Brazilian Journal of Chemical Engineering
Print version ISSN 01046632Online version ISSN 16784383
Braz. J. Chem. Eng. vol.17 n.47 São Paulo Dec. 2000
http://dx.doi.org/10.1590/S010466322000000400029
OPERATIONAL COST MINIMIZATION IN COOLING WATER SYSTEMS
M.M.Castro, T.W.Song and J.M. Pinto*
Department of Chemical Engineering, University of São Paulo, Av. Prof.
Luciano Gualberto, trav.3, n^{o}. 380, 05508900, São Paulo  SP, Brazil
(Received: December 12, 1999 ; Accepted: April 6, 2000)
Abstract  In this work, an optimization model that considers thermal and hydraulic interactions is developed for a cooling water system. It is a closed loop consisting of a cooling tower unit, circulation pump, blower and heat exchangerpipe network. Aside from process disturbances, climatic fluctuations are considered. Model constraints include relations concerning tower performance, air flowrate requirement, makeup flowrate, circulating pump performance, heat load in each cooler, pressure drop constraints and climatic conditions. The objective function is operating cost minimization. Optimization variables are air flowrate, forced water withdrawal upstream the tower, and valve adjustment in each branch. It is found that the most significant operating cost is related to electricity. However, for cooled water temperatures lower than a specific target, there must be a forced withdrawal of circulating water and further makeup to enhance the cooling tower capacity. Additionally, the system is optimized along the months. The results corroborate the fact that the most important variable on cooling tower performance is not the air temperature itself, but its humidity.
Keywords: cooling water systems; cooling towers; nonlinear optimization; cooler network.
INTRODUCTION
Cooling towers are usually present wherever water is used as a cooling medium. In wet or evaporative towers the water to be cooled comes in contact with the outdoor air.
The cooling process is accomplished by a combination of sensible heat transfer due to temperature difference and the evaporation of a small portion of the water. The second mechanism accounts for about 80 % of the total heat removed (Burger, 1993).
Water cooling towers are sized and selected based on economical considerations as well as constraints imposed by system components. However, the thermal performance must ensure a precise cooling water temperature. This variable plays a pivotal role in most industrial applications and slight deviations from design specifications may have a significant impact on overall plant economics.
Therefore, there has been a growing interest on cooling tower analysis. In the literature, many of the available studies have concerned the design of cooling towers (KintnerMeyer and Emery, 1995; ElDessouky et al., 1997). Other studies have presented operation and control topics of cooling towers (Evans, 1980; Burger, 1993; Pannkoke, 1996). Additional studies have considered the mathematical modeling and simulation of thermal performance of cooling towers (Fredman and Saxén, 1995; AlNimr, 1998). Moreover, Cheremisonoff and Cheremisinoff (1981) compiled an extensive list of references concerning cooling tower design and operation.
However, despite the present widespread and continually growing interest on cooling tower rating, to the authors' knowledge, investigations concerning a systemic analysis and overview are not yet available.
In this work, an optimization model for a cooling water system which supplies a heat exchanger network is developed. The model considers the thermal and hydraulic interactions in the overall process and is applied to the analysis of typical operational cases. The system behavior is studied under normal operating conditions and then compared to disturbances such as additional cooling load or cooler temperature requirement from the tower.
PROCESS DESCRIPTION
The cooling water system is shown schematically in Fig. 1. It is a closed loop consisting of a cooling tower unit, a water circulation pump, an air blower and a heat exchanger network.
The performance of a cooling tower is measured by how close it brings the cold water temperature to the wetbulb temperature of the surrounding air. The lower the wetbulb temperature (which indicates either cool air, low humidity or a combination of both), the colder the tower can make the water. Nevertheless, the desired temperature of the cooled water is essentially greater than the wetbulb temperature of the air. The difference between these temperatures is called "approach" and its value generally falls between 5^{o}F and 20^{o}F (Pannkoke, 1996).
Another performance parameter is the difference between the temperature of the hot water entering the cooling tower and the temperature of the colder water leaving the tower, usually referred to as "range".
The thermal performance of a cooling tower depends on the packing arrangements as well as the circulating water and air flowrates. "Rating factor" represents the number of tower units required for a given water rate and set of temperature conditions, usually expressed in ft^{2}/gpm (Evans, 1980).
During operation there is some loss of water. Firstly water vapor passes through the cooling tower and is discharged into the atmosphere. Another source of water loss is due to entrained water droplets that escape from the tower with the exhaust air. Water is also lost from intermittent purge of small amounts of circulating water to prevent an increase in the concentration of solids due to evaporation. Makeup refers to the water flowrate required to replace the circulating water that is lost by evaporation, drift and blowdown.
Occasionally, forced withdrawal of circulating water, upstream of the tower, may be imposed to relieve the heat load so as to achieve lower temperatures on the water stream that leaves the tower with the corresponding makeup.
Since air flow promotes evaporation in the tower, an increase in air throughput constitutes another expedient way to increase the cooling capacity. Tower air flowrate can be controlled through several ways: onoff fans operation, use of variablespeed fans, use of automatically adjustable pitch fans (Pannkoke, 1996).
The main process variables concerning the heat exchanger network are the individual cooling requirements, the inlet and outlet temperatures of each cooler, the circulating pump performance and the split of flows through each branch. Due to strong interactions in the system, slight deviations from design specifications intervene on the overall plant behavior.
The total circulating water flowrate depends on the pump operation point defined by its performance and the hydraulic characteristics of the system. The water flowrate through each branch of the pipecooler network is related to its individual flow resistance, given by the control valve adjustment and other fixed characteristics. However, any change in flowrate disturbs not only the flowrate through the pipecooler branches but also the coolers outlet temperatures. This in turn affects the temperature of hot water at the cooling tower inlet.
In fact, the thermal behavior of the system is even more complex. Aside from process disturbances, the cooling tower performance is influenced by climatic fluctuations (wetbulb temperature or humidity throughout the year). So, the actual temperature of the cooled water will vary in accordance to these inevitable oscillations and it affects again the overall system.
Therefore a realistic prediction of the operational conditions can only be done through a systemic analysis and overview, as will be seen in the remainder of this paper.
MATHEMATICAL MODEL
Model constraints
Cooling tower: A detailed phenomenological model of a cooling tower may become extremely complex. On the other hand, it is possible to establish correlation functions among the variables involved in typical performance diagrams, such as:

(1) 
For a given tower in operation, it is possible to write relation (2) for the rating factor that relates operating conditions with design conditions .

(2) 
The performance of a tower in operation, as mentioned, is affected by the water and air flowrates. The humid air flowrate can be calculated through a water mass balance, by relating the amount of water that evaporates and the air humidity, as seen in (3):
(3) 
The absolute humidity of the air entering the tower is calculated at ambient temperature:
(4) 
Note that in Eq. (4) the absolute humidity H_{w} is calculated at the wet bulb temperature as well as the enthalpy l_{w}. It is assumed that the air that leaves the tower is at the saturation condition (Evans, 1980). Thus:
(5) 
where P_{vap} is determined at the air outlet temperature, given by the average of the inlet and outlet temperatures of the water (Evans, 1980).
As described in Section 2, the makeup water must supply all water lost in the system:
w_{mu} = w_{evap} + w_{rem} + w_{entr} + w_{purge}  (6) 
It is important to note that the portion of the recycled water flowrate which is intentionally removed (w_{rem}) is eliminated upstream of the tower (see Fig. 1), in order to reduce its load. Thus, the amount of water cooled in the tower is given by Eq. 7, where w_{e} is the water flowrate in the closed circuit.
(7) 
The makeup water is at ambient temperature and therefore affects the temperature of the circulating water (T_{e}). This is more significant when makeup flowrate is high, as in Eq. (8).
(8) 
According to Perry and Green (1997), the water flowrate that evaporates is estimated as:
(9) 
where the temperatures are expressed in ^{o}C.
The amount of water purged is established from the concentration cycles, defined as the ratio between the amount of solids dissolved (mostly chlorides) in the recycled water and in the makeup water:
(10) 
The water loss by entrainment is assumed to be 0.1 % with respect to the water flowrate through the tower (Perry and Green, 1997).

(11) 
Pump and common line: The total pressure drop in the circuit (DP_{total}) comprises the contribution from the common line and the contribution of one of the individual branches which are in parallel, for instance the pressure drop in branch 1, DP_{1}:
(12) 
where v_{line} is calculated from w_{e} and the friction factor can be calculated, for instance, from the equation proposed by Chen (1979). In Eq. (12), the height variation between pump outlet and cooling tower inlet is neglected, since it is a fixed value.
Since the cooling tower operates at atmospheric pressure, this pressure variation corresponds to its increase in the circulation pump. From the characteristic curve of the pump, given in general form by Eq. (13), its operation point can be determined.
(13) 
Heat exchanger and individual branch: The heat load Q_{i} to be removed by the cooling water is specified for each heat exchanger Ei.
(14) 
In the proposed model, it is assumed that the controlling heat transfer resistance is at the process side. Therefore, neither the condition of the process fluids nor the thermal performance of the heat exchangers, such as the influence of the water flowrate in the overall heat coefficient, were considered explicitly. Nevertheless, the influence of the cooling water conditions on the system, such as specifications for the water outlet temperature and/or flowrate for a given heat exchanger (see Section 4.2) will be studied.
The total pressure drop in an individual branch consists of the pressure drop in the line and the pressure drop in the exchanger. The former is subdivided in a variable term (according to the valve opening) and a fixed part (pipe and fittings). In the latter, there is a contribution from the straight tube as well as from the direction changes. By neglecting again the height variation between the extremes of the branches in parallel yields:
(15) 
Again, in order to obtain the friction factor in the pipes (f_{i}), Chens correlation can be used (Chen, 1979). The tube side friction factor in the heat exchanger (f_{t,i}) and the term related to the tube side return pressure loss (K_{t,i}), can be determined with the equations in Kern (1965).
As a consequence of the stream split in the parallel branches, the pressure drop in the branches must be the same, that is:
(16) 
Node: The water streams that leave the branches are mixed before returning to the tower. The mass and energy balances in the mixing point are given respectively as:
(17), (18) 
Physical properties: The density and specific heat of the cooling water are calculated at the inlet temperature in the heat exchanger branches. These properties are assumed constant due to their small variation in the range of the operating temperatures. The viscosity values in the common line are calculated at the average inlet and outlet tower temperatures and, in each branch, calculated at the average inlet and outlet heat exchanger temperatures. The analytical relations for the physical properties can be found, for instance, in Yaws (1977).
Objective Function
The objective function is the minimization of the overall operating cost ($/unit time), given by adding the cost of electricity and the cost of cooling water. The former is composed of the pumping cost and the fan operating cost, while the latter is related to the makeup water.
(19) 
In Eq. (19), c_{elec} is the cost coefficient for electricity ($/unit energy) and c_{cw} is the cost coefficient for cooling water ($/unit mass). The power consumed by the fan is a function of the air flowrate that is calculated through a water mass balance (Eq. 3). The terms in parenthesis are in m^{3}/h; note that the second term is related with the characteristic curve of the pump and its efficiency h_{p}. Moreover, the makeup water flowrate is calculated from Eq. (6).
Optimization Model
In the optimization model, there are 10n + 24 variables and 9n + 22 equality constraints with additional single bound constraints, where n is the number of branches in parallel. The optimization variables are air flowrate in the cooling tower (w_{air}), forced withdrawal of water (w_{rem}) and equivalent length of the valves in each branch (L_{va,i} i = 1, n).
The model was implemented in a spreadsheet system and solved with the GRG2 code, based on the Generalized Reduced Gradient Method and the solver parameters utilized are the following: automatic scaling, forward difference derivative calculations, tangent estimates and conjugate search. The convergence tolerance is set to 1x10^{4}.
CASE STUDIES
The optimization model presented in Section 3 is rather general, since in principle any cooling tower algebraic representation as well as pump curve could be fit into the model. In Section 4.1, a base case is optimized for a given system configuration and climatic conditions. The effect of the water outlet temperature on the optimal cost is considered under different cost factors in Section 4.2. Finally, the tower performance is studied for different monthly climatic conditions. It is important to note that although the results are derived for a set of conditions, most of the conclusions can be extended towards other systems.
Base Case
A schematic diagram of the system is described in Fig. 1, which is composed of five heat exchangers which use cooling water that circulates through the tower. The main specifications of the equipment units are given as follows.
The cooling tower performance (Eq. 1) is represented through a polynomial, given in Castro et al. (1998). As for the periodical water purges, four concentration cycles were adopted (Perry and Green, 1997). The circulating water flowrate at design conditions (w'_{e}^{o}) was 200000 kg/h which corresponds to a rating factor (rf^{o}) of 1. In this base case, there is no forced withdrawal of circulating water and thus the only makeup required is related to evaporated water, purge and drift. The characteristic curve of the pump for circulating water corresponds to the one of the ETA 10026 / KSB model. With respect to the line common to all heat exchangers in the circuit, the total length adopted is 320 m, with internal diameter of 0.1541 m (6" Sch 40, carbon steel). The atmospheric conditions are as follows: dry and wet bulb temperatures of 23^{o}C and 21^{o}C respectively. These values correspond approximately to average annual conditions in Porto Alegre, Brazil.
Data for the coolers and the main specifications of pipes in parallel are shown in Table 1.
The optimal results of this case are shown in Fig. 2. Note that, since the amount of makeup water is not significant, the value of the water temperature that feeds the exchangers is approximately the same as the value of the stream that leaves the tower. The value of the approach in the tower is 3.75^{o}C, with range of 4.10^{o}C. Flow velocity as well as pressure loss values in both pipes and coolers are within the ranges recommended in the literature.
The apparently high value of the air flowrate through the cooling tower (approximately 140000 kg/h) is due to the unfavorable inlet air conditions (relative humidity above 80%).
Influence of Cost Factors and Water Outlet Temperature
In this study, the influence of different temperature specifications of the water that leaves the tower in the optimal solution was considered. This is a practical constraint imposed by the process, for instance, by the operating pressures of a distillation column. Additionally, the effect of the water and electricity cost coefficients was analyzed.
Figure 3a shows the total operating cost versus the temperature of the outlet stream. Note that the curves are parameterized by the cost coefficient of water and they all converge to the same point that corresponds to the case for which there are no constraints in the water outlet temperature (approximately 24.75 ^{o}C).
Note that in Fig. 3a the lower the outlet temperature target, the more sensitive the total cost is to the water cost coefficient. This can be also seen in Table 2 which shows the values of the process variables for c_{cw} = 1. When no constraint is imposed on the temperature, the cost of cooling water represents less than 2 % of the overall operating cost; however, this cost reaches 53 % when the outlet water temperature is set at 22.50 ^{o}C.
For temperatures lower than 24.50 ^{o}C, there is a forced withdrawal of cooling water and further makeup flowrate, caused by the incapacity of the tower to provide the cooling requirements. Therefore, for higher water cost coefficients the higher is the influence in the total cost. Similar behavior to that in Table 2 is observed for the remaining water cost coefficients. As the water flowrate through the tower decreases, it causes a reduction in the evaporated water flowrate (see Eq. 9). Consequently there is a smaller air flowrate in the tower. Also, the rating factor increases due to a smaller water flowrate. It is interesting to note that in the last entry of Table 2 the temperature of the water stream after makeup slightly increases, since in this case the ambient temperature (T_{amb} = 23^{o}C) is higher than the temperature of the water that leaves the tower.
The total cost is less sensitive to lower target temperatures of the outlet stream under different electricity cost coefficients. This is due to the fact that the electricity cost component plays a more significant role in the total operating cost. The process variables present similar behavior as that shown in Table 2.
Influence Of Tower Performance And Monthly Climatic Conditions
In this study the climatological chart of Porto Alegre was considered, due to the fact that it is a city with very well defined meteorological changes. Hence, for the same system described in the base case, the operation was optimized for the corresponding dry and wet bulb average temperatures along the months.
Figure 3b shows the total optimal cost as a function of the month as well as for constraints on the rating factor. As mentioned, the rating factor provides an estimate of the tower performance. In other words, for a given water flowrate high values of the rating factor indicate that a large amount of area is required for cooling the water; on the other hand, low values of the rating factor point to a high tower performance. The unconstrained case corresponds to the point of minimal total cost, which corresponds to a rating factor of 1.07 ft^{2}/gpm. By imposing constraints on the rating factor the results indicate an increase in the total cost due to different reasons which can be better seen by examining the main process variables for January shown in Table 3.
It is important to note that higher values of the rating factor, namely 1.2 and 1.5, require the forced withdrawal of water from the tower and subsequent addition of water at ambient temperature. This implies higher costs for water and therefore a significantly higher overall cost; in fact from the last entry of Table 3 it can be seen that the cooling water cost reaches approximately 34 % of the overall cost. On the other hand, lower values of the rating factor correspond to a higher tower capacity, which elevates water and air flowrates. Interestingly, the temperatures of the water leaving the tower (T_{e}) are lower than the water temperature after makeup (T_{e}). This may seem contradictory but, as mentioned, the main purpose of water forced withdrawal and further makeup is to increase the tower efficiency by reducing its load.
In Fig. 3b, one can observe that, for a given value of the rating factor, the highest cost corresponds to the month of July. This might seem in principle paradoxical, as it is the coldest month of the year (in Brazil). Note however that it is the month with the highest humidity (the smallest difference between the dry and wet bulb temperatures). This confirms the fact that the main mechanism for water cooling is evaporation.
CONCLUSIONS
As a result of the strong interaction among the several process variables involved, the operational analysis of their effects on a cooling water system is very complex. Besides, tight constraints on the temperature of the water stream that leaves the tower may increase the operating cost substantially, in particular for the case in which the cost of water is high. Finally, results obtained from different climatic conditions point to the fact that the most important influence on the cooling system performance is not the ambient temperature itself, but its humidity. In summary, the general trend observed is that forced withdrawal of water upstream of the tower is an important resource for fulfilling the cooling duty requirements.
NOTATION
c_{cw}, c_{elec}  cost coefficient for cooling water, electricity 
c_{p}, c_{s}  specific heat of cooling water, humid air 
C_{tot}  total operating cost 
D_{i}, D_{line}  pipe inside diameter of branch i, common line 
D_{t,i}  tube diameter of heat exchanger in branch i 
f_{i}, f_{line}  friction factor in pipe of branch i,common line 
f_{t,i}  tubeside friction factor of cooler in branch i 
H_{in}, H_{out }  absolute humidity of the inlet air, outlet air 
H_{w }  absolute humidity of the saturated air 
K_{t,i}  tubeside return pressure loss coeff. (branch i) 
L_{eq,i}, L_{eq,line}  equivalent length in branch i, common line 
L_{t,i}  tube length of heat exchanger in branch i 
L_{va,i}  equivalent length of the valve in branch i 
MW_{air}, MW_{w}  molecular weight of the air, water 
n  number of parallel branches in the system 
n_{cycles}  number of concentration cycles 
n_{t,i}  number of tube passes of cooler in branch i 
N_{t,i}  number of tubes of heat exchanger in branch i 
P  atmospheric pressure 
P_{vap}  water vapor pressure 
Q_{i}  heat load in exchanger i 
rf, rf^{o}  rating factor, at design conditions 
T_{amb}  ambient temperature 
T_{e }  temperature of the water stream at tower exit 
T_{e}  water stream temperature after makeup 
T_{o}  inlet temperature of the water stream in tower 
T_{o,i}  water outlet temperature at exchanger Ei 
T_{wb}  wet bulb temperature 
v_{i}, v_{line}  flow velocity in branch i, common line 
v_{t,i}  flow velocity in the tubeside of cooler in Ei 
w_{air}  air flowrate through the tower 
w_{e}  total water flowrate in the common line 
w^{o}_{e}  design water flowrate through the tower 
w_{e}  water flowrate after forced removal 
w_{entr}  water flowrate lost by entrainment 
w_{evap}  water flowrate lost by evaporation 
w_{i}  water flowrate in branch i 
w_{mu}  makeup water flowrate 
w_{purge}  purged water flowrate 
w_{rem}  forced withdrawal water flowrate 
Greek Symbols
DP_{i,} DP_{total }  pressure drop in branch i, total 
h_{p}  pump efficiency 
l_{w}  specific enthalpy of the saturated air 
r, r_{air}  density of cooling water, air 
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*To whom correspondence should be addressed