Constants for transfer coefficients
1. Introduction
Recirculating cooling water systems are generally used to remove waste heat from hot process streams in conditions above the ambient temperature in many types of industries such as chemical and petrochemical, electric power generating stations, refrigeration and air conditioning plants, pulp and paper mills, and steel mills. Typical recirculating cooling water systems are constituted by a mechanical draft wetcooling tower that provides the cooling water that is used in a set of heat exchangers operated in parallel as can be seen in Figure 1. The economic optimization of recirculating cooling water systems includes the simultaneous selection of the optimal design variables of the cooling tower and each heat exchanger in the cooling network, as well as the optimal structure of the cooling water network. The question is then how to reach this goal. Earlier work on cooling water systems has concentrated on the optimization of standalone components, with special attention given on the individual heat exchangers of the cooling water network. Other publications have dealt with the problem of designing minimumcost cooling towers for a given heat load that must be dissipated (see Söylemez 2001, 2004; SernaGonzález et al., 2010). Most of the methodologies previously reported have concentrated their attention in the optimal synthesis of cooling water networks (see Kim and Smith, 2001; Feng et al., 2005; PonceOrtega et al., 2007). All previous formulations simplified the network configurations because they consider the installation of only one cooling tower; however, the industrial practice shows that sometimes it is preferable to use a set of cooling towers connected in series, parallel, and seriesparallel arrangements to improve the performance of the cooling towers reducing the operational cost and, hence, to decrease the overall total annual cost for the cooling water system. In addition, previous methodologies do not have considered several arrangements for the cooling water that can improve the performance in the coolers and reduce their capital costs. Another limitation for the previously reported methodologies is that they are based on the use of simplified formulations for the design of cooling towers.
This chapter presents an optimization model for the simultaneous synthesis and detailed design of recirculating cooling water systems based on the superstructure of Figure 2. The model considers all the potential configuration of practical interest and the results show the significant savings that can be obtained when it is applied.
2. Model formulation
This section presents the relationships for the proposed model, which is based in the superstructure of Figure 2. In the next equations, the set NEF represents the cooling medium streams leaving the cooler network
The heat of each hot process stream (
are parameters with known values. The heat absorbed by the cooling medium in the matches is equal to the transferred heat by the hot process streams, where
A heat balance for each match of the superstructure is required to determine the intermediate temperatures of the hot process stream and the cooling medium as well as the cooling medium flowrate at each match. For problems with
where
Mass and energy balances are required to calculate the inlet flow rate and temperature for the cooling water to each stage. For the
To calculate the temperature of the bypass coolingwater stream of each stage (TOk), the following heat balance in the splitters is required,
Note that using the heat balance given in equation (4) is possible to know the inlet cooling medium temperature (
The above set of equations is necessary for
The following mass balance must be included for the first stage to determine the flowrate of the cold water provided by the cooling tower network to the cooler network, considering that only cold water is used in the first stage.
The inlet temperatures of the hot process streams define the last location for the superstructure. In other words, the inlet temperature of the hot process stream
The outlet temperatures of hot process streams give the first location for the superstructure. Therefore, the outlet temperature of the hot process stream
In addition, the inlet temperature for the cooling medium (
To ensure a monotonically decrement for the temperatures through the stages of the superstructure, the next constraints are included. It is necessary to specify that the temperature of each hot process stream in the stage
The inlet cooling medium temperature to stage
The temperature inlet cooling medium to the cooling network must be less or equal than the inlet cooling medium temperature in the stage
Finally, the outlet cooling medium temperature in the match
Logic constraints and binary variables are used to determine the existence of the heat exchangers between the hot process stream
Here
Because the area requirements for each match (
where
Equations (18) and (19) are written as inequalities because the heat exchanger costs decrease when the temperature differences increase. Note that the use of binary variables allows the feasibility because if the match does not exist, the parameter
The flowrates (Fw_{j}) and temperatures (Tw_{j}) of the cooling water streams that are directed to the splitters at the inlet of the cooling tower network are determined in the last stage of the cooler network. It is important to note that a problem with
The outlet cooling water stream flowrate from the cooling network can be sent to each tower of the cooling tower network (
the inlet water flowrate (
There is a loss of water in the cooling towers by evaporation (
while the drift loss of water is 0.2 percent of the inlet water flowrate to the cooling tower (Kemmer, 1988):
Thus, for each cooling tower, the outlet water flowrate is given by the following expression:
and the outlet cooling tower flowrate (
The flowrate (
To avoid salts deposition, usually a little blowdown flowrate (
Note that the last term is the total drift loss of water by the air in the cooling tower network. Also,
but the cooling medium temperature in the outlet of cooling network (
To maintain the cooling medium flowrate constant in the cooling system, it is necessary a makeup flowrate to replace the lost water by evaporation, drift and blowdown,
Note that the total water evaporated and drift loss of water in the cooling tower network are considered. The flowrate required by the cooling network (
and the inlet cooling medium temperature to the cooling network is obtained from,
To avoid mathematical problems, the recycle between cooling towers is not considered; therefore, it is necessary to specify that the recycle in the same cooling tower and from a cooling tower of the stage
The following relationships are used to model the design equations for the cooling towers to satisfy the cooling requirements for the cooling network. First, the following disjunction is used to determine the existence of a cooling tower and to apply the corresponding design equations,
Here
where
where
and the algebraic equations to calculate the enthalpy of bulk airwater vapor mixture and the water temperature corresponding to each Chebyshev point are given by,
where
To calculate the available Merkel number, the following disjunction is used through the Boolean variable
Notice that only when the cooling tower
Values for the coefficients
The corresponding disjunction is given by,
Using the convex hull reformulation (Vicchietti et al., 2003), previous disjunction is modeled as follows:



e=1 (splash fill) 
e=2 (trickle fill) 
e=3 (film fill) 

0.249013  1.930306  1.019766  
2  0.464089  0.568230  0.432896 
3  0.653578  0.641400  0.782744 
4  0  0.352377  0.292870 
5  0  0.178670  0 
Values for the coefficients
where



e=1 (splash fill) 
e=2 (trickle fill) 
e=3 (film fill) 

1  3.179688  7.047319  3.897830 
2  1.083916  0.812454  0.777271 
3  1.965418  1.143846  2.114727 
4  0.639088  2.677231  15.327472 
5  0.684936  0.294827  0.215975 
6  0.642767  1.018498  0.079696 
The airvapor flow at the fill inlet and outlet
where
where
where
If the arrangement is in parallel, the total fill height is equal to the fill height of the tallest cooling tower, but if the arrangement is in series, the total fill height is the sum of the cooling towers used in the cooling tower network. This decision can be represented by the next disjunction,
This last disjunction determines the existence of flowrates between cooling towers. Following disjunction is used to activate the arrangement in series,
here
If a series arrangement does not exist, then a parallel arrangement is used. In this case, the total fill height is calculated using the next disjunction based on the Boolean variable
The reformulation for the disjunction is:
Notice that when
Finally, an additional equation is necessary to specify the fill height of each cooling tower depending of the type of arrangement,
According to the thermodynamic, the outlet water temperature in the cooling tower must be lower than the lowest outlet process stream of the cooling network and greater than the inlet wet bulb temperature; and the inlet water temperature in the cooling tower must be lower than the hottest inlet process stream in the cooling network. Additionally, to avoid the fouling of the pipes, 50º C usually are specified as the maximum limit for the inlet water temperature to the cooling tower (SernaGonzález et al., 2010),
here
The local driving force (
The maximum and minimum water and air loads in the cooling tower 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, 2005),
Although a cooling tower can be designed to operate at any feasible
The flowrates of the streams leaving the splitters and the water flowrate to the cooling tower have the following limits:
The objective function is to minimize the total annual cost of cooling systems (
where
where the capital cooling network cost is obtained from the following expression,
Here
where
In addition, the operational cost for the cooling network is generated by the makeup flowrate used to replace the lost of water in the cooling towers network,
where
where
This disjunction is algebraically reformulated as:
where the parameters
This section shows the physical properties that appear in the proposed model, and the property correlations used are the following. For the enthalpy of the air entering the tower (SernaGonzález et al., 2010):
For the enthalpy of saturated airwater vapor mixtures (SernaGonzález et al., 2010):
For the massfraction humidity of the air stream at the tower inlet (Kröger, 2004):
where
where
For the density of the airwater mixture (SernaGonzález et al., 2010):
where
3. Results
Two examples are used to show the application of the proposed model. The first example involves three hot process streams and the second example involves five hot process streams. The data of these examples are presented in Table 3. In addition, the value of parameters
For the Example 1, the optimal configuration given in Figure 3 shows a parallel arrangement for the cooling water network. Notice that one exchanger for each hot process stream is required. In addition, only one cooling tower was selected; consequently, the cooling tower network has a centralized system for cooling the hot process streams. The selected packing is the film type, and the lost water is 13.35 kg/s due to the evaporation lost (75%), and the drift and blowdown water (4.89% and 20.11%), while a 70.35% of the total power consumption is used by the fan and the rest is used by the pump (29.64%). The two above terms represent the total operation cost of the cooling system; therefore, both the evaporated water and the power fan are the main components for the cost in this example. Notice that the water flowrate in the cooling network is 326.508 kg/s, but the reposition water only is 13.25 kg/s, which represents a save of freshwater of 95.94% respect to the case when is not used a cooling tower for thermal treatment of the cooling medium. The total annual cost is 468,719.906$US/year. The contribution to total annual cost for the cost of cooling network is 66%, while for cooling tower network and the pump are 31% and 2.96%, respectively. These results are given in the Table 4.
Respect to the Example 2, Figure 4 presents the optimal configuration, which shows a parallel arrangement to the cooling water network, while the cooling towers network is formed by a distributed system composed by two cooling towers to treat the effluents from the cooling network and to meet the cooling requirements. The selected fill is the film type, and the lost water by evaporation, drift and blowdown represent a 74.99%, 3.94% and 21.07% of the total water lost, respectively. Respect the total power consumption in the cooling system, the fan demands a 65.37% and the pump use a 34.62% of the total cost. The economical results are given in the Table 4. The optimal cooling system shows costs for the cooling network, cooling tower and water pump equal to 61.21%, 36.34% and 2.44%, respectively, of the total annual cost. In addition, for the case that only one cooling tower is selected, the total annual cost is 143,4326.66$US/year, which is 7% more expensive than the optimal configuration. The savings obtained are because the distributed system is able to find a better relationship between the capital cost and the operation cost, which depends of the range, inlet water flowrate and inlet air flowrate to the cooling tower network; therefore,
in the distributed systems there are more options. In this case, the use of freshwater by the cooling network is reduced by 94.92% with the use of the cooling towers. Other advantage of use a distributed system is that depending of the problem data just one cooling tower could not meet with the operational and/or thermodynamic constraints and could be necessary to use more than one cooling tower.



THIN (ºC)  THOUT (ºC)  FCP (kW/ºC)  Q (kW)  h (kW/m^{2}ºC) 

40  76.6  100  3660  1.089 

60  82  60  1320  0.845 

45  108.5  400  25540  0.903 



THIN (ºC)  THOUT (ºC)  FCP (kW/ºC)  Q (kW)  h (kW/m^{2}ºC) 

80  60  500  10000  1.089 

75  28  100  4700  0.845 

120  40  450  36000  0.903 

90  45  300  13500  1.025 

110  40  250  17500  0.75 




468,719.906  1,334,977.470 

309,507.229  817,192.890 

145,336.898  485,196.940 

6,131.013  16,588.250 

32,958.635  61,598.140 

13,875.780  32,587.640 

303,376.216  800,604.640 

112,378.262  181,000.330 
4. Conclusion
This chapter presents a new model for the detailed optimal design of recirculating cooling water systems. The proposed formulation gives the system configuration with the minimum total annual cost. The model is based on a superstructure that considers simultaneously series and parallel arrangements for the cooling water network and cooling tower network, in which the cooling medium can be thermally treated using a distributed system. Significant savings were obtained with the distributed cooling systems for the interconnection between cooling water network and cooling towers. Evaporation represents the main component for the lost of water (7075%); while the drift and blowdown represent the 35% and 2025%, respectively. The fan power consumption usually represents the 6570% of the total power consumption in the cooling system; and the pump represents around the 3035%. For recirculating cooling water systems the costs of cooling network, cooling tower network and the water pump represent the 6070%, 3040% and 25% of the total cooling system cost, respectively. When recirculating cooling water systems are used, the use of freshwater in the cooling network is significantly reduced (i.e., 95%).
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