in the cooling tower, this type of system, in contrast to a once-through mode of operation, reduces considerably the amount of make-up water needed, resulting in signiﬁcant fresh water savings. In addition, the amount of re-circulating water discharged (blowdown) than can require waste treatment is very small, so that the ecological effect is reduced. Over the last ten years, several studies on re-circulating cooling water systems have been reported in the literature, ranging from insight-based pinch analysis to mathematical-based optimization approaches. The seminal paper in this area was the one by Kim and Smith [1], who developed and applied a two-stage strategy based on the pinch analysis for the optimal design of cooling water systems. In the ﬁrst stage, the target for minimum ﬂow rate of re-circulating cooling water is calculated using a graphical technique, while in the second stage a network of coolers with a series arrangement is developed to reach the target by following heuristic design rules. In later works, research in re-circulating cooling water systems through pinch analysis has addressed different targets: debottlenecking [2], make-up water reduction [3], and minimization of the environmental impacts using ozone treatment in the cooling tower [4]. Also, a conceptual approach has been reported by Picón-Núñez et al. [5] to study the effect of the conﬁguration of cooler networks upon the total exchanger area. From a similar perspective, recently Picón-Núñez et al. [6] developed a procedure based on pinch technology to examine all of the cost factors and principles involved in the design of cooling water systems. Regarding the mathematical programming approaches, Castro et al. [7] presented a nonlinear model to minimize the operating cost of a cooling water system with ﬁxed conﬁguration using a regression model of the cooling tower. Kim and Smith [8] formulated an optimization model for retroﬁt design of cooling water-systems, where they included various actions such as heat load redistribution, hot water removal, use of air coolers and the increase of air ﬂow rate in the tower. Feng et al. [9] proposed a solution approach based on a mixed integer nonlinear programming (MINLP) model to obtain a cooler network conﬁguration with lowest consumption of cooling water. Ponce-Ortega et al. [10] developed a novel network superstructure for optimizing cooler networks including both capital investment and cooling water

consumption costs. In a later work, Ponce-Ortega et al. [11] ﬁrst proposed a model based on a generalized disjunctive programming formulation for simultaneous synthesis and detailed design aspects of cooler networks. Recently, other studies have presented the alternatives to improve the operation of existing cooling water systems using an optimization model that was validated through experimental tests in a pilot plant [12,13]. More recently, in contrast to the sequential design strategies used by other authors, PonceOrtega et al. [14] have considered a simultaneous optimization model for the detailed design of overall recirculation cooling water systems, which included the interaction of the cooler network and the cooling tower performance. It is worthy of note that all the above-mentioned work was developed for cooling water systems with a single cooling tower. However, most large-scale industrial systems (e.g. reﬁneries, petrochemicals, chemicals), where several processes, streams, supporting systems (e.g. energy) and their complex interactions are involved, feature multiple cooling towers for removing waste heat from hot streams above ambient temperature. A mathematical optimization technique for debottlenecking cooling water systems consisting of multiple cooling towers supplying cold water at different temperatures was ﬁrstly proposed by Majozi and Moodley [15]. This work was further extended in a very recent paper to incorporate the detailed performances of the cooling towers involved [16]. These authors proposed the minimization of the overall cooling water supply from all the sources as the objective function. This optimization problem is solved by an iterative two-level decomposition procedure, in which the cooler network is ﬁrst designed in an inner loop with assumed values for the inlet cooling water temperatures and ﬂow rates to it; the cooling towers are then detailed designed in an outer loop using as input data the cooling water return temperatures and ﬂow rates provided by the cooler network. This allows recalculating the inlet cooling water temperatures and ﬂow rates to the cooler network. The procedure continues until in two consecutive iterations these new values agree with the previous ones. Gololo and Majozi [16] demonstrated that cooling water systems with multiple cooling towers can achieve maximum cooling water reuse that, in turn, results in up to 22% savings in the circulating-water requirement compared with base cases. Two limitations of this approach, however, are that it does not consider the capital costs of cooler units and cooling towers and that the cooler network and the cooling towers are not designed simultaneously. In this paper, an integrated superstructure is developed for representing the large number of cooling system conﬁgurations with multiple cooling towers. This superstructure is constructed from smaller superstructures associated with the cooler and cooling tower networks. The cooler network superstructure is based

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

upon an adaptation of the stage-wise representation proposed by Ponce-Ortega et al. [10]. A cooling tower network superstructure is developed in which all structural alternatives of interconnected cooling towers are considered for reducing the temperature of the heated water streams that leave the cooler network. Different conﬁgurations with series, parallel, serieseparallel and paralleleseries arrangement of cooling tower units in the cooling tower network are considered. Based on the integrated superstructure, an optimization model is proposed to simultaneously account for all the relevant design decisions resulting from the coupling of the cooler and cooling tower network. The model is formulated as a mixed integer nonlinear programming (MINLP) problem, in which the ﬂow rates and temperatures of cooling water streams are treated as decision variables to be optimized simultaneously for both the cooler network, as well as the cooling tower network. Binary variables are assigned to the potential cooler units and cooling towers in the cooling system. The objective function is the minimization of the total annual cost, which involves both operating and investment costs. The operating costs include the make-up water cost and the electricity costs for circulating-water pumps and tower fans, whereas the capital costs involve the investment cost of the cooler units, cooling towers and circulatingwater pumps. A detailed model for sizing the cooling towers based on the Merkel method [17] is used. The mass transfer and pressure drop characteristics of the types of packing considered are modeled with the empirical correlations given by Kloppers and Kröger [18,19]. As will be shown, the optimization model can account for feasibility constraints such as minimum and maximum values of water-to-air mass ratios, water and air mass velocities in cooling towers, and so on. The solution of the MINLP formulation provides an optimal or close to the optimal conﬁguration of the cooling water system.

2. Problem statement The addressed optimization problem of cooling water systems can be stated as follows: Given is a set of hot process streams to be cooled using only water as a cooling utility, with known inlet and outlet temperatures, and heat capacity ﬂow rates. Given are also the dry- and wetbulb temperatures of the entering air to the cooling towers, which are determined by the plant location, as well as the correlations of physical properties of water and airewater vapor mixtures and cost data. The goal of this problem consists of determining the cooling water system, most likely involving multiple cooling towers with different supply temperatures, that operates so that the total annual cost is minimized. The solution of such a problem should provide information about: (a) the cooler network (the optimal size of coolers and interconnection among the coolers as well as the optimal temperatures and mass ﬂow rates for the cold utility of each cooler unit); (b) the cooling tower network (the cooling tower arrangement involving multiple cooling water sources for different temperature ranges, as well as the design details of each cooling tower such as water mass ﬂow rate, water consumption, inlet and outlet temperatures of the cooling tower, packing height and tower cross-sectional area, type of packing, and power consumption of the tower fan); (c) the interconnection between the two networks via the allocations of the outlet streams from the cooler network along the cooling tower network; and the allocation of the cold water streams from the cooling towers back to the coolers; (d) the power consumption of the circulating-water pumps.

959

To properly address this problem, this paper proposes to formulate a cooling water system superstructure that embeds all system conﬁgurations of practical interest, transform the superstructure model as an MINLP problem, and develop a suitable solution methodology. The optimal design strategy is based on the following assumptions: 1. Constant heat capacities for hot process streams and cooling water. 2. Constant overall heat transfer coefﬁcients for the cooler units. 3. Countercurrent coolers of the shell-and-tube type. 4. Saturated air at the outlet of the cooling towers. 5. Mechanical draft counter ﬂow wet-cooling towers. 3. Superstructure representations In the formulation of the synthesis problem of re-circulating cooling water systems with multiple cooling towers many alternative conﬁgurations can be considered systematically by including them in an overall superstructure such as the one that is presented in Fig. 2 for three hot streams and three cooling towers. In this representation, the stage-wise superstructure for the network of coolers developed by Ponce-Ortega et al. [10] is adapted for embedding it in an integrated cooling water system that also consists of a cooling tower network, which includes all the possibilities for combining the heated water streams that leave the cooler network and interconnecting the several cooling towers. The basic components of the overall system superstructure are heat exchangers, stream mixers, stream splitters, cooling towers and circulating-water pumps. 3.1. Cooler network The task of the cooler network is to process a set of hot process streams from given inlet to speciﬁed target temperatures. In this work, the cooler network superstructure is constructed by singlephase, countercurrent shell-and-tube heat exchangers. Within the cooler network different stages may exist, each of which extends over a limited temperature interval that incorporates thermodynamic constraints for the heat transfer process. To include network conﬁgurations completely in series, the number of stages required to model cooler networks is implemented as the number of hot process streams. Depending upon the optimal solution, each of these stages may exist or not. Each stage features a parallel arrangement of heat exchangers through which any hot process stream can exchange heat with the cooling water. Such parallel arrangement is obtained by splitting the inlet water stream for any stage into a number of branches equal to the number of hot streams. Unlike the original superstructure by Ponce-Ortega et al. [10], for each stage the total water stream at the outlet of the parallel arrangement of coolers is split into a stream bypass that is directed to the cooling tower network and a reuse stream that is merged with the inlet streams of the subsequent stage. Also, the total inlet stream for a given stage consists of cooled water streams originated from the outlets streams of all the cooling towers, a reuse stream originated from the outlet of the previous stage, and a stream of make-up water. It is worth mentioning that the fresh water stream that enters the cooling system is split into smaller streams before being sent to the feed mixers of the cooler network stages to make up for water losses in the cooling tower network, which consist of evaporation, drift, and blowdown [20]. A single source of make-up water is assumed at a constant temperature. The superstructure of the cooler network consequently allows bypass of the cooling water from the cooling tower network, as

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

Location k=3 Th1,3

1,2

Th2,3

2,2

Th3,3

Th3,2

3,2

2,3

Scn 2

3,3

Fwh5 Twh5

Fwh4 Twh4

Fwh1 Twh1

Sctn 2

M1ctn

...

Fwh2 Twh2

fwt13,1

2,1

THOUT2

Th3,1

3,1

THOUT3

1,2

Tcin2

2,2

S1cn

fwx1,1 Tcout1,1 fwx2,1 Tcout2,1 fwx3,1 3,1 Tcout3,1

1,1

Tcin1

2,1

fwm1

M1cn

M cn 2

Fwin,1 Twin,1

fwe1

S1wm

Fwm Twm

fwm2

fwc1,2 fwc2,2 fwc3,2

fwc3,3 fwc2,3 fwc1,3 S3ctn

THOUT1

Th2,1

fresh water

fwx1,2 Tcout1,2 fwx2,2 Tcout2,2 fwx3,2 3,2 Tcout3,2

M3cn

Fwh3 Twh3

1,1

Th1,1

FO1 TO1

1,3

Tcin3

Location k=1

Stage k=1

Th2,2

FO2 TO2 fwx1,3 Tcout1,3 fwx2,3 Tcout2,3 fwx3,3 Tcout3,3

Location k=2 Th1,2

Stage k=2

fwc3,1

Location Stage k=3 k=4 FCP1 Th1,4 1,3 H1 THIN1 FCP2 Th2,4 2,3 H2 THIN2 FCP3 Th3,4 3,3 H3 THIN3

fwc2,1

960

fwm3

fwc1,1

Faout,1 TAout,1;haout,1

CT1

S1ctn

Sctn 4

Fwout,1 Twout,1

Fain,1

S5ctn

F1ctn TAin,1;hain,1

M ctn 4

Fwdis,1 ctn Twdis,1 P1

Sctn 6

fwb1

ftt1,2

...

fwt23,4

...

fwt

1 3,2

M ctn 2

Fwin,2 Twin,2

fwe2 Faout,2 TAout,2;haout,2

ftt3,1

ftt2,1

CT2

Fwout,2 Twout,2

Fain,2

F2ctn TAin,2;hain,2

Sctn 7

ftt2,3

...

Fwdis,2 P ctn Twdis,2 2

fwb2

fwt23,5

M5ctn

...

fwt13,3

M3ctn

Fwin,3 Twin,3

fwe3 Faout,3 TAout,3;haout,3

ftt3,2

ftt1,3

CT3

Fain,3

Fwout,3 Twout,3

M ctn 6

Fwdis,3 P ctn Twdis,3 3

S8ctn

fwb3

...

fwt23,6

F3ctn TAin,3;hain,3

MBD

Fwb

Fig. 2. Integrated superstructure for the cooling water system design problem.

well as bypass, mixing and splitting of previously used water. Therefore, the proposed superstructure allows the reuse of water before it returns to the cooling tower network, thus providing a better management strategy for the integrated cooling water system. Finally, it is interesting to note that the superstructure of Ponce-Ortega et al. [10] merges all the heated water streams (i.e. outlet streams) that leave the cooler network and assumes that the single outlet stream generated is sent to a single cooling tower. Whereas, in the superstructure proposed in this work, all potential

heated water streams with different ﬂow rates and temperatures appear as separated streams at the hot end of the cooler network. These streams return to the cooling tower network and are not merged to consider all possible conﬁgurations of multiple cooling towers. 3.2. Cooling tower network To ﬁnd the optimal cooling system, this paper considers all possibilities for combining the heated water streams from the

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

cooler network and interconnecting the several cooling towers. These are included in the superstructure of the cooling tower network, which is shown in Fig. 2 for the case of ﬁve heated water streams and three cooling towers. The ﬁve heated water streams ctn ctn ctn are divided by initial splitters ðSctn and Sctn 1 ; S2 ; S3 ; S4 5 Þ into smaller streams that are directed to the mixers at the inlet of each ctn cooling tower ðMctn and Mctn 1 ; M2 3 Þ and to the mixers at the ctn outlet of each cooling tower ðMctn and Mctn 4 ; M5 6 Þ. The smaller heated water streams enter the top of the cooling towers and ﬂow down over some packing, breaking into ﬁlms and droplets. For each cooling tower, ambient air is forced or induced into the tower by the action of an electrically driven fan and ﬂows in countercurrent fashion with respect to the water stream. The cooling process is accomplished by a combination of sensible heat transfer and evaporation of a small portion of circulating water. The moist air stream leaves at the top of the tower. That stream contains the water lost through evaporation and drift, which is replaced with fresh make-up water. After the cooling towers, splitters ctn ðSctn and Sctn 6 ; S7 8 Þ are placed to direct the cooled water streams to the mixers at the inlet of another cooling tower, to the mixers cn cn at the inlet of a stage of the cooler network ðMcn 1 ; M2 and M3 Þ and to the ﬁnal mixer (MBD) prior to the total outlet stream that constitutes the blowdown water of the cooling tower network. Also, each cooling tower is followed by a pump ctn ðPctn and Pctn 1 ; P2 3 Þ, which returns the cooled water to the cooler network and/or sends it to other cooling towers. It should be noted that recycles around the cooling towers are not allowed. This superstructure representation incorporates cooling towers arrangements in series, parallel, serieseparallel and paralleleseries and options on heated water stream splitting, mixing and bypassing. In this way, any heated water stream leaving the cooler network can, in principle, ﬂow to any cooling tower with the possibility of a partial bypass to reduce the mass load in the cooling towers. In addition, the heated water streams may be thermally processed in a distributed arrangement of several cooling towers instead of combining them into a single stream followed by a single cooling tower. Notice that the superstructure is generalized for any number of hot process streams and cooling towers. Next, the mathematical model developed based on the overall superstructure representation is presented in detail. 4. Mathematical formulation The fundamental index sets of the mathematical formulation are: the set of all heated water streams leaving the cooler network NEF, the set of all stages of the cooler network ST, the set of all hot process streams HP, and the set of all cooling towers NCT. The subscripts av, cu, d, dis, f, ﬁ, fr, i, j, k, e, l, m, in, n, ct, out, p, t and WB are used to denote average, cooling medium, drift, end of the cooling towers, fan, ﬁll, cross-sectional, hot process stream, heated water streams, stage in the cooler network, type of packing, constants for type of packing to calculate the Merkel number, constants for type of packing to calculate the loss coefﬁcient, inlet, temperature increment index, cooling tower, outlet, pump, total and wet-bulb, respectively. The superscript max is an upper limit and min is a lower limit. In addition, the scalar NOK is the total number of stages in the cooler network. Two types of variables appear in this formulation: continuous and integer variables. The continuous variables denote the stream ﬂow rates, their temperatures, as well as the design and operating variables of coolers and cooling towers. The integer variables represent the existence or not of coolers and cooling towers. If a particular cooler or cooling tower exists in the optimal cooling system, the corresponding binary variable takes the value of one and the value of zero otherwise.

961

4.1. Overall heat balance for each hot process stream in the cooler network The hot process streams need cooling from their inlet temperatures THINi to the respective outlet temperatures THOUTi. They are identiﬁed by the subindex of the hot stream’s number i. The product of the speciﬁc heat capacity and the mass ﬂow rate is deﬁned as FCPi. The heat QHPi to be removed from each hot process stream is expressed as:

ðTHINi THOUTi ÞFCPi ¼ QHPi ;

i˛HP

(1)

The heat load of hot stream i is equal the sum of the heat qi,k that it exchanges with the cooling water at each stage k

QHPi ¼

X

qi;k ;

i˛HP

(2)

k˛ST

The total heat load Q of the cooler network can be expressed as the sum of the heat loads of the hot process streams:

X

Q ¼

QHPi

(3)

i˛HP

4.2. Heat balance for each match in the cooler network Equations (4) and (5) below describe the energy balances for each match between hot stream i and the cooling water at the stage k,

Thi;kþ1 Thi;k FCPi ¼ qi;k ;

k˛ST; i˛HP

Tcouti;k Tcink fwxi;k CPcu ¼ qi;k ;

k˛ST; i˛HP

(4)

(5)

where fwxi,k is the cooling water ﬂow rate for each match between hot process stream i and cooling water in stage k, Thi,kþ1 and Thi,k are the temperatures of hot stream i at hot end (k þ 1) and cold end (k) of stage k, respectively, Tcink is the inlet water temperature to stage k, Tcouti,k is the outlet temperature of cooling water for each match between hot process stream i and cooling water in stage k, and CPcu is the heat capacity at constant pressure of cooling water. The cooling water ﬂows and the temperatures for cooler units are variables, while CPcu is ﬁxed prior to the optimization process. 4.3. Mass and heat balances of the cooling water at each stage of the cooler network Based on the proposed representation of the cooler network, the mass balances of the cooling water around each stage can be described as:

X

X

fwxi;k þ

fwcct;kþ1 þ fwmkþ1

ct˛NCT

i˛HP

¼ FOk þ

X

fwxi;kþ1 ;

k˛ST 1

(6)

i˛HP

and the corresponding energy balances are:

X

!

fwxi;k Tcink þ

i˛HP

¼ FOk TOk þ

X

fwcct;kþ1 Twdis;ct þ fwmkþ1 Twm

ct˛NCT

X fwxi;kþ1 Tcinkþ1 ; k˛ST 1

(7)

i˛HP

where fwcct,k and Twdis,ct are the ﬂow rate and temperature of the cooling water supplied by the cooling tower ct to the stage k of the

962

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

cooler network, fwmkþ1 and Twm are the ﬂow rate and temperature of the make-up water supply to the hotter adjacent stage k þ 1, FOk and TOk are the ﬂow rate and temperature of the stream sent to the cooling tower network from stage k (e.g., bypass of stage k), fwxi,kþ1 is the cooling water ﬂow rate of match between hot stream i and the cooling water at the stage k þ 1. The energy balances for the splitters in each stage are written as:

X

! fwxi;k TOk ¼

i˛HP

X fwxi;k Tcouti;k ;

k˛ST 1

(8)

max

qi;k Uqi;k z1i;k 0;

i˛HP; k˛ST

(18)

max

where Uqi;k is an upper limit equals to the heat content of the hot process stream i and z1i;k is a binary variable used to determine the existence of the cooler units. 4.8. Temperature difference constraints

i˛HP

4.4. Mass and energy balances at the inlet mixer of the ﬁrst stage It should be noted that only mixing of the various cooling-water supplies and make-up water is allowed at the mixer located prior to the ﬁrst stage of the cooler network. Therefore, mass and energy balance constraints for this mixer have not been included in equation (7). So for the inlet mixer of the ﬁrst stage:

fwmk þ

X

fwcct;k ¼

ct˛NCT

X

the cooling water in stage k. These constraints are written as follows:

X

fwxi;k ;

k ¼ 1

fwcct;k Twdis;ct þfwmk Twm ¼

ct˛NCT

fwxi;k Tcink ; k ¼ 1

(10)

i˛HP

4.5. Speciﬁcation of the inlet and outlet temperatures The ﬁxed inlet and outlet temperatures of the hot process streams deﬁne the highest location (k ¼ NOK þ 1) and lowest location (k ¼ 1) of temperature in the cooler network superstructure, respectively. These constraints are given by,

THINi ¼ Thi;NOKþ1 ; THOUTi ¼ Thi;1 ;

i˛HP

i˛HP

(11) (12)

Also, the temperature of the water leaving each match must be lower than an upper limit to avoid fouling, scaling and corrosion. This constraint is expressed as:

Tcouti;k Umax Tcouti;k ;

i˛HP; k˛ST

(13)

max

where UTcouti;k is the upper limit that commonly takes the value 50 C [20]. 4.6. Temperature constraints In the cooler network there must be a monotonic increase of temperature at each successive stage. These constraints can be written as:

Thi;k Thi;kþ1 ;

k˛ST; i˛HP

(14)

Tcink Tcouti;k ;

k˛ST; i˛HP

(15)

Tcink Tcinkþ1 ;

k˛ST; ck>1

(16)

Tcouti;k Tcouti;kþ1 ;

dtcali;k Thi;kþ1 Tcouti;k þ Gi 1 z1i;k ; dtfrii;k Thi;k Tcink þ Gi 1 z1i;k ;

k˛ST; i˛HP

(19) (20)

dtcali;k DTMIN ;

k˛ST; i˛HP

(21)

dtfrii;k DTMIN ;

k˛ST; i˛HP

(22)

where dtfrii,k and dtcali,k are the temperature differences in the cold and hot ends for the match between hot stream i and cooling water in stage k, DTMIN is the minimum temperature difference allowed between hot process streams and cooling water, and Gi is an upper limit for the temperature difference of the hot process stream i. The value of Gi is given by,

Gi ¼ maxð0; THINi TCUIN; THOUTi TCUin ; THINi Tcoutmax ; THOUTi Tcoutmax Þ

(23)

Equations (19) and (20) are written as inequalities because the heat exchanger costs decrease as the temperature differences increase. Note that if a cooler between the hot stream i and cooling water in stage k is selected, the binary variable z1i;k will take the value 1 and the constraints (19) and (20) are applied to calculate nonnegative temperature differences. Otherwise, if such cooler is not selected, the corresponding binary variable will take the value 0 and the constraints (19) and (20) are relaxed for that match to allow the temperatures to take any values (as determined by the rest of the model). Therefore, the binary variables are required so as to avoid causing infeasibilities. 4.9. Assignment of cooling tower network inlet ﬂow rates and temperatures The mathematical model for the cooling tower network involves a set of inlet streams j that will be denoted by the index set NEF, which is the set of all possible heated water streams ﬂowing to the cooling tower network from the cooler network. For a given problem with NH hot streams, NEF ¼ 2NH 1. Each of these streams j will have associated as variables the ﬂow rate Fwhj and the temperature Twhj deﬁned as,

k ¼ NOK; j˛SJ i ; j ¼ i

Fwhj ¼ fwxi;k ; k˛ST; i˛HP

k˛ST; i˛HP

(9)

i˛HP

X

To ensure thermodynamic feasibility for heat transfer, nonnegative temperature differences are required between the hot and cooling water streams for the various cooler units selected by the mathematical model. These constraints are written as follows:

(24)

(17) Fwhj ¼ FOk ;

j ¼ SJ b ; k ¼ j NOK

(25)

4.7. Existence of the cooler units in the cooler network Logic constraints and binary variables are deﬁned to represent the existence or nonexistence of a match between hot streams i and

Twhj ¼ Tcouti;k ; Twhj ¼ TOk ;

k ¼ NOK; j˛SJ i ; j ¼ i b

j ¼ SJ ; k ¼ j NOK

(26) (27)

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

where SJi is a subset of NEF and represents the number of heated water streams directed to the cooling tower network from the coolers in the last stage of the cooler network, and SJb is a subset of NEF that represents the number of heated water streams directed to the cooling tower network from the splitters in the stages of the cooler network. For example, from the superstructure for the cooling system shown in Fig. 2, the subset SJi is deﬁned by 1, 2 and 3 and the subset SJb is deﬁned by 4 and 5. 4.10. Mass balances for splitters of the cooling tower network inlet streams The superstructure for the cooling tower network consists of NCT cooling towers. Each inlet stream of the cooling tower network is split into 2NCT streams, NCT of which are directed to the inlet of 1 Þ and the other NCT are bypasses to the the cooling towers ðfwtj;ct 2 cooling towers ðfwtj;ct Þ,

X

Fwhj ¼

1 fwtj;ct þ

ct˛NCT

X

2 fwtj;ct ;

j˛NEF

(28)

Fwdis;ct ¼

Fwin;ct ¼

X

1 fwtj;ct þ

j˛NEF

X ct 0 ˛NCT

fttct0 ;ct ;

ct˛NCT

(29)

ct 0 sct Fwin;ct Twin;ct ¼

X

1 fwtj;ct Twhj

j˛NEF

þ

X

ct 0 ˛NCT ct 0 sct

ct˛NCT

4.12. Mass and energy balances for mixers at the outlet of each cooling tower

X

ct˛NCT

(31)

where win,ct and wout,ct are the mass-fraction humidity of the air entering and leaving each cooling tower ct, respectively. The mass ﬂow rate of drift can be expressed as [21],

fwdct ¼ 0:002Fwin;ct ;

ct˛NCT

fwect þ fwdct þ fwbct fwdct NCYCLES

(36)

where NCYCLES is the number of cycles of concentration required to limit the formation of scale in cooling equipment. The total blowdown ﬂow rate (Fwb) for the cooling systems is the sum of the blowdown ﬂow rates for each cooling tower ct,

Fwb ¼

X

fwbct

(37)

ct˛NCT

It should be noted that the streams leaving the mixers at the outlet of each cooling tower are split into NH þ NCT streams, NH of which are sent to the mixing points at the inlet of each stage k of the cooler network, NCT 1 are recycled to the other cooling towers, and the last one is directed to the blowdown mixer,

X ct 0 ˛NCT ct 0 sct

X

fttct0 ;ct þ

fwcct;k þ fwbct ; ct˛NCT

(38)

k˛ST

To maintain the total cooling water ﬂow rate constant in the cooling system, it is necessary to add a fresh water ﬂow rate to replace the overall loss of water due to evaporation, drift and blowdown:

X

fwect þ

X

fwdnct þ

ct˛NCT

X

fwbct

(39)

ct˛NCT

where Fwm is the water consumption or make-up water of the cooling system. The make-up water stream is split into NH streams, which go to the inlet mixers of each stage of the cooler network,

Fwm ¼

X

fwmk

(40)

k˛ST

(32)

Thus, the outlet water ﬂow rate (Fwout,ct) for each cooling tower ct is given by:

Fwout;ct ¼ Fwin;ct fwect fwdct ;

(35)

A portion of the circulating water that is called the blowdown is removed from each cooling tower to avoid the excess of dissolved solids/impurities in the water, such that deposits do not form on the surfaces within the heat-transfer equipment. The ﬂow rate of blowdown (fwbct) for each cooling tower ct can be expressed as [20],

ct˛NCT

fwect ¼ Fact wout;ct win;ct ;

2 fwtj;ct Twhj þFwout;ct Twout;ct ; ct˛NCT

4.13. Blowdown and make-up water ﬂow rates

Fwm ¼

In each cooling tower, the water is lost due to evaporation (fwect) and drift (fwdct). Conservation of mass yields the following relationship for the evaporation rate of water:

(34)

where Fwdis,ct are the ﬂow rates of the outlet streams from such mixers, and Twout,ct is the temperature of the outlet water from each cooling tower.

(30)

where ct 0 is an alias for index ct used to model the recirculation between cooling towers.

ct˛NCT

j˛NEF

Fwdis;ct ¼

fttct 0 ;ct Twdis;ct 0 ;

2 fwtj;ct þ Fwout;ct ;

Fwdis;ct Twdis;ct ¼

fwbct ¼

The inlet stream of each cooling tower consists of streams originated from the heated water streams that leave the cooler network and substreams of the outlets of all the others cooling towers, fttct 0 ;ct . Therefore, each mixer at the inlet of a cooling tower produces a ﬂow Fwin,ct at a given temperature Twin,ct that can be obtained by:

963

j˛NEF

ct˛NCT

4.11. Mass and energy balances in mixers prior to each cooling tower

X

ct˛NCT

(33)

The mass and energy balance constraints for mixers at the outlet of each cooling tower will then be given by:

4.14. Flow rate constraints in the cooling tower network To avoid mathematical problems, the recycle in the same cooling tower is not considered. In this regard, the next feasibility constraints is employed,

fttct;ct 0 ¼ 0;

ct; ct 0 ˛NCT; ct ¼ ct 0

(41)

964

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

4.15. Design of cooling towers The following equations represent the geometrical and thermodynamic constraints of the cooling towers that are necessary to meet the requirements of the cooler network. Therefore, the following disjunction is used to determine the existence of a cooling tower and to apply the corresponding design equations,

2

3 2 Zct 2 6 J Jmax 7 Zct n ; ct˛NCT 4 ct 5 ct Jct ¼ 0 Jct Jmin ct 2 is a Boolean variable used to determine the existence of where Zct the cooling towers, Jmax is an upper limit for the variables, Jmin ct ct is a lower limit for the variables, Jct is any design variable of the cooling tower like inlet ﬂow rate, mass air ﬂow rate, Merkel number, and others. For example, when inlet ﬂow rate to the cooling tower is used as a design variable, previous disjunction for the inlet ﬂow rate to each cooling tower ct is reformulated as follows:

max

Fwin;ct UFwin;ct z2ct 0; min

Fwin;ct UFwin;nct z2ct 0;

ct˛NCT

(42)

ct˛NCT

(43)

min where Umax Fwin;ct and UFwin;ct are upper and lower limits for the inlet ﬂow rate to each cooling tower, respectively. Notice that this reformulation is applied to each design variable of the cooling towers. The detailed thermal-hydraulic design of cooling towers is modeled with the Merkel’s method [17]. The required Merkel’s number in each cooling tower, Mect, is calculated using the fourpoint Chebyshev integration technique [22],

Mect ¼ 0:25CPcu Twin;ct Twout;ct

4 X

1=Dhn;ct ; ct˛NCT

(44)

n¼1

where n is the temperature increment index. For each temperature increment, the local enthalpy difference ðDhn;ct Þ is calculated as follows,

Dhn;ct ¼ hsan;ct han;ct ;

n ¼ 1; :::; 4; ct˛NCT

(45)

The heat and mass transfer characteristics for a particular type of packing are given by the available Merkel number correlation developed by Kloppers and Kröger [19]:

!c2;ct !c3;ct 1þc4;ct Fwav;ct Fact Lfi;ct Mect ¼c1;ct Afr;ct Afr;ct c Twin;ct 5;ct ; ct˛NCT

where Afr,ct is the packing area, Lﬁ,ct is the height packing, and c1, c2, c3, c4 and c5 are constants that depend on the type of ﬁll used for each cooling tower ct. To calculate the available Merkel number, the following e : disjunction is used through the Boolean variable Yct

2

CPcu Fwav;ct Twn;ct Twout;ct ; Fact n ¼ 1; .; 4; ct˛NCT

ð46Þ

Twn;ct ¼ Twout;ct þ TCHn Twin;ct Twout;ct ; n ¼ 1; .; 4; ct˛NCT

cl;ct 2

where TCHn is a constant that represents the Chebyshev points (TCH1 ¼ 0.1, TCH2 ¼ 0.4, TCH3 ¼ 0.6 and TCH4 ¼ 0.9), hain,ct and Fact are the enthalpy of inlet air and the air mass ﬂow rate for each cooling tower, and Fwav,ct is the average value of the water mass ﬂow rate. Thus,

Fwav;ct ¼

Fwin;ct þ Fwout;ct ; 2

ct˛NCT

(48)

ðsplash fillÞ ¼ c1l;ct ; l ¼ 1;.;5

cl;ct ¼

3

3

2 Yct

cl;ct

7 ðtrickle fillÞ 5 ¼ c2l;ct ; l ¼ 1;.;5

7 5; ct˛NCT

ðfilm fillÞ c3l;ct ;

2

7 6 5n4

3 Yct

6 n4

l ¼ 1;.;5

Notice that only when the cooling tower ct exists, its design variables are calculated and only one ﬁll type must be selected. Therefore, the sum of the binary variables referred to the different ﬁll types must be equal to the binary variable that determines the existence of the cooling towers. Then, this disjunction can be described with the convex hull reformulation [23] by the following set of algebraic equations:

y1ct þ y2ct þ y3ct ¼ z2ct ;

ct˛NCT

cl;ct ¼ c1l;ct þ c2l;ct þ c3l;ct ; cel;ct ¼ bel yect ;

(50)

l ¼ 1; .; 5; ct˛NCT

e ¼ 1; .; 3; l ¼ 1; .; 5; ct˛NCT

(51) (52)

Values for the coefﬁcients bel for the splash, trickle, and ﬁlm type of ﬁlls are given in Table 1 [19]. For each type of packing, the loss coefﬁcient correlation can be expressed in the following form [18]:

" Kfi;ct ¼ d1;ct þ d4;ct

Fwav;ct Afr;ct

!d2;ct

Fwav;ct Afr;ct

!d5;ct

Fact Afr;ct

!d3;ct

Fact Afr;ct

!d6;ct # Lfi;ct ; ct˛NCT

ð53Þ

where d1, d2, d3, d4, d5 and d6 are constants for calculating the loss coefﬁcient in the cooling towers (Kﬁ,ct). The corresponding disjunction to select the value of above-mentioned constants depending on the ﬁll type is given by,

2 ð47Þ

3

1 Yct

6 4

and the algebraic equations to calculate the enthalpy of bulk airewater vapor mixture (han,ct) and the water temperature (Twn,ct) corresponding to each Chebyshev point are given by the following relationships:

han;ct ¼ hain;ct þ

ð49Þ

6 6 4

1 Yct ðsplash fillÞ

3

2

7 6 7n6 5 4

2 Yct ðtrickle fillÞ

dm;ct ¼ d1m;ct ; m ¼ 1; .; 6 dm;ct ¼ d2m;ct ; m ¼ 1; .; 6 2 3 3 Yct 6 7 7; ct˛NCT ðfilm fillÞ n6 4 5 dm;ct ¼ d3m;ct ; m ¼ 1; .; 6

3 7 7 5

Using the convex hull reformulation [23], previous disjunction is modeled as follows:

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

4.16. Feasibility constraints for the cooling towers

Table 1 Constants for transfer coefﬁcients. bel

l

1 2 3 4 5

e ¼ 1 (splash ﬁll)

e ¼ 2 (trickle ﬁll)

e ¼ 3 (ﬁlm ﬁll)

0.249013 0.464089 0.653578 0 0

1.930306 0.568230 0.641400 0.352377 0.178670

1.019766 0.432896 0.782744 0.292870 0

dm;ct ¼ d1m;ct þ d2m;ct þ d3m;ct ; dem;ct ¼ cem yect ;

(54)

e ¼ 1; .; 3; m ¼ 1; .; 6; ct˛NCT

(55)

Values for the coefﬁcients for the three ﬁlls are given in Table 2 [18]. The total pressure drop of the air stream for each cooling tower (DPt,ct) is given by [20],

rav;ct A2fr;ct

Kfi;ct Lfi;ct þ 6:5 ;

ct˛NCT

(56)

where Favav,ct is the arithmetic mean air-vapor ﬂowrate through the ﬁll in each cooling tower and rav;ct is the harmonic mean density of the moist air through the ﬁll. These variables are calculated as:

Favav;ct ¼

Favin;ct þ Favout;ct ; 2

ct˛NCT

rav;ct ¼ 1= 1=rin;ct þ 1=rout;ct ;

(57)

ct˛NCT

(58)

where rin,ct and rout,ct are the densities for the inlet and outlet air which are estimated with the equation (A.7) given in the Appendix. The air-vapor ﬂow at the ﬁll inlet and outlet Favin,ct and Favout,ct are calculated as follows:

Favin;ct ¼ Fact þ win;ct Fact ;

ct˛NCT

Favout;ct ¼ Fact þ wout;ct Fact ;

(60)

The required power for the tower fan of each cooling tower (PCf,ct) is given by:

PCf ;ct ¼

Favin;ct DPt;ct

rin;ct hf ;ct

; ct˛NCT

(61)

where hf ;ct is the fan efﬁciency. The power consumption for the water pump of each cooling tower (PCp,ct) may be expressed as [24]:

PCp;ct

ct˛NTC

(64)

Twin;ct >Twout;ct ; TAout;ct >TAin;ct ;

ct˛NTC ct˛NTC

(65) (66)

The local driving force (hsact hact) must satisfy the following condition at any point in each cooling tower ct [20],

hsan;ct han;ct >0

n ¼ 1; .; 4; ct˛NTC

(67)

The maximum and minimum water and air loads in the cooling towers are determined by the range of test data used to develop the correlations for the lost and overall mass transfer coefﬁcients for the ﬁlls. The constraints are [18,19],

2:90

Fwav;ct 5:96; Afr;ct

1:20

Fact 4:25; Afr;ct

ct˛NTC

ct˛NTC

(68)

(69)

Although each cooling tower can be designed to operate at any feasible Fwav,ct/Fact ratio, Singham [25] suggests the following limits:

0:5

Fwav;ct 2:5; Fact

ct˛NTC

(70)

The ﬂow rates of the heated water streams leaving the previous splitters to the cooling tower network have the following limits: 1 0 fwtj;ct Fwhj ;

j˛NEF; ct˛NCT j˛NEF; ct˛NCT

(71) (72)

(62) 4.17. Objective function The objective function is to minimize the total annual cost of the cooling system (TACS) that consists in the total annual cost of cooler network (TACNC), the total annual cost of the cooling tower network (TACTC) and the pumping cost (PWC),

Table 2 Constants for loss coefﬁcients.

1 2 3 4 5 6

(63)

The ﬁnal set of temperature feasibility constraints arises from the fact that the water stream must be cooled and the air stream heated in the cooling towers. These constraints can be expressed as:

2 0 fwtj;ct Fwhj

#

"Fw dis;ct Lfi;ct þ 3:048 g ¼ ; ct˛NCT hp gc

where hp is the pump efﬁciency.

m

Twin;ct 50 C;

(59)

ct˛NCT

ct˛NCT

where TWBin,ct is the wet-bulb temperature of the entering air. To avoid fouling, scaling and corrosion, the temperature of the water entering to each cooling tower should not be greater than 50 C [20], as given in constraint (64).

m ¼ 1; .; 6; ct˛NCT

Fav2av;ct

The water outlet temperature of each cooling tower is chosen so that [20]:

Twout;ct TWBin;ct þ 2:8;

cem

DPt;ct ¼ 0:8335

965

cem e ¼ 1 (splash ﬁll)

e ¼ 2 (trickle ﬁll)

e ¼ 3 (ﬁlm ﬁll)

3.179688 1.083916 1.965418 0.639088 0.684936 0.642767

7.047319 0.812454 1.143846 2.677231 0.294827 1.018498

3.897830 0.777271 2.114727 15.327472 0.215975 0.079696

TACS ¼ TACNC þ TACTC þ PWC

(73)

4.17.1. Power cost for the water pump The cost of the electricity demanded by the water pumps in the cooling systems is generated by the power consumption in the pumps located at the outlet of each cooling tower,

966

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

X

PWC ¼ HY ce

PCp;ct

(74)

ct˛NCT

2 6 4

where HY is the yearly operating time and ce is the unit cost for the electricity.

1 CCTVct ¼ CCTVct

(75)

where the capital cooler network cost is obtained from the following expression,

" CAPCNC ¼ KF

X X

CFHEi z1i;k þ

i˛HP k˛ST

X X

# CAHEi Abi;k

(76)

i˛HP k˛ST

Here, CFHEi is the ﬁxed cost for the cooler unit i, CAHEi is the cost coefﬁcient for the area of cooler unit i, KF is the factor used for annualizing the capital costs, and b is the exponent for the capital cost function. The area for each match is calculated as follows,

3

2 Yct

7 6 5n4

ðsplash fillÞ

4.17.2. Total annual cost for the cooler network The total annual cost for the cooler network is formed by the annualized capital cost of cooler units (CAPCNC) and the make-up water cost (OPCNC).

TACNC ¼ CAPCNC þ OPCNC

2

3

1 Yct

7 5

ðtrickle fillÞ

2 CCTVct ¼ CCTVct 2 3 Yct 6 n4 ðfilm fillÞ

3 7 5; ct˛NCT

3 CCTVct ¼ CCTVct

This disjunction is algebraically reformulated as [23]: 1 2 3 CCTVct ¼ CCTVct þ CCTVct þ CCTVct ; e ¼ ae yect ; CCTVct

ct˛NCT

(82)

e ¼ 1; :::; 3; ct˛NCT

(83)

e

where the parameters a are shown in Table 3 for the splash, trickle, and ﬁlm types of ﬁll. Note that the investment cost expression properly reﬂects the inﬂuence of the type of packing, the air mass ﬂowrate (Fact) and basic geometric parameters, such as height (Lﬁ,ct) and area (Aﬁ,ct) for each tower packing. The electricity cost needed to operate the air fans is calculated using the following expression:

OPTNC ¼ HY ce

X

PCf ;ct

(84)

ct˛NCT

Ai;k ¼ qi;k = Ui DTMLi;k þ d ;

i˛HP; k˛ST

(77) 4.18. Physical properties

Ui ¼ 1=ð1=hi þ 1=hcu Þ;

i˛HP

(78)

where Ui is the overall heat-transfer coefﬁcient, hi and hcu are the ﬁlm heat transfer coefﬁcients for hot process streams and cooling water, respectively, DTMLi;k is the mean-logarithmic temperature difference of each match and d is a small parameter (i.e., 1 106 ) used to avoid divisions by zero. The Chen [26] approximation is used to estimate DTMLi;k ,

DTMLi;k ¼

1=3 dtcali;k dtfrii;k dtcali;k þ dtfrii;k 2 ; k˛ST; i˛HP

(79)

(80)

4.17.3. Total annual cost for the cooling tower network The total annual cost for the cooling tower network involves the investment cost for the cooling towers (CAPTNC) as well as the operational cost (OPTNC) of the tower fans. The investment cost for the cooling towers is represented by a nonlinear ﬁxed charge expression of the form [27]:

i X h CCTF z2ct þ CCTVct Afr;ct Lfi;ct þ CCTMA Fact

hsai;ct ¼ f Twi;ct

(85)

i ¼ 1; .4;

wout;ct ¼ f Pt;ct ; PVout;ct ; PVWB;in;ct ¼ f TWBin;ct ; TAout;ct ¼ f hsaout;ct ; PVout;ct ¼ f TAout;ct ;

where cw is the unit cost for the make-up water.

CAPTNC ¼ KF

hain;ct ¼ f TWBin;ct ; ct˛NCT ct˛NCT

win;ct ¼ f TWBin;ct ; TAin;ct ; Pt;ct ; PVWB;in;ct ;

The operational cost for the cooler network is generated by the make-up water used to replace the overall water loss in the cooling tower network,

OPCNC ¼ cwHY Fwm

The physical properties that appear in the proposed model are shown in this section. The correlations are listed in the Appendix and should be only applied if a cooling tower exists.

ct˛NCT

ct˛NCT

(89)

ct˛NCT

(90)

ct˛NCT

rout;ct ¼ f Pt;ct ; TAout;ct ; wout;ct

(87) (88)

ct˛NCT

rin;ct ¼ f Pt;ct ; TAin;ct ; win;ct ;

(86)

(91)

ct˛NCT

(92)

(93)

5. Solution procedure

(81)

ct˛NCT

where CCTF is the ﬁxed charge associated with the cooling towers, CCTVct is the incremental cooling towers cost based on the tower ﬁll volume, and CCTMA is the incremental cooling towers cost based on air mass ﬂowrate. The cost coefﬁcient CCTVct depends on the type of packing. To implement the discrete choice for the type of e is used as part of the following packing, the Boolean variable Yct disjunction,

It should be noted that this model takes explicitly into account the interactions among the cooler network and the cooling tower network (i.e., the two major components in the integrated cooling Table 3 e Cost coefﬁcients CCTV for each type of ﬁll. ae

e ¼ 1 (splash ﬁll)

e ¼ 2 (trickle ﬁll)

e ¼ 3 (ﬁlm ﬁll)

2006.6

1812.25

1606.15

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974 Table 4 Data for examples. Streams

THIN ( C)

Example 1 1 80 2 75 3 120 4 90 5 110 Example 2 1 45 2 66 3 45 4 50 5 98 6 65 Parameter Value ce HY KF

hf hp Pt CCTF CCTMA

THOUT ( C) FCP (kW/ C) Q (kW)

h (kW/m2 C)

60 28 40 45 40

500 100 450 300 250

10000 4700 36000 13500 17500

1.089 0.845 0.903 1.025 0.75

22 20 33 38 55 45

152 543 4167 4750 1395 850

3496 24978 50004 57000 59985 17000

1.1 1.125 0.95 0.875 0.95 1.25

Parameter

Value

Parameter Value

0.2983 year1 0.75

b

5.75 105 NCYCLES US$/kg water 4.193 hcu kJ/kg C 0.80 Twm

CFHE

100 US$

hain,ct

0.6 101325 Pa

CAHE TAin,ct

700 US$/m2 22 C

PVWB,in,ct win,ct

12 C 5 C

rin,ct

0.076 CUw US$/kW h 8000 h/year CPcu

31185 US$ TWBin,ct 1097.5 US$/ DTMIN (kg dry air/s)

4 2.5 kW/m2 K

20 C 34.188 J/kg dry air 1417.8 Pa 0.005 kg-water/ kg-dry-air 1.192 kg/m3

967

water system). Also, the model can provide the traditional cooling system conﬁguration that only requires a single cooling tower if this represents the optimal conﬁguration. The proposed MINLP model was implemented in the software GAMS [28], where the solvers DICOPT together with CPLEX and CONOPT [29] were used to solve the MINLP, MILP and NLP problems, respectively. DICOPT yields optimal solutions for convex MINLP models; however, for non-convex problems such as the one presented in this paper, no global optimal solution can be guaranteed using DICOPT. In these cases, the solver DICOPT [29] depends strongly on the initial guesses for the optimization variables to ﬁnd good solutions; therefore, the possibility to ﬁnd a solution that is either close or equal to the global optimum depends strongly on the starting point of the optimization procedure. To ﬁnd solutions close to the global optimum, in this paper the following initialization approach is implemented: 1. Solve the problem that only includes the minimization for the objective function subject to equations (1)e(23) and equations (76)e(80), which represent the cooler network. In this step, the variables Twdis,ct are ﬁxed (i.e., they are known parameters). 2. The results obtained in the above step are used as initial values to solve again the cooler network problem (equations (1)e(23) and equations (76)e(80)); however, in this step, also equations (24)e(27) are included to determine Fwhj and Twhj. 3. The values of Fwhj and Twhj so obtained in the previous step are used to solve the cooling tower network model (equations

Fig. 3. Conﬁguration with multiple cooling towers for Example 1.

968

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

(28)e(72) and equations (81)e(93)). In addition, the ﬁxed values for Twdis,ct used in the ﬁrst step are again employed due to they were previously used to obtain the initial conﬁguration of the cooler network. 4. Finally, the conﬁguration for the cooler network obtained in the ﬁrst step and the conﬁguration for the cooling tower network obtained in the third stage are joined to determine the initial overall conﬁguration, which is the starting point required to solve the integrated model (equations (1)e(93)). The conﬁguration provided by the solution of the integrated model is used iteratively as a new estimate of the optimization procedure until there are not improvements in the objective function. Here, the conﬁguration of the iteration t is the initial value for the iteration t þ 1. Therefore, the implemented approach gives good solutions which are close to the optimal one; however, as it was mentioned above, the global optimal solution cannot be guaranteed when the solver DICOPT is used. For the case when the global optimal solution is required, global solvers like BARON or LINDOGLobal [29] can be used to solve the proposed MINLP model. 6. Results Two examples are used to show the application of the proposed model. The ﬁrst example features ﬁve hot process streams while the second one includes six hot process streams. The process stream data of the examples are given in Table 4. For both examples, the values of parameters ce, HY, KF, hf, hp, Pt, CCTF, CCTMA, CUw, CPcu, b, CFHE, CAHE, TAin,ct, TWBin,ct, DTMIN, NCYCLES, hcu, Twm, hain,ct, PVWB,in,ct, win,ct and rin,ct are given in Table 4. In addition, four cooling towers were considered in the cooling system superstructure. The computational results were obtained using an Intel Core 2 Duo processor at 2.53 GHz. It should be noted that each example was solved using the proposed model for cooling water systems with multiple cooling towers and for cooling water systems with a single cooling tower and a set of coolers arranged in parallel (i.e. traditional cooling

systems) to establish a comparison between these strategies. In order to obtain the traditional conﬁguration from the proposed model, the number of stages of the cooler network and the number of cooling towers must be equal to 1.

6.1. Example 1 The MINLP formulation for this example has 33 binary variables, 543 continuous variables, and 722 constraints. Solution of this MINLP problem using GAMS/DICOPT required 21 s of CPU time. The cooling system conﬁguration corresponding to the MINLP solution is shown in Fig. 3, which includes the overall cooling water requirements to the cooler network and the ﬂow rates for evaporation, blowdown and make-up water as well as the design and operation variables of the cooling towers and cooler units. The optimal cooling system has a total annual cost of $773,959.3 yr1. Most of this cost is attributed to the cooling tower network requiring $422,225 yr1, while the total annual cost for the cooler network is $314,365.8 yr1 and the pumping power cost is $37,368.6 yr1. Table 5 summarizes the economic results obtained. Note that the cooler network consists of ﬁve cooler units to cool the hot process streams down. These units have a total area of 4892.72 m2. The cooler network is supplied by two cooling towers CT1 and CT2 with different supply temperatures, which are connected in parallel. The cooling water source with the lowest supply temperature (i.e. the water from the cooling tower CT1 at 23.723 C) is mixed with make-up water before it is used to supply the coolers C-1 and C-2, which have a parallel arrangement. This cooling water source removes 22,200 kW from hot process streams H2 and H5. On the other hand, coolers C-3, C-4 and C-5 are supplied with cooling water from cooling tower CT2, which is the source with the highest supply temperature (30.598 C). This cooling water is ﬁrst used in cooler C-3 and then reused in coolers C-4 and C-5 before being returned to the cooling tower network; thus, these coolers have a serieseparallel arrangement. The overall cooling water ﬂow rate to the cooler network is 927.472 kg/s.

Table 5 Results of examples. Example 1

Example 2

Traditional arrangement

Proposed formulation

Traditional arrangement

Proposed formulation

1,134,586.0

9,268,981.1 759,956.4 8,284,791.2 588,101.8 3,734,926.7 171,854.6 4,549,864.5 224,233.5 103.8 10 477 620 10

2,940,433.4 68.28% 1,225,943.7 1,611,867.7 1,060,943.3 1,098,645.1 165,000.4 513,222.6 102,622.0 99.7 44 641 835 23

Total annual cost of cooling system (US$/yr) Annual saving Total annual cost of cooler network (US$/yr) Total annual cost of cooling tower network (US$/yr) Capital cost of cooler network (US$/yr) Capital cost of cooling tower network (US$/yr) Operating cost of cooler network (US$/yr) Operating cost of cooling tower network (US$/yr) Pumping power cost (US$/year) Make-up water (kg/s) Binary variables Continuous variables Constraints CPU (s)

279,020.3 817,919.4 215,830.6 598,248.9 63,189.7 219,670.5 37,646.4 38.2 9 384 483 9

773,959.3 31.78% 314,365.8 422,225.0 251,306.7 326,093.1 63,059.1 96,131.9 37,368.6 38.079 33 543 722 21

Design of cooling towers

CT1

CT1

CT2

CT1

CT1

CT2

Heat transfer area (m2) Fill height (m) Merkel number (dimensionless) Total pressure drop of the air stream (Pa) Power consumption of the tower fan (kW) Power consumption of the water pump (kW) Loss coefﬁcient (m1) Water-to-air mass ratio (dimensionless) Type of packing

243.7 2.4 2.0 203.1 359.1 61.6 16.7 0.9 Film

52.2 1.6 2.1 151.6 54.4 12.0 22.3 1.0 Film

145.0 1.0 1.2 103.9 102.8 49.1 25.2 1.8 Film

916.1 5.5 6.7 969.3 7436.9 366.9 22.1 0.7 Film

71.3 3.7 5.0 630.0 386.0 22.412 22.1 0.7 Film

528.3 1.3 1.6 125.1 452.9 145.5 23.7 1.3 Film

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

It is interesting to note that the heated water streams from the cooler network are split and mixed before they are fed to the cooling towers CT1 and CT2. Since all hot process streams have inlet temperatures above 50 C þ DTMIN, the return temperature from the cooler network to both cooling towers is 50 C, which is the maximum acceptable temperature of the cooling water in the cooling system due to fouling, corrosion or problems with the cooling tower packing. Table 5 shows also a comparison of the optimal conﬁguration obtained from the proposed MINLP model with the traditional cooling system (a parallel arrangement of coolers that are supplied by a single cooling tower), which is shown in Fig. 4. Note that the air ﬂow rate to the cooling tower of this cooling system is 813.241 kg/s, 39.11% greater than the air consumption for the system with two cooling water sources. Also, note that the total annual cost for the traditional design is $1,134,586 yr1, 46.59% higher than the total annual cost for the cooling system with two cooling water sources.

969

6.2. Example 2 The resulting MINLP problem involves 44 binary variables, 641 continuous variables, and 835 constraints. Solution of this formulation took approximately 15 CPU seconds using GAMS/DICOPT. The capital and operating costs of the optimal solution are given in Table 5. The optimal cooling system, as shown in Fig. 5, consists of two cooling towers CT1 and CT2 in series with supply temperatures of 15 C and 26.922 C, respectively, and six coolers. The cooling tower CT2 with a cross-sectional area of 528.267 m2 is followed by the cooling tower CT1 with an area of 71.304 m2. The cooling water requirements for cooling tower CT1 and cooling tower CT2 are 204.51 kg/s and 2065.419 kg/s, respectively. Cooling tower CT1 supplies coolers C-1 and C-2, which are connected in parallel, and cooling tower CT2 supplies coolers C-3, C-4, C-5 and C-6. The third and fourth coolers are connected in parallel and both of them are in series with the parallel arrangement of coolers C-5 and C-6. There is a mixing between the water supplied by cooling tower CT2 and

Fig. 4. Traditional conﬁguration for Example 1.

970

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

make-up water before removing heat from hot streams H3, H4, H5 and H6. Also, the three heated cooling water streams from the cooler network are mixed before entering the cooling tower CT2. Thus, the water return temperature (49.141 C) to the cooling tower CT2 is obtained by calculating the temperature of the mixture of heated water streams from the cooler network. On the other hand, 99.638 kg/s of cooling water are lost in the system through evaporation, drift and blowdown. It is interesting to note that a large fraction of the cooling water supplied by the cooling tower CT2 is sent to the cooler network and the blowdown mixer (88.95% and 0.93%, respectively), while the rest of it (209.055 kg/s) is fed to cooling tower CT1 before it is directed to the cooler network at a lower temperature (15 C). Therefore, this system is unique in that all the cooling water required by the smaller cooling tower CT1 is supplied by the large

cooling tower CT2 at 26.922 C. This forms the main interaction between both cooling towers. The total annual cost required for this design is $2,940,433.4 yr1, which comprises of $1,611,867.7 yr1 for the cooling tower network, $1,225,943.7 yr1 for the cooler network and $102,622 yr1 for the pumping power. This ﬁgure corresponds to a 68.28% reduction in cost compared to the traditional solution of Fig. 6, which has a total annual cost of $9,268,981.1 yr1 as shown in Table 5. A comparison of the results between the cooling systems with multiple cooling towers and traditional cooling systems indicates a signiﬁcant difference in the air ﬂow rate to the cooling towers. For Example 1, examination of the two conﬁgurations depicted in Figs. 3 and 4 shows that the air ﬂow rate to the cooling tower of the traditional design is 813.241 kg/s, which is 39.11% greater than that

Fig. 5. Conﬁguration with multiple cooling towers for Example 2.

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

971

Fig. 6. Traditional conﬁguration for Example 2.

for the optimal cooling system with two cooling towers. For Example 2, as can be seen by comparing the cooling systems shown in Figs. 5 and 6, this difference is drastically increased. In fact, in the traditional design for Example 2 the optimal air ﬂow rate to the cooling tower is 105.68% higher (3893.272 kg/s vs. 1892.807 kg/s). This difference is reﬂected in the power consumption of the tower fan and the tower size, which is indicated mainly by the volume of the ﬁll. This is the product of the height and the cross-sectional area of the tower packing. In Example 2, for instance, as can be obtained from Figs. 5 and 6, the fan power consumption of the traditional design is 128.51% higher than that of the optimal cooling system

with two cooling water sources, which has a fan power consumption of 157.13 kW. In addition, the tower size (586.01 m3) of the traditional design for Example 2 is 154.12% greater than the total tower size (230.61 m3) of the optimal cooling system (Fig. 5) derived by the proposed method in this paper. The Example 1 presents a similar behavior. Therefore, as the air ﬂow rate increases for removing the required heat of the cooler network, the tower size and the power consumption of the tower fan also increase. As a result, the total annual cost of the cooling tower network increases with increase in total air ﬂow rate. So by using multiple cooling sources, a very large reduction (68.28%) in total annual cost

972

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

results for Example 2. The annual saving is also large (31.78%) for Example 1, as shown by Table 5. These signiﬁcant cost differences between the traditional and multiple cooling sources solutions are due to the fact that, in traditional cooling systems, all the cooling water used is supplied by a single cooling tower at the lowest temperature that the water must reach, Tlcwt ¼ Tcold,min DTMIN, where Tcold,min is the outlet temperature of the coldest hot process stream. Furthermore, since the wet-bulb temperature of ambient air and the supply temperature of the cooling tower are speciﬁed, the approach temperature (Tlcwt TWBin,ct) for the single cooling tower is predeﬁned. It is worthwhile to note that the air ﬂow rate and the tower size increase when the tower approach decreases, which also increases the total annual cost of the cooling tower [20]. Therefore, the traditional design method uses the smallest approach temperature to meet the total required cooling load without regard to the energy used by the cooling tower fan and the tower size. In contrast to the traditional method described above, the MINLP formulation for optimal synthesis of cooling water systems with multiple cooling towers takes into account that not all the hot process streams require cooling water at the lowest water temperature, Tlcwt. Thus, unless all the cooling demand is required at the lowest temperature, some of the demand is satisﬁed at intermediate temperature levels when using the proposed method. As a result, the overall performance of cooling water systems can be improved by considering multiple cooling towers with different supply temperatures (i.e. less air and volume of the ﬁll are required for the removal of the same amount of heat, which gives lower total annual costs).

The two examples presented have shown that the optimal air ﬂow rate to the cooling tower for the traditional cooling system is greater than that for the system with multiple cooling water sources. Therefore, for removing the same amount of process heat, cooling water systems consisting of multiple cooling towers with different supply temperatures require less energy consumption of tower fans and packing area than traditional cooling systems, which gives lower total annual costs. In Examples 1 and 2, the cooling systems with multiple cooling sources result in an annual saving of 31.78% and 68.28%, respectively. Thus, the fact that improved solutions have been obtained using the proposed MINLP formulation illustrates the importance of considering multiple cooling water sources in the synthesis of cooling water systems.

Acknowledgements The authors acknowledge the ﬁnancial support from the Scientiﬁc Research Council of UMSNH (CIC Grant 20.1).

Appendix. Property correlations For the enthalpy of the air entering the tower [20]:

hain ¼ 6:4 þ 0:86582*TWBin þ 15:7154expð0:0544*TWBin Þ (A.1) For the enthalpy of saturated airewater vapor mixtures [20]:

7. Conclusions This paper proposes an MINLP formulation for the optimal design of integrated cooling systems that involves multiple cooling water sources with different supply temperatures. The formulation is based on superstructures of cooler and cooling tower networks for all possible ﬂow connections. This approach, therefore, accounts explicitly for the interactions among the two main components of cooling water systems to simultaneously trade-off capital investment and operating costs. The cooler network superstructure is a stage-wise representation that allows the bypass of either fresh or previously used cooling water, while the cooling tower superstructure is a novel representation based upon counter ﬂow wet-cooling towers that includes all the different cooling tower conﬁgurations (i.e. series, parallel, serieseparallel, and paralleleseries) and contains all possible options for splitting, mixing, and bypassing of heated water streams. Potential cooling water sources (streams at different temperatures originated from the outlets of cooling towers) can be assigned to each of the mixers prior to the stages of the cooler network. Make-up water can also be fed to each mixer of the cooler network. In the problem formulation, integer variables were used to represent the existence of coolers and cooling towers. Also, a detailed modeling based on the Merkel method was used to take into account all important design parameters of wet-cooling towers, such as the cross-sectional area for packing section, tower ﬁll height, type of packing, pressure drop of air through the ﬁll, and so on. The model treats as optimization variables the outlet and inlet temperatures of each cooling tower as well as the outlet temperature for each stage of the cooler network superstructure, and imposes an upper limit on the outlet temperature of each cooler for the cooling water to avoid operational problems due to fouling conditions (return temperature to the cooling tower network). A set of operational and geometric constraints are incorporated in the model.

hsai ¼ 6:3889 þ 0:86582*Twi þ 15:7154expð0:054398*Twi Þ;

i ¼ 1; .4

ðA2Þ

For the mass-fraction humidity of the air stream at the tower inlet (Kröger, 2004):

2501:6 2:3263 ðTWBin Þ 2501:6 þ 1:8577 ðTAin Þ 4:184 ðTWBin Þ ! 0:62509 PVWB;in Pt 1:005 PVWB;in

1:00416 ðTAin TWBin Þ 2501:6 þ 1:8577 ðTAin Þ 4:184 ðTWBin Þ

win ¼

ðA:3Þ

where PVWB,in is calculated with Equation (A.5) and evaluated at T ¼ TWBin. For the mass-fraction humidity of the saturated air stream at the cooling tower exit [30]:

wout ¼

0:62509PVout Pt 1:005PVout

(A.4)

where PVout is the vapor pressure estimated with Equation (A.5) evaluated at T ¼ TAout, and Pt is the total pressure in Pa. Equation (A.5) was proposed by Hyland and Wexler [31] and it is valid in the range of temperature of 273.15e473.15 K,

ln ðPVÞ ¼

3 X

cn T n þ 6:5459673ln ðTÞ

(A.5)

n ¼ 1

where PV is the vapor pressure in Pa, T is the absolute temperature in Kelvin, and the constants have the following values: c1 ¼ 5.8002206 103, c0 ¼ 1.3914993, c1 ¼ 4.8640239 103, c2 ¼ 4.1764768 105 and c3 ¼ 1.4452093 107. For the outlet air temperature, Serna-González et al. [20] proposed:

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

hsaout þ 6:38887667 0:86581791*TAout 15:7153617expð0:05439778*TAout Þ ¼ 0

ðA:6Þ

For the density of the airewater mixture [20]:

r¼

i h Pt w ½1 þ w 1 w þ 0:62198 287:08 T

(A.7)

where Pt and T are expressed in Pa and K, respectively. The density of the inlet and outlet air are calculated from the last equation evaluated in T ¼ TAin and T ¼ TAout for w ¼ win and w ¼ wout, respectively. Nomenclature A bel CAHE ce CFHE CAPCNC CAPTNC CCTF CCTMA CCTV CCTVe CP cl cel cem cw dm dem dtcal dtfri Fa Fav FCP FO ftt Fw fwb Fwb fwc Fwdis fwd fwe Fwh fwm Fwm fwt1 fwt2 fwx g gc h ha hsa HY K KF

heat transfer area, m2 coefﬁcients for the correlation for Me area cost coefﬁcient, US$/m2 unity cost of electricity, US$/kgWatts, US$/Joules ﬁxed cost for exchangers, US$/year capital cost of cooling network, US$/year capital cost of cooling tower network, US$/year initial cost of the cooling tower, US$ cost related to air ﬂow rate, US$ s/kg cost of the cooling tower due to packing volume, US$/m3 disaggregated variables for CCTV heat capacity, J/kg C constants for the correlation for Me, l ¼ 1,2,., 5. disaggregate variable for cl coefﬁcients for Kﬁ for packing type unit cost for cooling medium, US$/kg constants for the correlation for Kﬁ, m ¼ 1,2,., 6 disaggregated variable for dm temperature difference for the hot side in the matches, C temperature difference for the cold side in the matches, C dry air mass ﬂow rate, kg/s arithmetic airevapor ﬂow rate through the ﬁll, kg/s heat capacity ﬂow rate for hot process streams, kW/ C segregated ﬂowrate of the cooling medium on stages, kg/s ﬂowrate between cooling tower, kg/s water ﬂowrate in the cooling towers, kg/s blowdown water ﬂowrate in the cooling towers, kg/s blowdown water ﬂowrate in the cooling system, kg/s cooled water ﬂowrate from the cooling tower to the cooler network, kg/s water ﬂowrate in the end of the cooling towers, kg/s drift water in the cooling tower, kg/s evaporated water in the cooling tower, kg/s ﬂowrate of heated water streams, kg/s make-up water ﬂowrate of the cooling medium on stages, kg/s make-up water ﬂowrate of the cooling medium in the cooling systems, kg/s ﬂowrate of heated water streams in the inlet of the cooling towers, kg/s ﬂowrate of heated water streams in the outlet of the cooling towers, kg/s ﬂowrate of the cooling medium in match, kg/s acceleration due to gravity, m/s2 conversion factor for acceleration due to gravity ﬁlm heat transfer coefﬁcient, kW/m2 C enthalpy of bulk airewater vapor mixture, J/kg dry air enthalpy of saturated airewater vapor mixture, J/kg dry air yearly operating time, hr/year loss coefﬁcient, m1 factor used to annualize the inversion, 1/year

L Me NCYCLES OPCNC OPTNC PC PV PWC q QHP TA TACNC TACS TACTC TCH Tcin Tcout Th THIN THOUT TMPI TMPO TO Twh TWB Tw Twdis U w

973

height, m Merkel number, dimensionless cycle of concentration, dimensionless operational cost of cooling network, US$/year operational cost of cooling tower network, US$/year power consumption, W vapor pressure, Pa power cost of pump, US$/year heat exchanged between hot streams and the cooling medium, kW total heat of hot process stream, kW air dry-bulb temperature, C total annual cost of cooling network, US$/year total annual cost of cooling system, US$/year total annual cost of cooling tower network, US$/year constants of the Chebyshev technique, dimensionless inlet temperature for the cooling medium on stages, C outlet temperature for the cooling medium in the matches, C temperature of hot stream in the matches, C inlet temperature for hot process streams, C outlet temperature for hot process streams, C inlet of the hottest hot process stream, C inlet temperature of the coldest hot process streams, C outlet temperature for the segregated ﬂowrate of cooling medium, C temperature of heated water streams, C air wet-bulb temperature, C water temperature in the cooling towers, C water temperature in the end of the cooling towers, C overall heat-transfer coefﬁcient for the hot process streams, kW/m2 C mass-fraction humidity, kg-water/kg-dry-air

Binary variables binary variable to determine the existence of heat z1i;k exchangers z2ct binary variable to determine the existence of cooling towers ye binary variable for selection of packing type Greek symbols d parameter (for example 1 106) to avoid singularities in the objective function U parameter to ﬁx the upper limits G upper bound for temperature difference of hot process streams Jct any variable for the design of cooling tower Dh local enthalpy difference, J/kg of dry air DTMIN minimum approach temperature difference, C DTML mean-logarithmic difference on matches DP pressure drop, Pa h efﬁciency r density of the airewater mixture, kg/m3 Sets NEF ST HP NCT

{jjj is a cold process stream, j ¼ 1, ., NEF} {kjk is a stage in the superstructure, k ¼ 1, ., NOK} {iji is a hot process stream} {ctjct is a cooling tower, ct ¼ 1, ., NCT}

Scalars NOK

total number of stages in the cooling network

974

E. Rubio-Castro et al. / Applied Thermal Engineering 50 (2013) 957e974

Subscripts av average cu cooling medium dis end of the cooling tower network f fan ﬁ ﬁll fr cross-sectional i hot process stream j cold process stream k index for stage (1, ., NOK) and temperature location (1, ., NOK þ 1) e type of packing, e ¼ 1,2,3 l constants for type of packing, l ¼ 1, ., 5 m constants for type of packing, m ¼ 1, ., 6 in inlet n temperature increment index, n ¼ 1, ., 4 ct cooling tower out outlet p pump t total WB wet-bulb Superscripts max upper limit min lower limit References [1] J.K. Kim, R. Smith, Cooling water system design, Chemical Engineering Science 56 (12) (2010) 3641e3658. [2] J.K. Kim, G.C. Lee, F. Zhu, R. Smith, Cooling system design, Heat Transfer Engineering 23 (6) (2002) 49e61. [3] J.K. Kim, R. Smith, Cooling system design for water and wastewater minimization, Industrial and Engineering Chemistry Research 43 (2) (2004) 608e613. [4] M.H. Panjeshahi, A. Ataei, M. Gharaie, R. Parand, Optimum design of cooling water systems for energy and water conservation, Chemical Engineering Research and Design 87 (2) (2009) 200e209. [5] M. Picón-Nuñez, A. Morales-Fuentes, E.E. Vázquez-Ramírez, Effect of network arrangement on the heat transfer area of cooling networks, Applied Thermal Engineering 27 (16) (2007) 2650e2656. [6] M. Picón-Núñez, G.T. Polley, L. Canizalez-Dávalos, J.M. Medina-Flores, Short cut performance method for the design of ﬂexible cooling systems, Energy 36 (8) (2011) 4646e4653. [7] M.M. Castro, T.W. Song, J.M. Pinto, Minimization of operational costs in cooling water systems, Chemical Engineering Research and Design 78 (2) (2000) 192e201. [8] J.K. Kim, R. Smith, Automated retroﬁt design of cooling-water systems, AIChE Journal 49 (7) (2003) 1712e1730. [9] X. Feng, R.J. Shen, B. Wang, Recirculating cooling-water network with an intermediate cooling-water main, Energy & Fuels 19 (4) (2005) 1723e1728.

[10] J.M. Ponce-Ortega, M. Serna-González, A. Jiménez-Gutiérrez, MINLP synthesis of optimal cooling networks, Chemical Engineering Science 62 (21) (2007) 5728e5735. [11] J.M. Ponce-Ortega, M. Serna-González, A. Jiménez-Gutiérrez, A disjunctive programming model for simultaneous synthesis and detailed design of cooling network, Industrial and Engineering Chemistry Research 48 (6) (2009) 2991e3003. [12] G.F. Cortinovis, J.L. Paiva, T.W. Song, J.M. Pinto, A systematic approach for optimal cooling tower operation, Energy Conversion and Management 50 (9) (2009a) 2200e2209. [13] G.F. Cortinovis, M.T. Ribeiro, J.L. Paiva, T.W. Song, J.M. Pinto, Integrated analysis of cooling water systems: modeling and experimental validation, Applied Thermal Engineering 29 (14e15) (2009b) 3124e3131. [14] J.M. Ponce-Ortega, M. Serna-González, A. Jiménez-Gutiérrez, Optimization model for re-circulating cooling water systems, Computers and Chemical Engineering 34 (2) (2010) 177e195. [15] T. Majozi, A. Moodley, Simultaneous targeting and design for cooling water systems with multiple cooling water supplies, Computers and Chemical Engineering 32 (3) (2008) 540e551. [16] K.V. Gololo, T. Majozi, On synthesis and optimization of cooling water systems with multiple cooling towers, Industrial and Engineering Chemistry Research 50 (7) (2011) 3775e3787. [17] F. Merkel, Verdunstungskuhlung, VDI Zeitchriff Deustscher Ingenieure 70 (1926) 123e128. [18] J.C. Kloppers, D.G. Kröger, Loss coefﬁcient correlation for wet-cooling tower ﬁlls, Applied Thermal Engineering 23 (17) (2003) 2201e2211. [19] J.C. Kloppers, D.G. Kröger, Reﬁnement of the transfer characteristic correlation of wet-cooling tower ﬁlls, Heat Transfer Engineering 26 (4) (2005) 35e41. [20] M. Serna-González, J.M. Ponce-Ortega, A. Jiménez-Gutiérrez, MINLP optimization of mechanical draft counter ﬂow wet-cooling towers, Chemical Engineering Research and Design 88 (5e6) (2010) 614e625. [21] F.N. Kemmer, The NALCO Water Handbook, McGraw-Hill, New York, USA, 1988. [22] A.K.M. Mohiudding, K. Kant, Knowledge base for the systematic design of wet cooling towers. Part I: selection and tower characteristics, International Journal of Refrigeration 19 (1) (1996) 43e51. [23] A. Vecchietti, S. Lee, I.E. Grossmann, Modeling of discrete/continuous optimization problems characterization and formulation of disjunctions and their relaxation, Computers and Chemical Engineering 27 (3) (2003) 433e448. [24] S.A. Leeper, Wet Cooling Towers: Rule-of-thumb Design and Simulation. Report, U.S. Department of Energy, 1981. [25] J.R. Singham, Heat Exchanger Design Handbook, Hemisphere Publishing Corporation, New York, USA, 1983. [26] J.J. Chen, Letter to the editors: comments on improvement on a replacement for the logarithmic mean, Chemical Engineering Science 42 (1987) 2488e2489. [27] M. Kintner-Meyer, A.F. Emery, Cost-optimal design for cooling towers, ASHRAE Journal 37 (4) (1995) 46e55. [28] A. Brooke, D. Kendrick, A. Meeraus, GAMS User’s Guide, The Scientiﬁc Press, USA, 2011. [29] J. Viswanathan, I.E. Grossmann, A combined penalty function and outer approximation method for MINLP optimization, Computers and Chemical Engineering 14 (7) (1990) 769e782. [30] D.G. Kröger, Air-cooled Heat Exchangers and Cooling Towers, PennWell Corp., Tulsa, Oklahoma, 2004. [31] R.W. Hyland, A. Wexler, Formulation for the thermodynamic properties of the saturated phases of H2O from 173.15 K and 473.15 K, ASHRAE Transactions 89 (2A) (1983) 500e519.

Copyright © 2023 C.COEK.INFO. All rights reserved.