1、300 2008 ASHRAE ABSTRACTTypically, catalog information for cooling towers is avail-able only for a limited range of operating conditions for sealevel applications. The information is often not suitable for theselection of a tower at other operating conditions (e.g., highaltitude, different ambient t
2、emperatures), the evaluation ofmeasured performance, or the simulation over a wide operat-ing range. The analogy approach (Braun et al., 1989) providesa general method for representing the performance of coolingtowers over a wide range of conditions. The accuracy of thismethodology is 2% compared to
3、 catalog values. The method-ology is able to extend catalog information to other operatingconditions, including water inlet and entering temperatures,wet bulb temperature, air and water flow rates, and altitude.INTRODUCTIONCooling towers are widely used in commercial air-condi-tioning applications.
4、The selection of a cooling tower for agiven application is based on the heat rejection for designconditions for the specific location. Design conditions varywidely depending on the location, and the available cataloginformation is usually for sea level operation and limited interms of the range of o
5、perating variables. Additionally, it isoften desired to evaluate the measured performance of a toweragainst that expected, and available catalog data need to beextended to cover the experimental conditions. Further, insimulating the performance of an HVAC system for buildingdesign or evaluation, the
6、 dependence of the performance of acooling tower on operating variables needs to be available overa wide range of conditions. These considerations lead to theneed to develop a methodology to extend available informa-tion available from catalogs or measurements to cover theexpected range of operation
7、.The analogy approach (Braun et al., 1989) provides ageneral method for representing the performance of coolingtowers over a wide range of operation. The results from theanalog approach have been shown to agree with those from the“exact” solution of the governing heat and mass transfer equa-tions wi
8、thin about 2%. The analogy approach provides a meth-odology for extending catalog information to other operatingconditions, including water inlet and entering temperatures,wet bulb temperature, air and water flow rates, and altitude.METHODOLOGYThe analogy method for cooling towers is based on thefun
9、damental differential equations for heat and massexchange in a cooling tower (1). The analogy method will besummarized, with the details and verification of the approachgiven in Reference 1. The control volume showing mass andenergy flows for a counterflow cooling tower section is givenin Figure 1.
10、The fill volume measured from the top of the toweris a convenient coordinate. The relevant conservation relationsare an overall tower energy balance and an air stream energybalance that relates the increase in the air enthalpy to theenergy transfer due to the evaporating water.A simplifying assumpti
11、on is that since the water loss istypically 1 to 5% of the total flow the water flow rate isconstant throughout the tower. Assuming that the water flowrate is constant allows the overall energy balance relation forthe tower to be written as:(1)mwcwdTwdV- madhadV-=Using the Analogy Approach to Extrap
12、olate Performance Data for Cooling TowersJohn W. Mitchell, PhD James E. Braun, PhDFellow ASHRAE Fellow ASHRAEJohn W. Mitchell is the Kaiser Professor Emeritus of Mechanical Engineering at the University of Wisconsin, Madison, WI. James E. Braunis a professor of Mechanical Engineering at Purdue Unive
13、rsity, West Lafayette, IN.NY-08-0362008, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in ASHRAE Transactions, Volume 114, Part 1. For personal use only. Additional reproduction, distribution, or transmission in either print or digital fo
14、rm is not permitted without ASHRAEs prior written permission.ASHRAE Transactions 301To develop the analogy relations, the energy balance isformulated in terms of enthalpy. An effective specific heat isintroduced to allow the water temperature Tw to be replacedby the saturated air enthalpy hw,sat eva
15、luated at the watertemperature Tw. The effective specific heat is defined so thatthe water temperature and the saturated air enthalpy at thewater temperature are related as(2)The effective specific heat, cs, is evaluated as the changein enthalpy with temperature along the saturation line. Theappropr
16、iate value for the entire cooling tower process is basedon the water and air inlet and outlet states, and is evaluatednumerically as:(3)Incorporating the effective specific heat allows the overallenergy balance, Equation 1, to be rearranged and written interms of enthalpies as:(4)It is convenient to
17、 define an equivalent capacitance ratem* that is analogous to the thermal capacitance rate C* usedin sensible heat exchanger analysis. (5)The energy balance, Equation 4, is rewritten in terms ofenthalpies using the equivalent capacitance ratio m* as:(6)The energy balance on the air stream relates th
18、e change ofenthalpy of the air to the transfer of energy from the watersurface:(7)It is convenient to introduce a non-dimensional transfercoefficient defined as:(8)(9)Equations 6 and 9 are analogous to those for a sensibleheat transfer exchanger (see Reference 2) with the enthalpiesreplacing the tem
19、peratures. This allows the effectiveness-Nturelations that were developed for heat exchangers to bedirectly used for cooling towers. In a heat exchanger, the heat transfer is given in terms ofeffectiveness and maximum heat transfer rate. The totalenergy transfer for the tower can then also be repres
20、ented byan effectiveness and a maximum energy transfer rate. Themaximum transfer would occur when the air leaving the toweris saturated at the water inlet temperature, and is given by. (10)Effectiveness deleted from Equation 10The tower energy transfer rate is given by the product ofthe effectivenes
21、s and the maximum energy transfer rate:(11)The energy transfer rate is also given by an energy balanceon the water, using the inlet water flow rate, as(12)The correspondence between the cooling tower and thesensible heat exchanger parameters is given in Table 1:In reference 1 the results using the a
22、nalogy method arecompared to those obtained by integrating the governing heatFigure 1 Mass and energy flows for a cooling towersection.csdTwdV-dhwsat,dV-=csdhwsat,dTw-saturationhwsati,hwsato,Twi,Two,-=dhwsat,dV-macsmwcw-dhadV-=m*=macsmwcw-dhadV-1m*-dhwsat,dV-=madhadV-hccp-A hwsat,ha()=NtuhcAVmacp-=d
23、hadV-NtuV- hwsat,ha()=Qmahwsati,hai,()=Qmahwsati,hai,()=QmwcwTwi,Two,()=302 ASHRAE Transactionsand mass transfer equations through the cooling tower. Theenergy transfer rate from the analogy method has been foundto agree with the exact solution within 2%. The analogyapproach is established as an acc
24、urate representation of cool-ing tower performance. EXTENSION OF CATALOG INFORMATION TO DIFFERENT OPERATING CONDITIONSCooling tower manufacturers provide enough informa-tion to select a tower to reject a given amount of heat at differ-ent ambient and operating conditions. In general, they do notprov
25、ide sufficient information to determine the basic coolingtower parameters. The analogy approach provides a methodfor estimating capacity at different operating conditions.Catalog data for two sizes (Models 1 and 2) of cross-flowcooling towers made by a manufacturer for two different fanpowers (A and
26、 B) are given in Table 2. or each model, thecapacity, which is the maximum water flow rate (in gpm) forwhich the inlet (Ti,) and outlet (To) temperatures (F) will beachieved at the indicated atmospheric wet bulb temperature(Twb) is given. The power of the tower fan is also given. Forexample, the fif
27、th column shows that at a wet bulb temperatureof 80 F, model 1 A will cool a water flow of 53 gpm from 95F to 85 F. The fan power required for the air flow is 1 horse-power. The catalog information in Table 2 includes the effects ofthree variables: wet bulb temperature, inlet temperature, andoutlet
28、temperature. The data correspond to two different towerwater temperature differences (i.e., ranges) of 10F and 15F.The approach to the wet bulb temperature is the same for bothranges at a given wet bulb temperature. The tower capacity(energy transfer rate) is the product of the flow rate, specifiche
29、at, and range. If the condition for which the capacity is desired is fordifferent inlet states or wet-bulb temperature but the same airand water flow rates, then the only parameter that needs to bechanged is the effective specific heat cs. However, if either theair or water flows rates for the desir
30、ed condition are different,a new value of the overall conductance needs to be determinedbecause the conductance is a function on the air and water flowrates. A correlation that relates the overall conductance to theflow rates and design values uses a power relation (Braun etal., 1989):(13)where the
31、exponent n may be determined from data at differ-ent operating conditions. If data are not available, a value forn of 0.4 is a satisfactory approximation. The relation is writtenin terms of the Ntu using Equation 8. (14)An example will be carried out to illustrate how cataloginformation at one opera
32、ting condition can be used to estimatethe performance at other ambient conditions. A representationof a cooling tower can then be developed and the performanceevaluated over the range of operating conditions.The example will use as a base the performance informa-tion for Model 1 B with a 2 hp fan (T
33、able 1) operating at a wetbulb temperature of 75F with an inlet temperature of 95Fand a range of 10F. The capacity is 107 gpm. The perfor-mance for the three conditions listed below will be estimated.The catalog capacity given in Table 1 for these conditions islisted in parentheses. a. A wet bulb te
34、mperature of 64F and a range of 10F (131gpm).b. A wet bulb temperature of 64F and a range of 15F (96gpm).1. A wet bulb temperature of 64F and a range of 15F with1 hp fan (75 gpm).The base operating conditions are used to determine thebase value of the Ntu. The water flow rate capacity is 107 gpm,whi
35、ch corresponds to 53,553 lb/hr. The cooling capacity,which is the product of the flow rate, specific heat, and rangeis 535,530 Btu/hr. The effectiveness is determined from the relation for themaximum heat transfer to the air, Equation 11. However, theair flow rate for this tower is not known so the
36、effectivenessand value of m* cannot be determined. The assumption ismade that the value of m* is unity. For well-design coolingtowers the value of m* is on the order of unity and so this is areasonable assumption. This allows an air flow to be deter-mined from the definition of m*, Equation 5. The v
37、alue of theeffective specific heat csis found using the saturated airenthalpy at the water inlet (Btu/lbm) and outlet temperature(Btu/lbm) divided by the temperature difference (Equation 3):The air flow rate is thenThe effectiveness is then determined from Equation 11.The enthalpy of the saturated a
38、ir at the inlet water temperatureand the entering air are 63.2 and 38.4 Btu/lb, respectively.hcAVhcAV()basemwmw base,-mamabase,-n=Ntu Ntubasemwmwbase,-nmama base,-n 1=cshwsati,hwsato,()Range-=63.2 49.3()Btu/lbm()10 F()- 1 . 3 8 4 B t u / l b m=mam*mwcwcs- 1*53 553 lbm/hr()*1.00(Btu/lb F),1.384 Btu/l
39、bm F()-=38 700 lbm/hr,=Qmahwsatin,hain,()-=535 530 Btu/hr(),38 700 lb/hr()* 63.2 38.4 Btu/lb(),-= 0.558=ASHRAE Transactions 303The Ntu can then be determined from the expression forcross flow heat exchangers (Kays and London, 1964). Thevalue of Ntu corresponding to an m* of unity and an effective-ne
40、ss of 0.558 is 1.504. Although the values of effectivenessand Ntu are not “correct” since they are based on the assump-tion of an m* of unity, the combination of these values is foundto yield the correct total heat transfer.The extension can now be made to situation a), in whichthe wet bulb temperat
41、ure is 64F. Using the enthalpies of satu-rated air at the new water inlet and outlet conditions of 55.8and 43.6 Btu/lb, respectively, the new value for csof 1.22 Btu/lb-F is computed. The three coupled equations that need to besolved simultaneously for the new condition are Equation 5 form*, Equatio
42、n 8 for Ntu that includes the effect of the newwater flow rate from Equation 14, and the expression for theeffectiveness of a cross-flow exchanger. The solution of thesethree equations yields m* = 0.723, Ntu = 1.629, and = 0.634.The energy transfer can then be computed from Equation 11using the valu
43、e of the inlet air enthalpy of 29.2 Btu/lb for thiscondition. The water flow rate is then determined from the expres-sion for capacity, Equation 12.The flow rate of 65,300 lb/hr corresponds to 130.6 gpm.This is essentially the same value as given in Table 1 for theseconditions of 131 gpm. The same p
44、rocedure was followed for the 15F range,condition b). The flow rate was found to be 93.1 gpm, whichis within 3% of the catalog value of 96 gpm. For the conditions represented by c), the air flow rate isdifferent from the base conditions. The actual values of theflow rates are not given, and so the f
45、an law relation betweenpower and flow rate is used to estimate the relative change inflow rate. Fan power is proportional to the cubic power of flowrate and the air flow rate at condition c) relative to the assumedvalue for the base case is then:Following the calculations described earlier, the capa
46、cityis determined to be 76.6 gpm, which is within 2% of the cata-log value for that condition of 75 gpm.The results for the extension of catalog information toother design inlet conditions and other air and water flow ratesfor Model 1 of Table 2 are summarized in Figure 2. The basecase conditions ar
47、e for Model 2 B with a capacity of 107 gpm.The extrapolated values cover a range from 35 to 130 gpm.The standard deviation of the extrapolations for the coolingcapacity agree within about 4% (0.4 gpm) compared to thecatalog values. The analogy approach provides an accurateTable 1. Analogous Parameters for Sensible Heat Exchangers and Cooling TowersParameterSensible Heat ExchangerCooling TowerCapacitance rate ratioC* m*Number of Transfer UnitsNtu Ntu*Effectiv
