using the analogy approach to extrapolate performance data for cooling towers.pdf

300 2008 ASHRAE

ABSTRACT

Typically, 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 temperatures), 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 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.

INTRODUCTION

Cooling towers are widely used in commercial air-condi-tioning applications. 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 operating 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 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.

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 theexact solution of the governing heat and mass transfer equa-tions within 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.

METHODOLOGY

The analogy method for cooling towers is based on thefundamental 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. 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 assumption 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)m wcwdTwdV

---------- m adhadV--------=

Using the Analogy Approach to Extrapolate Performance Data for Cooling Towers

John W. Mitchell, PhD James E. Braun, PhDFellow ASHRAE Fellow ASHRAE

John 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 University, West Lafayette, IN.

NY-08-036

2008, 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 form is not permitted without ASHRAEs prior written permission.

Copyright American Society of Heating, Refrigerating and Air-Conditioning EngineProvided by IHS under license with ASHRAE Licensee=Istanbul Teknik Universtesi/5956919001

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ASHRAE Transactions 301

To 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 evaluated 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. Theappropriate 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 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 the 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 temperatures. 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 represented 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 of

the effectiveness 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 analogy method arecompared to those obtained by integrating the governing heat

Figure 1 Mass and energy flows for a cooling towersection.

csdTwdV

----------dhw sat,dV

------------------=

csdhw sat,dTw

------------------ saturationhw sat i,, hw sat o,,Tw i, Tw o,

-------------------------------------------- = =

dhw sat,dV

------------------m acsm wcw--------------

dhadV--------=

m*=m acsm wcw--------------

dhadV-------- 1

m*-------dhw sat,dV

------------------=

m adhadV--------

hccp-----A hw sat, ha( )=

NtuhcA Vm acp

------------------=

dhadV-------- Ntu

V--------- hw sat, ha( )=

Q m a hw sat i,, ha i,( )=

Q m a hw sat i,, ha i,( )=

Q m wcw Tw i, Tw o,( )=



--`,,,`,,``````,``,````,``,,,`,`-`-`,,`,,`,`,,`---

302 ASHRAE Transactions

and 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 accurate representation of cool-ing tower performance.

EXTENSION OF CATALOG INFORMATION TO DIFFERENT OPERATING CONDITIONS

Cooling 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 notprovide 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 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 fifth 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 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, specificheat, 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 desired 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 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 operating 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 (Table 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 temperature 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,which 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 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 value of theeffective specific heat cs is 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 then

The effectiveness is then determined from Equation 11.The enthalpy of the saturated air at the inlet water temperatureand the entering air are 63.2 and 38.4 Btu/lb, respectively.

hcA V hcA V( )basem w

m w base,-------------------

m am a base,------------------

n=

Ntu Ntubasem w

m w base,-------------------

n m am a base,------------------

n 1=

cshw sat i,, hw sat o,,( )

Range------------------------------------------------=

63.2 49.3( ) Btu/lbm( )10 F( )-------------------------------------------------------- 1.384Btu/lbm=

m a m*m wcwcs

-------------- 1*53 553 lbm/hr( )*1.00(Btu/lb F), 1.384 Btu/lbm F( )---------------------------------------------------------------------------------= =38 700 lbm/hr,=

Q

m a hw sat in,, ha in,( )---------------------------------------------------=

535 530 Btu/hr( ),38 700 lb/hr( )* 63.2 38.4( ) Btu/lb( ),------------------------------------------------------------------------------------------= 0.558=



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The 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-ness 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 temperature 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 cs of 1.22 Btu/lb-F is computed. The three coupled equations that need to besolved simultaneously for the new condition are Equation 5 form*, Equation 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 value 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 procedure 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 fan 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 capacityis 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 are 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 accurate

Table 1. Analogous Parameters for Sensible Heat Exchangers and Cooling Towers

ParameterSensible Heat

ExchangerCooling Tower

Capacitance rate ratio

C* m*

Number of Transfer Units

Ntu Ntu*

Effectiveness

Energy flow

Table 2. Catalog Values of Cooling Tower Performance

Model Fanpower(hp)

TowerVol(ft3)

Ti (F) 95 100 95 100 90 95 90 95

To (F) 85 85 85 85 80 80 80 80

Twb (F) 80 80 75 75 70 70 64 64

1 A 1 140 (gpm) 53 42 84 63 74 56 102 75

1 B 2 140 (gpm) 68 53 107 80 95 71 131 96

2 A 2 210 (gpm) 87 69 137 103 122 91 168 123

2 B 3 210 (gpm) 101 79 160 120 142 106 197 144

f C*,Ntu( )= f m*,Ntu*( )=Cmin Th i, Tc i,( ) m a hw sat i,, ha i,( )

V

V

V

V

Q 0.634*38,700(lb/hr)*(55.8 29.2(Btu/lb)=653,000 Btu/hr=

m Q

cw Tw in, Tw out,( )---------------------------------------------=

653 000 Btu/hr( ),1.00 Btu/lb F( )* 90 80( ) F( )--------------------------------------------------------------------------- 65 300 lb/hr,==

m a c, m a base,PcPbase-------------

1 338 700 (lb/hr),= =

1 hp( )2 hp( )--------------

1 3 30 700 lb/hr,=



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method of extrapolating performance over a wide range ofdesign conditions.

In extrapolating the data to other conditions, it is impor-tant to ensure that the nozzles and sump are appropriatelysized. The catalog data presented in Table 2 for a given modelapply to a fixed tower geometry except for the orifice sizes inthe distribution nozzles and the pipe diameters used for inletand outlet piping connections. The sizes of these componentsdepend on the design flow rate. If during operation the waterflow is significantly higher or lower than the design flow (onthe order of 10 to 20%), then the performance may be affected.For water flow rates lower than the design value the head overthe nozzles may be too low for uniform flow over the mediaand for higher water flow rates the basins may overflow. Fora given tower in which the flows vary significantly the perfor-mance may deviate from that predicted by extension of themodel relations presented here. Accurate performance at off-design conditions needs to be obtained from the manufacturer.

The results show that the analogy method can be used todetermine the basic parameters for a cooling tower. Theassumption that the value of m* is unity yields values for theair flow rate, Ntu, and effectiveness that are not correct butthat do combine to give the actual heat transfer at the baseconditions. The parameters determined in this mannerprovide accurate estimates of tower performance at designconditions. Even with limited the catalog information orexperimental measurements, a performance map over a widerrange of ambient and operating conditions can be developedusing the analogy approach. This would facilitate the selec-tion of a cooling tower for a given application where thedesign conditions are different from those available. A modelbased on the analogy method could useful in simulating the

performance of a tower. The impact of different control strat-egies on the performance of other HVAC equipment in asystem could be evaluated.

EXTENSION OF CATALOG INFORMATION TO DIFFERENT ALTITUDES

Data provided by cooling tower manufacturers is for sealevel conditions and may not accurately reflect the perfor-mance at high altitude conditions. The analogy method canalso be used to extrapolate catalog data to different altitudes.First of all, the analogy method will be used to study the influ-ence of altitude on performance.

There are several effects of altitude. Lower air densities athigher altitude lead to lower air mass flow rates and heat trans-fer coefficients for a given fan and tend to reduce the heatrejection capacity from that stated in the catalog. However, thedriving potential for heat and mass transfer (the enthalpypotential) actually increases with increasing altitude due toreduced partial pressure of the water vapor.

A cooling tower fan delivers a constant volume flow rate.Since the air density decreases with increasing altitude themass flow rate is approximately proportional to the air density.It is useful to consider a volumetric heat rejection capacity toevaluate the effect of altitude. The heat rejection per unitvolume of air flow is defined as

(15)

To determine the errors associated with using sea-leveldata at higher altitudes, it is necessary to consider the effect ofaltitude on air density, air enthalpies, and device effectiveness.

Air density is proportional to air pressure at a constanttemperature. To generically evaluate the effect of altitude, thechange in atmospheric pressure with elevation relative to thesea level value is modeled as

(16)

At 10,000 ft (3000 m) above sea level, the air pressure anddensity are about 30% less than at sea level. The air densitydecreases a little more than the air pressure because the humid-ity ratio increases as the total pressure decreases for a givendry bulb and wet bulb temperature. Without considering othereffects, Equation 15 indicates that the capacity of a coolingtower would decrease by about 30% at this altitude.

The enthalpy potential is the difference between theenthalpy of saturated air at the inlet water temperature and theenthalpy of the atmospheric air. Figure 3 shows the influenceof altitude on enthalpy potential for a fixed air inlet dry bulband different wet bulb and water inlet temperatures. For agiven rating condition, the enthalpy potential increases byabout 40% at 10,000 ft as compared with sea level. The effectof altitude is more pronounced for higher water and wet bulbtemperatures.

Figure 2 Plot of predicted value of capacity vs. catalogvalue.

Q v a hw sat i,, ha i,( )=

p p0 ea z=



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The enthalpy potential increases with altitude because ofthe dependence of humidity ratio on pressure. Using propertyrelations for humid air, the enthalpy potential is

(17)

The first term in Equation 17 is associated with the sensi-ble heat transfer from the water surface to the air. For givenwater surface and air dry bulb temperatures, this term is essen-tially constant with changing altitude. For given dry bulb andwet bulb temperatures, the specific heat increases by about1.5% at 10,000 ft. The second term in Equation 17 is associ-ated with the evaporation of moisture from the water surfaceto the air. The humidity ratio potential (wsat,w,i wa,i) increaseswith altitude due to a decrease in pressure.

The increase in humidity ratio difference with altitude canbe shown using the definition of humidity ratio.

(18)

At the water surface, the vapor pressure is the saturationpressure associated with the water temperature and does notvary with altitude. For the atmospheric air, the vapor pressureis evaluated as a function of dry bulb and wet bulb temperatureand also does not vary with altitude. Therefore, the humidityratio increases with altitude since the total pressure decreases.The increase in humidity ratio is greater for the vapor at thewater surface because of the higher vapor pressure, and theeffect is greater for the higher vapor pressures associated withhigher water temperatures and inlet wet bulb temperatures.

The mass transfer coefficient is related to the heat transfercoefficient through the Lewis relation

(19)

For turbulent flow such as found over cooling towersurfaces, the convection transfer coefficients generally vary

with Reynolds number raised to a power that is less than unity.As a result, the convection coefficient can be estimated fromthe convection coefficient at sea level according to

(20)

The exponent m is generally between about 0.3 and 1.0,and was taken to be 0.8 for this study.

Using Equations 17 and 14 with the assumption ofconstant volumetric flow, the value of Ntu is related to thevalue at sea level with

(21)

The exponent (1-m) is equal to 0.2 for turbulent flow. Forthe same volume air flow rate and the same water mass flowrate, the cooling tower effectiveness then increases with alti-tude as the density decreases.

Figure 4 shows the impact of altitude on convection coef-ficient and number of transfer units. At 10,000 feet above sealevel, the convection coefficient is reduced by about 25%,whereas Ntu increases by less than 10%.

The impact of the change in Ntu with altitude on effec-tiveness depends upon the magnitude of the Ntu at sea level.The results shown in Figure 5 are based on the counter-flowrelation for effectiveness (Kays and London, 1964). For a highvalue of Ntu and effectiveness, an increase in altitude resultsin a small increase in effectiveness. However, for a less effec-tive cooling tower the increase is more significant. Thisincrease counteracts the reduction due to air density depictedin Figure 4.

For the same volume flow rate, an increase in altitudeleads to a reduced mass flow rate, a somewhat increased effec-tiveness and a significant increase in enthalpy potential.Figure 6 shows the impact of altitude on the volumetric capac-ity for three different situations. The three cases were selectedto bound the range of performance effects of altitude for cool-ing towers. For low values of Ntu0 and high water and wet bulbtemperatures, the volumetric capacity increases slightly withaltitude. For high values of Ntu0 and low water and wet bulbtemperature case, the volumetric capacity decreases by 10% at10,000 ft. However, for most design conditions, the volumetriccapacity decreases only slightly with altitude from that statedin the catalog for sea level conditions.

In summary, the first step in correcting sea-level perfor-mance or catalog data for different altitudes is to determine acorrelation for Ntu at sea level (Ntu0) using Equation 14 asoutlined earlier. The Ntu at higher elevations can then be esti-mated from the correlation determined at sea level butcorrected for air property effects using Equation 21. The towerNtu is used along with an appropriate effectiveness relation(crossflow or counterflow) to determine tower effectiveness.Equation 11 is then used to determine the tower capacity forany altitude with air enthalpies calculated at the local ambient

Figure 3 Effect of altitude on enthalpy potential.

hw sat i,, ha i, cpm Tw i, Ta i,( ) ww sat i,, wa i,( )hfg+=

w 0.622pvp pv--------------=

hmhccpm--------=

hc hc 0,aa 0,----------

m=

Ntu Ntu0cpm 0,cpm

-------------a 0,a

---------- 1 m

=



--`,,,`,,``````,``,````,``,,,`,`-`-`,,`,,`,`,,`---


pressure and the entering water and ambient wet bulb temper-atures. Further, the mass flow rate used in Equation 11 and incalculating m* for the heat exchanger effectiveness(Equation 5) needs to be corrected for the lower density occur-ring at higher altitudes according to

(22)

With these corrections, sea-level catalog information canbe extrapolated to allow an estimation of the performance athigher altitudes to be made, and an appropriate tower selectedfor the application.

As an example of the effect of altitude, the performancefor Model 1 B with a 2 hp fan (Table 1) operating at a wet bulbtemperature of 75F with inlet and outlet temperatures of 95Fand 85F will be estimated for an altitude of 10,000 ft. At thisaltitude the pressure is 10.1 psia and the air density is0.0476 lbm/ft3.

The enthalpy potential, which is the difference betweenthe enthalpy of saturated air at the inlet water temperature andthe enthalpy of the atmospheric air, for sea level conditions is:

Whereas at 10,000 ft the enthalpy potential is

This is a 40% increase in potential, as shown in Figure 3.The effect of altitude on the Ntu is given by Equation 21,

which shows that the main effect is that of the density. The Ntuof the Model 1 B at sea level was found to be 1.504. At altitude,the Ntu is

The Ntu is increased by 6.6%, as shown in Figure 4. Theeffectiveness is increased from the sea level value of 0.558 to0.574, which is an increase of 3%, as shown in Figure 5. Thevolumetric heat capacity is increased

The volumetric heat capacity for sea level conditions is0.0975 Btu/ft3, which is 2% greater than that at 10,000 ft, asshown by Figure 6 for these conditions.

The implication for this change in volumetric heat capac-ity on the design point capacity is that the design water flowrate is reduced 2% (from 107 gpm to 105 gpm) to provide thesame range for these conditions. This is a relatively small

Figure 4 Effect of altitude on convection coefficient andNtu.

Figure 5 Effect of altitude and Ntu at sea level oneffectiveness.

Figure 6 Effect of altitude on volumetric heat rejectioncapacity relative to sea level catalog values.

m aaa 0,----------m a 0,=

Enthalpy potential hw sat i,, ha i,( )=63.2 38.4( )= Btu/lbm( ) 24.8 Btu/lbm=

Enthalpy potential hw sat i,, ha i,( )=83.2 43.1( )= Btu/lbm( ) 35.0 Btu/lbm=

Ntu 1.504*0.247 Btu/lbm F( )0.251 Btu/lbm F( )-----------------------------------------------

0.0705 lbm/ft3( )0.0476 lbm/ft3( )--------------------------------------

0.21.60= =

Q v 0.574*0.0476 lbm/ft3( )*(83.2-48.1) Btu/lbm( )=

0.958 Btu/ft3=



--`,,,`,,``````,``,````,``,,,`,`-`-`,,`,,`,`,,`---


difference, but as shown in Figure 6, there would be signifi-cantly greater differences for lower wet-bulb temperatures,lower water inlet temperatures, and exchangers with higherNtu and effectiveness.

CONCLUSIONS

The analogy method provides a general representation ofthe performance of cooling towers over a wide range of oper-ation. The parameters of the analogy method can be cali-brated using baseline data at one operating condition.Together with correlations for the transfer coefficients, theanalogy method can be used to extend the baseline data tocover a wide range of operating conditions. The approach isdemonstrated to accurately extend catalog data at one condi-tion to cover a wide range of entering and leaving watertemperatures, ambient air wet bulb temperatures, air and waterflow rates, and altitude.

NOMENCLATURE

a exponent for altitude effect

area per unit volume

cs effective specific heat

cp specific heat of air

cpm specific heat of air-water vapor

cw specific heat of water

Cmin min

C* capacitance rate ratio

h enthalpy

hc con

hfg latent heat of vaporization for water

hm mass transfer coefficient

m exponent

mass flow rate

m* mass flow rate ratio

n exponent

Ntu Number of Transfer Units

p pressure

pv partial pressure of vapor

maximum energy transfer rate

capacity or heat transfer rate

heat transfer rate per unit volume

T temperature

V volume

w humidity ratio

z elevation above sea level

effectiveness

density of air

Subscripts

a air

base base conditions

i inlet

o outlet

w water

w, sat saturated air at water temperature

0 sea level conditions

REFERENCES

1.Braun, J. E., S. A. Klein, and J. W. Mitchell, EffectivenessModels for Cooling Towers and Cooling Coils,ASHRAE Transactions, 95, Part 2, 164, 1989

2.Kays, W. M. and A. L. London Compact Heat Exchang-ers, McGraw Hill, 1964

3.Marley Cooling Tower, Marley, Mission, Kansas, 2003

A

m

Q maxQ

Q v

a



--`,,,`,,``````,``,````,``,,,`,`-`-`,,`,,`,`,,`---

ABSTRACTINTRODUCTIONMETHODOLOGYFigure 1 Mass and energy flows for a cooling tower section.EXTENSION OF CATALOG INFORMATION TO DIFFERENT OPERATING CONDITIONSTable 1. Analogous Parameters for Sensible Heat Exchangers and Cooling TowersTable 2. Catalog Values of Cooling Tower PerformanceFigure 2 Plot of predicted value of capacity vs. catalog value.EXTENSION OF CATALOG INFORMATION TO DIFFERENT ALTITUDESFigure 3 Effect of altitude on enthalpy potential.Figure 4 Effect of altitude on convection coefficient and Ntu.Figure 5 Effect of altitude and Ntu at sea level on effectiveness.Figure 6 Effect of altitude on volumetric heat rejection capacity relative to sea level catalog values.CONCLUSIONSNOMENCLATURESubscriptsREFERENCES