MINLP optimization of mechanical draft counter flow wet-cooling towers

chemical engineering research and design 8 8 ( 2 0 1 0 ) 614–625 Contents lists available at ScienceDirect Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd MINLP optimization of mechanical draft counter flow wet-cooling towers Medardo Serna-González a , José M. Ponce-Ortega a,b , Arturo Jiménez-Gutiérrez b,∗ a b Facultad de Ingeniería Química, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Mexico Departamento de Ingeniería Química, Instituto Tecnológico de Celaya, Celaya, Mexico a b s t r a c t In this paper, the problem of the optimal design of mechanical draft counter flow cooling towers that meets a set of specified constraints is formulated as a mixed-integer nonlinear programming (MINLP) problem. The Merkel’s method is used to specify the characteristic dimensions of cooling towers, together with empirical correlations for the loss and overall mass transfer coefficients in the packing region of the tower. Water-to-air mass ratio, water mass flow rate, water inlet and outlet temperatures, operational temperature approach, type of packing, type of draft, height and area of the tower packing, total pressure drop of air flow, power consumption of the fan, and water consumption provide the set of optimization variables. The MINLP problem is formulated so as to minimize the total annual cost of the cooling tower. The performance of the proposed procedure is shown with the solution of six examples. © 2009 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Counter flow cooling towers; Optimization; MINLP; Disjunctive programming 1. Introduction Closed-cycle cooling water systems are widely used to dissipate the low-grade heat of chemical and petrochemical process industries, electric-power generating stations, refrigeration and air conditioning plants. In closed-cycle systems, water is used to cool the hot process streams, then it is cooled by evaporation and direct contact with air in a wet-cooling tower and recycled to the cooling network. The wet or evaporative cooling towers may be classified as natural draft and mechanical draft types (Singham, 1983). In both cases, as shown in Fig. 1, warm water from the cooling network of a plant enters the top of the tower and flows downward over an internal labyrinth-like packing, or fill, while air can flow upward (counter flow) or horizontally (cross-flow). The fill distributes the water flow uniformly and provides a large air–water interface area for the simultaneous heat and mass transfer processes. 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 enthalpy is increased. The cooled water is then collected in a cold water basin below the fill from which it is returned to the cooling network. The blow-down water and the water lost through evaporation and drift are replaced with fresh make-up water. The moist air leaves at the top of the tower. In the mechanical draft cooling tower, air is circulated through the tower by means of electrically driven fans. On the other hand, the natural draft cooling tower uses the natural buoyancy of the warmed air to circulate it through the tower. Mechanical draft towers can be either induced draft (fan located at the top of the tower) or forced draft (fan located at the bottom of the tower). In practice, large natural draft cooling towers are used in power plants for cooling the water supply to the condenser. Mechanical draft cooling towers are preferred for oil refineries and other process industries, as well as for central air-conditioning systems because they cover a wider range of sizes, are generally more compact, give more uniform cooling and have lower water loss than natural draft towers (Mills, 1999; Najjar, 1988; Singham, 1983). Though the fundamentals of mechanical draft counter flow cooling towers are presented in a number of references (see, for example, Foust et al., 1979; Mills, 1999; Singham, 1983; Kloppers and Kröger, 2005b), few studies are available for the optimal economic design of this type of cooling Corresponding author. E-mail address: arturo@iqcelaya.itc.mx (A. Jiménez-Gutiérrez). Received 10 September 2008; Received in revised form 25 September 2009; Accepted 30 September 2009 0263-8762/$ – see front matter © 2009 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2009.09.016 ∗ chemical engineering research and design 8 8 ( 2 0 1 0 ) 614–625 615 Nomenclature w a afi Afr b c c1 –c5 Ccap,CT CCTF CCTMA CCTV CkCT,V Cop ce cj cjk cp cw d d1 –d6 dl dkl ek G ha hsa hd HY K Kf L m mav Me ncycles P Ptot PV Q T TA TAC TCH TMIP TMOP TW TWB v correlation coefficient for the estimation of enthalpy of saturated air, hsa surface area per unit volume (m−1 ) cross-sectional area of the cooling tower (m2 ) correlation coefficient for the estimation of enthalpy of saturated air, hsa correlation coefficient for the estimation of enthalpy of saturated air, hsa correlation coefficients for the estimation of Merkel number, Me installed capital cost of the cooling tower (US$) fixed cooling tower cost (US$) incremental cooling tower cost based on air mass flow rate (US$ s/kg) incremental cooling tower cost based on tower fill volume (US$/m3 ) disaggregated variables for the capital cost coefficients of cooling towers annual operating cost (US$/yr) unit operating cost of electricity (US$/J) variables for the Merkel number calculation disaggregated variables for the Merkel number calculation specific heat at constant pressure (J/kg K) unit operating cost of make-up water (US$/kg) correlation coefficient for the estimation of enthalpy of saturated air, hsa correlation coefficients for the estimation of loss coefficient, Kfi variables used in the calculation of the loss coefficient disaggregated variables for the calculation of the loss coefficient coefficient cost for different fill types mass velocity (kg/m2 s) enthalpy of air (J/kg dry air) enthalpy of saturated air at the local bulk water temperature (J/kg dry air) mass transfer coefficient (kg/m2 s) yearly operating time (s/yr) loss coefficient annualized factor of capital cost (yr−1 ) fill height (m) mass flow rate (kg/s) air-vapor flow rate (kg/s) Merkel number number of cycles of concentration electric power (W) total pressure (Pa) vapor pressure of water (Pa) heat load (W) temperature (K or ◦ C) dry-bulb temperature of the air (◦ C) total annual cost (US$/yr) Chebyshev constant minimum process inlet temperature (◦ C) minimum process outlet temperature (◦ C) water temperature (◦ C) wet-bulb temperature of the air (◦ C) velocity (m/s) yk ymfd ymid mass-fraction humidity of moist air (kg of water/kg of dry air) binary variable for the fill type binary variable for the selection of mechanical forced draft binary variable for the selection of mechanical induced draft Greek symbols h local enthalpy difference (J/kg dry air) P pressure drop (Pa) Tmin minimum allowable temperature difference (◦ C) f fan efficiency  density of air–water vapor mixtures (kg/m3 ) Subscripts a dry air av air-vapor bw blow-down water fi fill fin fan inlet i index to denote the temperature interval, i = 1, . . ., 4 in at tower inlet j index used for the constants in the fill type, j = 1, . . ., 5 k superscript to denote the fill type, k = 1, 2, 3 l index used for the constants in the fill type, l = 1, . . ., 6 m mean n index used in Eq. (40), n = −1, ..., 3 mcl miscellaneous component min minimum misc miscellaneous mw water make-up out at tower outlet t total vp velocity pressure w water wb wet-bulb temperature wbin wet-bulb temperature of the inlet air wd water drift wev water evaporated towers. Given a heat load and other design conditions, Kintner-Meyer and Emery (1994, 1995) presented a method for the optimum sizing of cooling systems, which includes the cost optimal selection of stand-alone cooling tower range and approach. Recently, using the one-dimensional effectiveness-NTU method, Söylemez (2001, 2004) presented thermo-economic and thermo-hydraulic optimization analyses, which yield simple algebraic formulas for estimating iteratively the optimum heat and mass transfer area, as well as the optimum performance point for forced draft counter flow cooling towers. However, these studies give optimum cooling towers under prescribed thermal conditions of the system because some of the design variables are specified, not optimized. Therefore, they may have limited applications for a general problem in which one needs to consider simultaneously the effect of continuous and discrete variables such as 616 chemical engineering research and design 8 8 ( 2 0 1 0 ) 614–625 Fig. 1 – Closed loop cooling tower system. wet-bulb temperature of the ambient air, water-to-air mass ratio, water outlet temperature, water inlet temperature, type of draft, and type of packing. In addition, the design of cooling towers is also subjected to a number of nonlinear design equations and feasibility constraints such as minimum and maximum values of water-to-air mass ratios, water and air mass velocities, and so on. Thus, mixed-integer nonlinear programming (MINLP) techniques are suited for solving this general problem. In this paper, an MINLP formulation is developed for optimizing mechanical draft counter flow cooling towers, which takes into consideration the above mentioned group of independent variables together with a set of constraints imposed on the problem. The objective is to minimize the total annual cost of the tower. Merkel’s method (Merkel, 1926) is used for estimating the size and performance of the tower. This method has been widely applied to design these units and is recommended by international standards (Kloppers and Kröger, 2005a,b). The design of cooling towers also requires the optimal selection of the type of packing and, in this work, the choice is limited to film, splash, and tickle types of fills. The mass transfer and pressure drop characteristics of these types of packing are modeled with the empirical correlations given by Kloppers and Kröger (2003, 2005a). In addition, mechanical cooling towers can be either induced or forced. Disjunctive programming is used to formulate the discrete choices of type of packing and type of draft. For thermodynamic properties, the enthalpy of saturated air–water vapor mixture is calculated as an exponential function of the local bulk water temperature. To develop this correlation, the ASHRAE property table (ASHRAE, 2001) was used as the source of the enthalpy of saturated air data from 8 to 55 ◦ C for standard atmospheric pressure. The fitted equation reproduces the data from ASHRAE with an average absolute deviation of 0.047%. The optimization problem was implemented into the GAMS software environment, using the DICOPT solver. 2. narily entails not only compromises in the detailed design of the cooling tower itself but also in the selection of the design conditions (Fraas, 1989) such as the air mass flow rate, the water outlet and inlet temperatures, as well as the type of packing. Thus, all or any number of these variables can be considered as independent variables to be optimized, and that can be restricted between specified maximum and minimum limits. To include these types of factors, in this paper the problem for designing a cost-optimal counter flow draft cooling tower can be stated as follows: For a given heat load of the cooling circulating-water system that must be dissipated, and knowing the wet- and dry-bulb temperatures of air entering the tower, determine the mechanical draft counter flow cooling tower operating in steady state with the minimum total annual cost (that includes power and annualized investment costs) subject to the following assumptions (Mills, 1999; Singham, 1983; Webb, 1984): 1. The Lewis number relating heat and mass transfer is equal to one. 2. The water mass flow rate through the tower is constant because the loss of water to evaporation, drift, and blowdown is small compared with the total water load. 3. The liquid-side heat transfer resistance is negligible, that is, the temperature of the air at the interface is equal to the local bulk temperature of the water. 4. The air exiting the tower is saturated with water vapor. 5. The cooling tower operates adiabatically. 6. The cross-sectional area of the tower is uniform. 7. Thermodynamic properties are constant across any horizontal section of the tower. 3. Cooling tower model Fig. 2 shows the general arrangement and variables used for the representation of a counter flow cooling tower; the subscripts out and in refer to the outlet and inlet streams Problem definition A typical problem that often arises in cooling tower design is the determination of the required size and cost of a tower when the heat rejection load, the water outlet temperature, the water inlet temperature, the dry- and wet-bulb temperatures of the entering air, the air mass flow rate, and the type of packing are given. However, the design of a cooling tower ordi- Fig. 2 – A differential control volume of a counter flow cooling tower. chemical engineering research and design 8 8 ( 2 0 1 0 ) 614–625 617 conditions. Let TW represent the water temperature, TA the dry-bulb temperature of the air, TWB the wet-bulb temperature of the air, and ha the enthalpy of bulk air–water vapor mixture passing through the packing. Let Lfi represent the tower fill height, Afr the cross-sectional area of the tower, and mw and ma the water and dry air mass flow rates, respectively. Theoretically, the lowest temperature that the water can reach is the wet-bulb temperature of the air entering the wetcooling tower. As a result, cooling towers are able to lower the water temperature more than indirect heat transfer devices that use only ambient air to remove process heat. The temperature difference between the water outlet temperature and the wet-bulb temperature of the air entering the tower is called the tower approach. Approach = TWout − TWBin (1) The designer must select the highest geographic wetbulb temperature for sizing a cooling tower. As the wet-bulb temperature decreases, so does the available water outlet temperature of the tower. Hence, the entering wet-bulb temperature determines the minimum operating temperature level throughout the process that can be achieved by using cooling water as cold utility. Thus, in cooling tower design, the wet-bulb temperature of entering air is a critical factor. The temperature difference between the cooling tower water inlet and outlet temperatures defines the range of the tower. Range = TWin − TWout (2) Heat and mass balances taken over a differential section of the cooling tower fill (Fig. 2) using the assumptions listed above leads to Merkel’s method (1926) for the design of mechanical wet-cooling towers. In the integrated form, Merkel’s equation is presented as (Kloppers and Kröger, 2005a,b): Me =  TWin TWout cpw dTW (hsa − ha) (3a) where the Merkel number, Me, is given by Me = hd afi Afr Lfi mw = hd afi Lfi Gw (3b) In these equations afi is the surface area of the fill per unit volume of fill, hd is the mass transfer coefficient, Afr is the tower fill area, Gw is the mass velocity of water, and hsa is the enthalpy of saturated air–water vapor mixture at the local bulk water temperature. From assumption 1, the potential for heat and mass transfer at any point in a tower is expressed as the local enthalpy difference (hsa − ha) in the right-hand side of Eq. (3a). This is best illustrated in Fig. 3, which shows both operating and equilibrium lines of a counter flow cooling tower in a plot of air enthalpy versus local bulk water temperature. The upper curve represents the enthalpy of saturated air (hsa) as a function of the water temperature. The straight line is the operating line, which results from an overall energy balance on the fluid streams entering and leaving the tower. cpw mw (TWin − TWout ) = ma (haout − hain ) (4) Fig. 3 – Equilibrium and operating lines on enthalpy-temperature diagram. From assumption 2, the reduction of the water mass flow rate due to evaporation is neglected. Thus, the change in enthalpy in the liquid phase is taken as a result of temperature change alone and the cpw mw /ma ratio is the slope of the operating line. It should be noted that the operating line is the locus of corresponding TW and ha points along the length of the tower. This property is useful to derive Eq. (9) below. The integration of the right-hand side of Merkel’s equation requires a knowledge of the local enthalpy difference (hsa − ha) along the air flow path in the tower and design conditions. These are given by the water mass flow rate, the water outlet temperature, the water inlet temperature, the wet-bulb temperature of the entering air, and the cpw mw /ma ratio. As seen in Fig. 3, the vertical distance (dashed lines) between the two curves represents the driving force (hsa − ha) at any point in the tower (assumption 3), which is needed for the integration of the right-hand side of Eq. (3a). The integrated value of Eq. (3a), known as the required Merkel number, is a measure of the degree of difficulty of the tower for meeting the required design conditions. Higher values of Eq. (3a) imply a greater degree of difficulty. It should be noted that the required Merkel number represents the theoretical analysis of the design problem. The required Merkel number can be determined without knowledge of the characteristics of the tower packing. However, the heat and mass transfer characteristics for a particular fill type are also needed in the thermal-hydraulic design of mechanical draft cooling towers. As seen in Eq. (3b), the Merkel number contains both the cross-sectional area and the height of the fill, as well as the surface area of the fill per unit volume of fill, and the overall mass transfer coefficient for a particular type of packing. Hence, the left-hand side of the Merkel’s equation is a measure of the quality and quantity of the packing being used. For each type of packing, there is a fixed type of empirical correlation between the Merkel number and the ratio of water flow rate to air flow rate (Li and Priddy, 1985; Webb, 1984). Typically, this information is reported in the literature or presented by cooling tower manufacturers. Thus, for a given packing and flow rate ratio mw /ma , the left-hand side of Eq. (3a) can be determined without knowledge of the properties of the air and water at the inlet and exit from the cooling tower (Singham, 1983). The resulting value is known as the available Merkel number. For specified design conditions and type of packing, the sizing calculation requires equality of the two sides of the 618 chemical engineering research and design 8 8 ( 2 0 1 0 ) 614–625 Merkel’s equation through a basic search procedure on the water-to-air mass flow ratio mw /ma (Li and Priddy, 1985; Singham, 1983). This method gives the tower size and the value of mw /ma for accomplishing the required cooling. The tower size is indicated mainly by the volume of the fill, which includes both the cross-sectional area and the height of the fill. Review of the literature suggests that this design method has come into common use in the cooling tower industry (Kloppers and Kröger, 2005a; Mills, 1999; Mohiudding and Kant, 1996; Webb, 1984). The method is accurate up to temperatures of about 60 ◦ C; comparison with more exact results are usually within 3–5%, and seldom shows errors greater than 10% (Mills, 1999). 4. MINLP model formulation With the implementation of Merkel’s method, the optimal problem of the mechanical draft counter flow cooling tower is formulated as an MINLP optimization model, with the minimization of the total annualized cost as the objective function. The major variables are the air and water mass flow rates, water inlet and outlet temperatures, wet- and dry-bulb temperatures of the air entering the tower, heat load of the tower, type of packing, type of draft, height and area of the tower packing, total pressure drop due to the air flow rate, power consumption of the fan and water consumption. Thus, the model includes the selection of type of packing and type of draft as the only logical decisions, which are modeled using disjunctive programming. Constraints are implemented from the energy balances, heat load equation, Merkel’s equation, transfer coefficient and loss coefficient correlations for the tower, relationships of thermodynamic properties and pressure drops along the air path, water consumption equation, practical limits for water mass flow rates, and operational limits and logical constraints. 4.1. Heat load The heat load Q is the amount of heat transferred from the water to the air stream, and is determined from the range of the tower and the incoming water mass flow rate: Q = cpw mw (TWin − TWout ) (5) The heat load must meet the required heat rejection of the cooling network of the process. Thus, the heat load imposed on a cooling tower is determined by the process being served. 4.2. Required Merkel’s number To formulate the model as an MINLP problem, the cooling tower model in Eq. (3a) is transformed into an algebraic equation using the four-point Chebyshev integration technique, as recommended by the British Standard (1988) and the Cooling Tower Institute (1990, 1997). Application of this numerical method requires four water temperature increments for the determination of the integral, as shown in Figure 8 of Mohiudding and Kant (1996). Kloppers and Kröger (2005c) found that the Chebyshev procedure is generally very accurate when compared to the composite Simpson rule with 100 intervals, which has a fourth-order error. In algebraic form, the four-point Chebyshev procedure applied to Merkel’s equation gives (Mohiudding and Kant, 1996) Me =  TWin TWout 4  cpw dTW 1/hi = 0.25cpw (TWin − TWout ) (hsa − ha) (6) i=1 where i is the temperature-increment index. For each temperature increment, the local enthalpy difference (hi ) is calculated from hi = hsai − hai , i = 1, . . . , 4 (7) and the algebraic equations to calculate the water temperature and air enthalpies corresponding to each Chebyshev point are given by TWi = TWout + TCHi (TWin − TWout ), hai = hain + cpw mw (TWin − TWout ), ma i = 1, ..., 4 (8) i = 1, ..., 4 (9) hsai = −6.38887667 + 0.86581791 × TWi + 15.7153617exp (0.0543977 × TWi ), i = 1, ...4 (10) In Eq. (8) TCHi is a constant that represents the four-points of the Chebyshev technique (TCH1 = 0.1, TCH2 = 0.4, TCH3 = 0.6, TCH4 = 0.9). The enthalpy of bulk air–water vapor mixture is related to the local water temperature at each temperature-increment through Eq. (9), which indicates that the heat removed from the water being cooled is transferred to the air at any point within the packing region. As shown by Eq. (9), the air enthalpy follows a straight line that corresponds to the operating line for the tower given by Eq. (4); such equations explain the two forms that are implemented for the energy balance. Thus, when Chebyshev technique is used, the points of the operating line are set by Eqs. (8) and (9) together with both end points or one end point and the slope cpw mw /ma . As seen in Fig. 3, the enthalpy of a saturated air–water vapor mixture is a nonlinear function of the local bulk water temperature. This relation can be expressed empirically by the exponential function given by Eq. (10). The values of the constants in this correlation were obtained as shown in Appendix A. It was found that the average absolute deviation of this correlation is 0.047% over the water temperature range of 8–55 ◦ C for standard atmospheric pressure. Thus, enthalpy of saturated air–water vapor mixture at water temperature corresponding to each temperature-increment can be obtained with suitable accuracy from Eq. (10). 4.3. Packing performance: transfer and loss coefficients The heat and mass transfer characteristics of wet-cooling tower fills are a function of the surface area of the fill per unit volume of fill, afi , and the overall mass transfer coefficient, hd . These values must be known in order to determine the cross-sectional area and the height of the fill. However, it is often difficult to theoretically predict or empirically obtain afi because the two-phase flow in cooling tower fills is quite complex. In fact, it is not necessary to do so. Eq. (3b) clearly shows that the product of hd and afi is all that is needed to define the available Merkel number, in addition to the water mass flow 619 chemical engineering research and design 8 8 ( 2 0 1 0 ) 614–625 rate and the cross-sectional area and height of the tower fill. Thus, for a particular fill type, the available Merkel number is the quantity that must be determined to find the tower size. Usually this information is derived from test data on specific packing geometries and supplied for manufacturers of cooling towers. However, because manufacturers treat this information as proprietary, the published literature does not contain heat and mass transfer data for most of the commercially used fills (Li and Priddy, 1985; Webb, 1984). In this paper, the selection of the type of packing is limited to the splash, trickle, and film type fills tested by Kloppers and Kröger (2005b) for counter flow wet-cooling towers. The measured transfer coefficients are correlated in terms of the available Merkel number for a particular fill type as, Table 1 – Coefficients in correlation for available Me (Eq. (11)). cjk j k = 1 (splash fill) 1 2 3 4 5 0.249013 −0.464089 0.653578 0 0 Me = c1 mw Afr  c2  ma Afr  c3 (Lfi ) 1+c4 (TWin ) c5 (11) This empirical correlation is expressed as a function of the fill air and water inlet mass flow rates, the water inlet temperature, the fill height, Lfi , and the cross-sectional area of the cooling tower, Afr . Transfer correlations given in the literature for wet-cooling tower fills typically depend only on the air and water mass flow rates (Kloppers and Kröger, 2005b; Li and Priddy, 1985; Webb, 1984). Thus, compared with the existing traditional form of the Merkel number correlations, the new correlation provides a much better fit to experimental data (Kloppers and Kröger, 2005b). To implement the discrete choice for the type of packing, a Boolean variable Yk is used as part of the following disjunction, ⎡ ⎤ Y1 ⎢ (splash fill) ⎣ cj = cj1 , j = 1, ..., 5 ⎡ Y2 ⎥ ⎢ (trickle fill) ⎦∨⎣ cj = cj2 , ∨ ⎡ j = 1, ..., 5 ⎢ (film fill) ⎣ cj = cj3 , ⎥ ⎦ ⎤ Y3 j = 1, ..., 5 ⎥ ⎦ Yk is true if the packing is selected, and false otherwise. The above disjunction is transformed using the convex hull reformulation (Vecchietti et al., 2003) as follows: 2 3 y +y +y =1 cj = cj1 + cj2 + cj3 , cjk = akj yk , (12) j = 1, ..., 5 k = 1, ..., 3. j = 1, ..., 5 (13) 1.930306 −0.568230 0.641400 −0.352377 −0.178670 1.019766 −0.432896 0.782744 −0.292870 0 mw Afr d2  ma Afr d3 + d4  mw Afr d5  ma Afr  d6 Lfi (15) The corresponding disjunction is given by, ⎡ ⎤ Y1 ⎢ ⎣ (splash fill) dl = d1l , l = 1, ..., 6 ⎡ Y2 ⎥ ⎢ ⎦ ∨ ⎣ (trickle fill) dl = d2l , ∨ ⎡ l = 1, ..., 6 ⎢ ⎣ (film fill) ⎤ ⎥ ⎦ ⎤ Y3 dl = d3l , l = 1, ..., 6 ⎥ ⎦ Using the convex hull reformulation, one can express the disjunction as dl = d1l + d2l + d3l , l = 1, ..., 6 k = 1, ..., 3. (16) l = 1, ..., 6 (17) Values for the coefficients d1 to d6 for the three counter flow tower fills are given in Table 2 (Kloppers and Kröger, 2003). It must be pointed out that the processing technique of packing performance data is based on the Merkel method (Kloppers and Kröger, 2003, 2005b). For consistency, in this paper the same method is used to obtain empirically determined transfer characteristic correlation for a given fill and tower design, although it should be noted that other forms of empirical correlations for transfer and loss coefficients can be inserted into the MINLP model. 4.4. 1  dkl = bkl yk , ⎤ k = 3 (film fill) form (Kloppers and Kröger, 2003) Kfi = d1  k = 2 (trickle fill) Tower pressure drop In mechanical draft cooling towers, the total pressure drop along the air path, Pt , is the sum of the static pressure drop and the velocity pressure (Li and Priddy, 1985). The static pressure drop includes the pressure drop through the fill, Pfi , (14) Table 2 – Coefficients in correlation for Kfi (Eq. (15)). where yk is the set of binary variables associated with the type of packing. Values for the coefficients c1 , c2 , c3 , c4 , and c5 for each type of packing are given in Table 1 (Kloppers and Kröger, 2005b). The variables akj are used for modeling purposes. The tower fill performance is described not only by the transfer coefficient, but also by the loss coefficient per meter depth of fill, Kfi , that is required to calculate the pressure drop through the fill. For the splash, trickle, and film type fills, the loss coefficient correlations can be expressed in the following dkl l k = 1 (splash fill) 1 2 3 4 5 6 3.179688 1.083916 −1.965418 0.639088 0.684936 0.642767 k = 2 (trickle fill) 7.047319 0.812454 −1.143846 2.677231 0.294827 1.018498 k = 3 (film fill) 3.897830 0.777271 −2.114727 15.327472 0.215975 0.079696 620 chemical engineering research and design 8 8 ( 2 0 1 0 ) 614–625 and the sum of miscellaneous pressure losses, Pmisc , such as those due to drift eliminators, air inlet, and water distribution piping. The pressure drop through the fill matrix is coupled to the loss coefficient by the following relationship (Kloppers and Kröger, 2003) Pfi = Kfi Lfi mav2m (18) 2m A2fr where mavm is the arithmetic mean air-vapor flow rate through the fill mavin + mavout mavm = 2 (19) and m is the harmonic mean density of the moist air through the fill m = 1/ 1 in + 1 out  (20) To calculate the density of the inlet and outlet air in and out , Eq. (41) is used. The air–vapor flow at the fill inlet and outlet mavin and mavout are calculated as follows mavin = ma + win ma (21) mavout = ma + wout ma (22) where win is the humidity (mass fraction) of the inlet air, and wout is the humidity of the outlet air. These properties are calculated from Eqs. (36) and (39). As pointed out by Mills (1999), the miscellaneous pressure losses in components such as drift eliminators, air inlet, water distribution piping, column supports, etc., are calculated with a velocity head equation: Pmcl = Kmcl Pvp = (2/3)(Pfi + Pmisc ) (24) By adding Eqs. (18), (23), and (24), one can express the total pressure drop of the air stream for counter flow cooling towers as Pt = 1.667(Pfi + Pmisc ) 4.5. (25) Power consumption The required power for a cooling tower fan can be determined by multiplying the total pressure drop times the air volume flow rate at the fan location. Hence, the power consumption is calculated in terms of the air volume flow rate at the packing inlet for forced draft towers, or at the packing outlet for induced draft towers. The following disjunction is written for the discrete decision associated with the optimal selection of draft type: ⎡ mfd ⎤ ⎡ YCT mid YCT ⎤ ⎢ (mechanical forced draft) ⎥ ⎢ (mechanical induced draft) ⎥ ⎥∨⎢ ⎢ ⎥ ⎦ ⎣ ⎣ ⎦ mavout Pt mav P P= in t in f P= out f The disjunction can be modeled through the following set of equations: P≤ mavin Pt mfd + MPmax (1 − yCT ) in f (26) P≥ mavin Pt mfd − MPmax (1 − yCT ) in f (27) P≤ P≥ mcl v2mcl   mavout Pt + MPmax 1 − ymid CT out f mavout Pt − MPmax (1 − ymid CT ) out f (28) (29) 2 where Kmcl is the component-loss coefficient, and vmcl and mcl is the air–vapor velocity and density, respectively. Typical values for Kmcl are given by Li and Priddy (1985) and Mills (1999). The equation for the total miscellaneous pressure losses can be expressed in terms of a miscellaneous loss coefficient, Kmisc , which can be written as Kmisc = Kdrift + Kair inlet + Ksupports + · · ·. For mechanical draft cooling towers, an estimated value of 6.5 can be used for Kmisc . This estimate is based on the nominal values of the loss coefficients Kmcl used in the computer program CTOWER (Mills, 1999). Thus, the total miscellaneous pressure losses can be represented by Pmisc calculate the effective velocity pressure, m v2m mav2m = 6.5 = 6.5 2 2m A2 mfd yCT + ymid CT = 1 where MPmax is the variable from the big-M formulation, and mfd (23) where the mean velocity of air–vapor flow vm has been replaced by mavm /(m Afr ). This approach is only an approximation, and it should be used only in the absence of data provided by manufacturers for loss coefficients Kmcl . Another source for the total pressure drop in cooling towers is the velocity pressure, Pvp . According to Li and Priddy (1985), the velocity pressure may be as much as 2/3 of the total static pressure drop. In this work, such an upper value is used to mfd yCT and ymid CT are binary variables. The binary variable yCT reflects the selection of a mechanical forced draft cooling tower, whereas the binary variable ymid CT denotes the selection of a mechanical induced draft cooling tower. The fan mechanical efficiency, f , is generally obtained from manufacturer’s data. In absence of such an information, the fan efficiency can be approximated as 75% (Li and Priddy, 1985). 4.6. fr (30) Water consumption Make-up water (mmw ) is constantly added to the cooling tower basin to compensate for the loss of water due to evaporation, drift, and blowdown. Drift (mwd ) accounts for water losses in the form of suspended droplets entrained in the exit air stream. To minimize this type of loss, eliminators are usually fitted at the packing exit to intercept the larger droplets (Singham, 1983). As water evaporates, dissolved solids and other impurities concentrate in the cooling water. Blowdown (mbw ) is the portion of the circulating water that is removed from the system to avoid the excess of dissolved 621 chemical engineering research and design 8 8 ( 2 0 1 0 ) 614–625 solids/impurities contained in the water, such that deposits do not form on the surfaces within the tower (Fraas, 1989). From the conservation of mass, the rate of water evaporated (mwev ) into the air stream is calculated from the dry air flow rate and the air inlet and outlet water vapor content, mwev = ma (wout − win ) (31) Since the evaporation water is essentially free of dissolved solids and impurities, any such species introduced in the make-up water must be equal to those in blowdown and drift losses. Thus, the amount of blowdown can be expressed as mmw − mwd ncycles mbw = (32) where ncycles is the number of cycles of concentration required to limit scale formation in cooling equipment. Cycles of concentration are the ratio of the solid concentration in the circulating water to the solid concentration in the make-up water. Usually, this ratio has an average value between 2 and 4 (Li and Priddy, 1985). In a well-designed tower, the drift loss should be no more than 0.2 percent of the total circulating water (Kemmer, 1988) mwd = 0.002mw (33) To maintain a steady circulating water flow, the makeup water must equal the sum of the water losses, that is mmw = mmev + mbw + mwd . By combining this balance with equations (31) to (33), one can estimate the water consumption as, ncycles mwev mmw = (34) ncycles − 1 4.7. Physical properties For a total pressure of 1 atm, the enthalpy of the air entering the tower, hain , is equal to the enthalpy of saturated air at the inlet air wet-bulb temperature, TWBin (Singham, 1983). Thus, hain can be calculated from the following correlation (derived in Appendix A) evaluated at TWBin hain = −6.38887667 + 0.86581791 × TWBin + 15.7153617 exp(0.05439778 × TWBin ) (35) can be taken as saturated (assumption 4), and the saturated enthalpy, hsaout , can be found from an overall energy balance. Thus, Eq. (4) can be rearranged to give hsaout = hain + win = −   2501.6 − 2.3263(TWBin ) 2506 + 1.8577(TAin ) − 4.184(TWBin ) 1.00416(TAin − TWBin ) 2506 + 1.8577(TAin ) − 4.184(TWBin )    0.62509(PVwbin ) Ptot − 1.005(PVwbin )  (36) where PVwbin is the vapor pressure of water from Eq. (40) evaluated at the wet-bulb temperature of the inlet air, T = TWBin . The values of hain and win are then dependent only on the site conditions given by the problem specifications. Merkel’s method does not give the state of the moist air leaving the fill; however, in practical situations, the outlet air (37) Assumption 4 allows the air temperature leaving the fill to be calculated from hsaout using Eq. (A1). However, this equation is expressed in terms of temperatures instead of enthalpies. Hence, the outlet air temperature can be represented by the following functional relationship TAout = f1 (hsaout ) (38) The mass-fraction humidity of the saturated air stream at the tower exit (wout ) is given by (Kröger, 2004) wout = 0.62509 PVout Ptot − 1.05 PVout (39) where PVout is the vapor pressure of water from Eq. (40), evaluated at T = TAout , and Ptot is the total pressure of the ambient moist air in Pa. The vapor pressure of water corresponding to a specified temperature is calculated from the correlation given by Hyland and Wexler (1983), which is valid over a temperature range of 273.15–473.15 K, ln(PV) = 3  cn T n + 6.5459673 ln(T) (40) n=−1 where PV is the vapor pressure expressed in Pa, T is the absolute temperature in Kelvin, and the constants have the following values: c−1 = 5.8002206 × 103 , c0 = 1.3914993, c1 = −4.8640239 × 10−3 , c2 = 4.1764768 × 10−5 and c3 = −1.4452093 × 10−7 . The density of air–water vapor mixtures can be calculated from the ideal gas law as = w Ptot 1− 287.08 T w + 0.62198   [1 + w] (41) where Ptot is in Pa and T in Kelvin. The density of the inlet (outlet) air, in (out ), is obtained from this equation using T = TAin (T = TAout ) and w = win (w = wout ). 4.8. The mass-fraction humidity of the air stream at the tower inlet, win , is a function of the inlet air dry- and wet-bulb temperatures, as given by (Kröger, 2004) cpw mw (TWin − TWout ) ma Feasibility constraints In this section, inequality constraints are written to define the feasible region for the optimization problem, as well as feasible operating conditions for an optimal cooling tower. The wet-bulb temperature of the air entering the cooling tower is the lowest temperature at which the water can be cooled. In practice, however, the water outlet temperature should be at least 2.8 ◦ C above the site’s wet-bulb temperature (Li and Priddy, 1985), which sets a minimum temperature approach, TWout − TWBin ≥ 2.8 (42) From a thermodynamic viewpoint, the water outlet temperature of the cooling tower must be lower than the outlet temperature of the coldest hot process stream, TMPO. This 622 chemical engineering research and design 8 8 ( 2 0 1 0 ) 614–625 constraint can be expressed as TWout ≤ TMPO − Tmin (43) where Tmin is the minimum allowable temperature difference. It should be noted that TMPO must be greater than TWBin . Two temperature constraints limit the cooling water inlet temperature for the cooling tower. First, it must not exceed the inlet temperature of the hottest hot process stream, TMPI, at the cooling network of the process: TWin ≤ TMPI − Tmin (44) Since the process side temperatures are fixed, both TMPO and TMPI are known. Second, an upper limit on the cooling water inlet temperature is specified to avoid fouling, scaling and corrosion. Usually, this constraint limits the maximum cooling water inlet temperature to 50 ◦ C (Douglas, 1988): o TWin ≤ 50 C (45) The final set of temperature constraints arise from the fact that the water stream must be cooled and the air stream heated, Table 3 – Cost coefficients CkCTV for splash, trickle and film fills. ek k = 1 (splash fill) k = 2 (trickle fill) 2006.6 k = 3 (film fill) 1812.25 1606.15 Finally, the constraints below define nonnegativity of mass flow rates: mw > 0 (52) ma > 0 (53) 4.9. Objective function The objective function involves the minimization of the total annual cost, TAC, which is the sum of the annualized capital cost of the cooling tower and the annual operating costs Cop : (54) minimize TAC = Kf Ccap,CT + Cop where Kf is the annualized factor for investment, and Ccap,CT is the installed capital cost of the cooling tower. The annual operating costs are determined by the make-up water consumed and the fan power, (55) Cop = HY cw mmw + HY ce P TWin > TWout TAout > TAin (46) (47) Because of these inequalities, constraints are not needed to specify a monotonic increase of local bulk water temperature at each successive Chebyshev point of the tower operating line. The cooling tower must operate at some cpw mw /ma ratio to prevent the operating line from pinching the equilibrium curve. When a maximum cpw mw /ma ratio occurs, the driving force becomes zero and the Merkel number infinite. To avoid this limiting condition, the local driving force (hsa − ha) must be always positive at any point in the tower, hsai − hai > 0 i = 1, ..., 4 (48) A cooling tower can be designed to operate at any point within maximum and minimum cpw mw /ma ratios. However, most applications restrict the design of cooling towers to even narrower limits given by (Singham, 1983) mw 0.5 ≤ ≤ 2.5 ma (49) The maximum and minimum water and air loads are determined by the range of test data used to develop the correlations for the loss and overall mass transfer coefficients for the fills. The constraints are (Kloppers and Kröger, 2003, 2005b), 2.90 ≤ mw ≤ 5.96 Afr (50) 1.20 ≤ ma ≤ 4.25 Afr (51) Here, cw is the unit cost of make-up water, ce is the unit cost of electricity, and HY is the yearly operating time. The formula for the installed capital cost of cooling towers was taken from Kintner-Meyer and Emery (1995): (56) Ccap,CT = CCTF + CCTV Afr Lfi + CCTMA ma where CCTF is the fixed cooling tower cost, CCTV is the incremental cooling tower cost based on tower fill volume, and CCTMA is the incremental cooling tower cost based on air mass flow rate. The cost coefficient CCTV depends on the type of packing and is represented by the following disjunction: ⎡ ⎤ ⎡ Y1 Y2 ⎢ ⎥ ⎢ ⎣ (splash fill) ⎦ ∨ ⎣ (trickle fill) CCT,V = C1CT,V CCT,V = C2CT,V ⎤ ⎡ Y3 ⎥ ⎢ ⎦ ∨ ⎣ (film fill) CCT,V = C3CT,V ⎤ ⎥ ⎦ This disjunction can be implemented with the following equations: CCT,V = C1CT,V + C2CT,V + C3CT,V (57) CkCT,V = ek yk , (58) k = 1, ..., 3 The parameters CkCTV are shown in Table 3. 5. Some issues about the model solution The software DICOPT included in the GAMS optimization package (Brooke et al., 2006) was used to solve the proposed MINLP model given by Eqs. (5)–(58). The algorithm used within the DICOPT solver is the outer approximation with augmented penalty method by Viswanathan and Grossmann (1990). DICOPT yields optimal solutions for convex MINLP models; however, for non-convex problems (such as the one 623 chemical engineering research and design 8 8 ( 2 0 1 0 ) 614–625 Table 4 – Design specifications and process constraints for the examples. Examples Q (kW) TAin (◦ C) TWBin (◦ C) TMPI (◦ C) TMPO (◦ C) Tmin (◦ C) 1 2 3 4 5 6 3400 22 12 65 30 10 3400 17 12 65 30 10 3400 22 7 65 30 10 3400 22 12 55 30 10 3400 22 12 65 25 10 3400 22 12 65 30 5 presented in this paper) no global optimal solution can be guaranteed using this algorithm. Local optimum solutions depend on the starting point of the optimization procedure, and one aims in these cases to obtain a solution that is either close or equal to the global optimum. To help the numerical solution of the model here presented, almost all of the disjunctions were formulated with linear relationships using the convex hull reformulation, which avoids non-convex mixed-integer relationships. The only disjunction that could not be written with linear relationships was the one regarding the power consumption; in this case, the big-M technique was used. From the set of disjunctions, this is the only one that may affect the quality of the optimal solution because it involves nonlinear terms of continuous variables together with linear terms of binary variables multiplied by the big-M parameter. The value of the big-M parameter (i.e., Mpmax ) is a main problem for this formulation since it affects the relaxation (or new search space) for the model. In this case, the value of such parameter was simply chosen as the maximum allowed value for the left-hand side of constraints (26)–(29). Another important issue in the modeling task for this problem was to avoid divisions by zero and infeasibilities in the logarithmic terms. This was prevented by adding feasible lower limits higher than zero for variables that appear in the denominator of some equations, and including constraints to avoid values lower than one for the terms that appear in the arguments of logarithms. In addition, constraints given by Eqs. (42)–(53) provide limits that avoid potential infeasibilities during the search process. Given these aspects, a systematic initialization procedure for the model solution was not a crucial factor. What was used in this work was a simple initial set of reasonable values, given the physical nature of the system. Then, several other values were tried trying to improve the local solution to obtain another one closer to the global optimum one for this non-convex model. 6. Numerical examples In Table 4, the design specifications and process constraints for six examples are presented, with Example 1 taken as a base case. In the other examples, one input variable or design constraint was changed while the other conditions were kept at their base values. For all examples, the cooling tower was required to remove 3400 kW at an ambient air pressure of 101,325 Pa. The cpw for the water was taken as 4.187 kJ/kg ◦ C, the HY parameter was assumed as 2.934 × 107 s/year, and the annualizing factor for the capital cost as 0.2983/year. The values of cw , ce , CCTF , CCTMA , f and ncycles , were taken as US$5.283 × 10−4 /kg, US$0.085/kWh, US$31,185, US$1097.5/(kg of dry air/s), 0.75 and 4, respectively. The results for the six examples are presented in Table 5. For each case, the selected type of filling material is the film packing that offers the best combination of heat transfer and pressure losses, so that the lowest total annual cost is obtained. For all examples, the force draft type was selected. Water and air loadings fall within the ranges given by constraints (49) and (50). It should be noted that the total annual cost is dominated by the operating costs for all examples, and in general the cost of make-up water was greater than the cost of electric power required for the fan performance. The effect of changing the inlet air dry-bulb temperature, TAin , on the total annual cost can be seen in Tables 4 and 5 (Examples 1 and 2). As TAin is decreased from 22 to 17 ◦ C, the optimum total annual cost shows a 2.21% decrease, from US$66,065.14/yr to US$64,604.64/yr. Thus, the total annual cost is fairly insensitive to the inlet air dry-bulb temperature for the range of conditions under consideration. On the other hand, changing the tower approach temperature has a significant effect on the optimum value of the total annual cost. Reducing the approach temperature (Example 5) or increasing the approach temperature (Examples 3 and 6) noticeably affects the total annual cost of the system. For instance, as shown in the results of Table 5, if we take the optimum solution of Example 1 as a reference and decrease the approach temperature from 8 to 3 (Example 5) the total annualized cost increases by 77.52%. In general, as the approach temperature is reduced, the tower size increases exponentially (Li and Priddy, 1985). This happens because the driving forces become more limiting as the tower approach decreases (see Fig. 3), so that a larger Me or tower size is required to process the same heat load. Examples 3 and 6 show accordingly that as the tower approach temperature increases the total annual cost decreases significantly. Thus, it is clear that the tower approach temperature is an important optimization variable. Examples 3 and 6 also show that the optimal tower approach increases when TWBin and Tmin decrease. On the other hand, Example 5 shows that as TMPO decreases the optimal tower approach also decreases. Since TWBin is sitespecific and TMPO is limited by the temperature conditions of the process, these variables are taken as input parameters. Thus, only the value for Tmin could be subject to optimization; such an optimization task should take into account the interaction between the tower performance and the cooling network performance. The effect of the cooling range on the cooling tower cost is illustrated in Examples 3 and 6 for a constant tower approach and inlet water temperature. It was found that a reduced cooling range results in a higher water outlet temperature, with a corresponding increase in cooling driving forces. Thus, as the cooling range is decreased, the tower size (or Merkel number), and therefore the tower cost, becomes smaller. In addition, Examples 2 and 4 (or 1 and 4) show the effect changing the cooling range for a fixed tower approach and a fixed water outlet temperature. It can be observed that as 624 chemical engineering research and design 8 8 ( 2 0 1 0 ) 614–625 Table 5 – Optimization results for the examples. Example mw (kg/s) ma (kg/s) mw /ma mmw (kg/s) mwev (kg/s) mbw (kg/s) mwd (kg/s) TWin (◦ C) TWout (◦ C) TAout (◦ C) Range (◦ C) Approach (◦ C) Lfi (m) Afr (m2 ) Kfi Pfi (Pa) Pmisc (Pa) Pt (Pa) P (hp) Type of packing Type of draft Me Cmw Cpower Cop KCcap,CT TAC 1 2 3 4 5 25.720 31.014 0.829 1.541 1.156 0.334 0.051 50 20 37.077 30 8 2.294 8.869 21.946 280.331 36.189 527.640 24.637 Film Forced 3.083 23885.109 12737.595 36622.703 29442.436 66065.139 25.794 31.443 0.820 1.456 1.092 0.312 0.052 50 20 36.871 30 8 2.239 8.894 21.950 277.727 36.740 524.216 24.474 Film Forced 3.055 22566.366 12653.677 35220.043 29384.597 64604.640 25.700 28.199 0.911 1.564 1.173 0.340 0.051 50 20 36.998 30 13 1.858 8.862 21.926 186.621 29.782 360.744 15.205 Film Forced 2.466 24239.785 7861.037 32100.822 26615.995 58716.817 30.973 36.950 0.838 1.547 1.160 0.325 0.062 45 20 34.511 25 8 2.154 10.680 21.942 254.540 35.011 482.683 26.852 Film Forced 2.923 23983.449 13882.754 37866.203 32667.705 70533.909 22.127 32.428 0.682 1.542 1.157 0.341 0.044 50 15 36.411 35 3 6.299 7.630 22.066 1139.529 53.288 1988.425 97.077 Film Forced 7.335 23901.657 50190.495 74092.153 43186.526 117278.68 the cooling range increases, the inlet water temperature also increases. This result decreases the water and air mass flow rates, which gives lower operating and capital costs. It should be noted that the best optimal solution (Example 6) has a range and an approach temperature of 25 and 13 ◦ C, respectively. This result is in agreement with the above discussion. The operating and capital costs for this case are US$29,425.752/yr and US$25,030.274/yr, which reflects a reduction of 14.98% for the capital cost and 19.65% for the operating cost when compared to the base case. For all examples, the optimal water outlet and inlet temperatures correspond to the upper bounds on TWout and TWin given by constraints (43) and (44) or (45), since such values provide higher driving forces for the cooling operation. We have also found these results for other optimization problems of stand-alone cooling towers that we have solved. Based on these results, it is possible to develop a simpler procedure for the optimal design of individual mechanical draft counter flow cooling towers that avoids the optimization of TWout and TWin . Since the process side temperatures TMPI, TMPO and Tmin are fixed and the air inlet temperatures are determined by site conditions, the simplified procedure begins with the calculation of TWout and TWin using their upper bounds given by constraints (43) and (44) or (45). The upper bound of TWin is given by constraint (44) if TMPI − Tmin is less than 50 ◦ C. Otherwise, the upper bound of TWin is set at a temperature of 50 ◦ C. Then, the cooling approach and range are determined from Eqs. (1) and (2), respectively, and the water flow rate is determined by the heat load and tower range from Eq. (5). This implementation would simplify the MINLP model. The computations were performed on a PC with an Intel Celeron 650-MHz processor and 256 MB RAM. The solution took eight iterations (number of MILP problems) for all examples, and required between 0.015 and 0.062 CPU time (s), which 6 30.749 27.205 1.130 1.540 1.155 0.323 0.061 50 25 39.083 25 13 1.480 9.296 22.639 131.908 25.596 262.560 10.754 Film Forced 1.858 23865.877 5559.875 29425.752 25030.274 54456.026 shows the computational efficiency of the proposed formulation. A remark on the effect of different initializations for the optimization variables is in place. It was found in all the examples here presented that changing the starting points yielded the same optimal solutions, with no convergence problems observed for any of the optimization runs. 7. Conclusions An MINLP formulation has been presented for the optimal design of mechanical counter flow cooling towers. The Merkel method was used for sizing cooling towers because of its ample industrial application (Kloppers and Kröger, 2005a; Mohiudding and Kant, 1996). This method relates the specified cooling requirement to the heat transfer performance of a given packing material (Singham, 1983). The required Merkel number was calculated using the Chebyshev integration technique. Film, splash, and trickle-type fills were included, and the MINLP model incorporated empirical correlations for calculating the available Merkel number and the air pressure drop depending on the type of packing. Also, the model considered induced or forced mechanical cooling towers. These design decisions were represented by disjunctive programming models. For a given heat load and inlet air conditions, the model yields the optimal fill height and cross-sectional area of the tower, and the optimal operating values of water and air mass flow rates, water consumption, power consumption, water outlet and inlet temperature, as well as the optimal type of packing and type of draft. It has been shown that the approach temperature is a critical parameter for the optimal design of cooling towers. Finally, we can note that the wet-cooling tower model here presented could be viewed as one of the components for chemical engineering research and design 8 8 ( 2 0 1 0 ) 614–625 an integrated cooling system, which would include a cooling network, a water pumping system, and a mechanical draft wet-cooling tower. Models for cooling networks have been reported (see for instance Ponce-Ortega et al., 2007); those types of models and the one here developed could be extended and incorporated into a global framework aiming to optimize an integrated cooling system. Acknowledgment The authors gratefully acknowledge the Council of Scientific Investigation of the UMSNH, México for partial financial support to develop this work (Grant 20.1). Appendix A. Correlation equation for the temperature dependence of hsa To determine the required Merkel number it is necessary to evaluate the local value of (hsa − ha) along the air flow path. Therefore, calculations of the enthalpy of saturated air–water vapor mixtures, hsa, are needed. For optimization models of cooling towers, calculations of hsa should be fast and accurate. For this application, the following four-parameter correlation was developed, hsa = a + bT + c exp(dT) (A1) For the estimation of a, b, c, and d in Eq. (A1), minimization of the average absolute deviation (AAD) was used, AAD =  1   DEVf  (%) N (A2) f where N is the number of data points, and the relative deviation (DEV) is defined as DEV = hsacalc − hsatbl × 100(%) hsatbl (A3) where the subscript ‘calc’ and ‘tbl’ denote calculated value and table value, respectively. A data set consisting of 48 enthalpies of saturated air for a temperature range of 8–55 ◦ C for standard atmospheric pressure was used. The data were taken from the Thermodynamic Properties of Moist Air table published by the American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE, 2001). The nonlinear least-squares curve fitting was performed with the Microsoft Excel solver. The resulting values for the constants were: a = −6.38887667, b = 0.86581791, c = 15.7153617 and d = 0.05439778. The correlation reproduces the data from ASHRAE property table with 0.047% of average absolute deviation. Thus, Eq. (A1) provides a suitable correlation for the estimation of enthalpies for saturated air–water vapor mixtures. References ASHRAE., (2001). ASHRAE Handbook—Fundamentals. (American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc, USA). British Standard 4485., (1988). Water Cooling Towers. Part 2: Methods for Performance Testing. Brooke, A., Kendrick, D. and Meeraus, A., (2006). GAMS User’s Guide. 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