Simulation of a direct evaporative cooler and a cooling ... · PDF fileInternational Journal...

International Journal of Mechanical Engineering Education 35/3

Simulation of a direct evaporative cooler and a cooling tower by theeffectiveness–NTU methodP. A. F. Bougleuxa, F. E. M. Saboyab, R. Pitanga Marquesc and J. A. R. Parised (corresponding author)a Department of Mechanical Engineering, Pontificia Universidade Católica do Rio de Janeiro, Brazilb Late Professor of Thermal Engineering, Department of Mechanical Engineering,Universidade Federal Fluminense, Niterói, Brazilc Postgraduate Programme in Metrology, Pontificia Universidade Católica do Rio de Janeiro, Brazild Department of Mechanical Engineering, Pontificia Universidade Católica do Rio de Janeiro,Rua Marquês de Sao Vicente, 225, 22453-900 Rio de Janeiro RJ, BrazilE-mail: [email protected]

Abstract A direct evaporative cooler and a counter-flow cooling tower were simulated by theeffectiveness–NTU method. One-dimensional energy and mass balance as well as heat and masstransfer equations were applied. The resulting algebraic equations were solved numerically. To validatethe simulation model, numerical results of the evaporative cooler and cooling tower were comparedwith available experimental data. This simple application of the effectiveness–NTU method to suchimportant heat transfer devices may prove pedagogically useful to undergraduate mechanicalengineering students attending heat transfer courses.

Keywords Cooling tower; evaporative cooler; effectiveness–NTU method

Notation

a heat and mass transfer area per unit volume (m2/m3)A heat transfer area (m2)cp fluid specific heat (J/kg·K)CR capacity rate ratiof ′ slope of saturated air enthalpy versus temperature curve (J/kg·K)h specific enthalpy (J/kg)K mass transfer coefficient (kg/m2·s)m· mass flow rate (kg/s)m· min smaller value between m· a and n iteration numberNTU number of transfer unitsQ·

heat transfer rate (W)T temperature (K)U air relative humidity

m c

fw1 pw1 kg s

′( )

V mass transfer volume zone (m3)W humidity ratio (kg of water/kg of dry air)

Greek symbolsd enthalpy correction factore effectivenesss standard deviationz tolerance

Subscriptsa dry airm moist airs saturatedw water0 make-up water1 inlet2 outlet3 water outlet

Introduction

Electric power generation, manufacturing and large air-conditioning systems doreject, as required by the second law of thermodynamics, substantial amounts of heatto a heat sink. More often than not, in the absence of natural heat sinks (e.g. oceans,lakes and rivers), the water supply is limited. A plausible alternative is the atmos-phere, the only natural heat sink present everywhere. But the low heat capacity ofatmospheric air generally requires an intermediate thermodynamic device equippedwith a recirculating water system to serve as the transport medium for heat betweenthe source and the sink. This thermodynamic artifact is generically called a (wet,atmospheric) cooling tower. Cooling towers are either direct (open-circuit) or indirect (closed-circuit), counter-flow or cross-flow (according to the direction inwhich the air flows) heat ejection devices. However, for the sake of brevity we limitourselves here to direct counter-flow devices.

In a counter-flow cooling tower, warm water from the condenser (of a thermo-dynamic cycle) is pumped to the top of a closed structure equipped with power-driven fans to draw air through the tower (see Fig. 1). The warm water is thensprayed and falls by gravity. The spraying substantially increases the surface areaof the water in contact with the air, and the larger surface area (of the water asdroplets) greatly augments the exchange of heat between the two. This favours evap-oration. A small fraction (a few per cent) of the water evaporated cools down boththe air and the remaining water, which is collected at the bottom of the tower. Beforepumping the remaining cooled water back to the cycle, some additional water, theso-called make-up water, must be added to account for the water lost to the atmos-phere during the process. The make-up water must equal the total water lost (throughthe exhaust air, evaporation and leakage) in order to maintain a steady level of water

208 P. A. F. Bougleux et al.


in the (whole) cycle. A cross-flow cooling tower is exactly the same device exceptthat the air flows horizontally.

(Direct) evaporative coolers are based on the same principle. The high latent heatof vaporization of the water and the difference in temperature between the water and the air are responsible for the cooling process by a simple heat exchangemechanism: the water absorbs heat from the air as it evaporates. As a result, thetemperature of the air stream decreases while its humidity increases.

Now, if no heat is exchanged between the air stream and the surroundings, theevaporative cooling nearly coincides with the adiabatic cooling process (an approx-imation justifiable only if the temperature of the water supplied is not much differ-ent from the exit temperature of the air stream). Then the evaporative process followsa line of constant wet-bulb temperature on a psychrometric chart. And it may evenreach an extreme value, where the air leaving the cooler is saturated.

Walker et al. [1] and Robinson [2] were probably the first to account for anelementary theory of cooling towers. However, the practical use of a governingrelationship was introduced by Merkel [3]. He suggested the use of the enthalpypotential, combining heat and mass transfer processes. Over the years several mod-ifications of Merkel’s theory have been proposed [4, 5]. In 1956, Zivi and Brand [6]extended Merkel’s analysis to include cross-flow cooling towers. In 1961, Lowe andChristie [7] carried out some studies with different types of tower packings. Later,Benton and Waldrop [8] introduced a computational model for the simultaneousheat, mass and momentum transfer of the cooling process inside the tower based onthe finite-integral method. Jaber and Webb [9] proposed the effectiveness methodfor modeling evaporative coolers and cooling towers using a correction factor, sug-gested by Berman [10]. Berman [10] approximated the saturated air enthalpy versusthe temperature of the water curve to a straight line, hence his correction factor.

Effectiveness–NTU method of simulation 209


Mass transfer zone

Fan

Moisture separator

Water manifoldSpray nozzles

Fill

Louvres

Water basin

Fig. 1 Counter-flow induced draft cooling tower.

Theoretical and experimental investigations of the performance of cooling towershave been conducted by Verdu-Estevez and Torella-Alcarez [11], as well as Gomeset al. [12] and Bernier [13]. In 1996, an interesting survey of several methods for aproper thermal design of cooling towers was presented by Mohiuddin and Kant [14,15]. Gan and Riffat [16] simulated closed wet cooling towers based on a two-phaseflow of gas and water droplets model. More recently, Lu and Cai [17] formulatedan ‘engineering model’ for both the counter-flow and cross-flow cooling towers ofMerkel [3], and Braun et al. [18] derived fundamental mass and energy balanceequations with polynomials to calculate the effectiveness of the heat exchange.

Detailed information on the characteristics of cooling towers is seldom known, inwhich case the use of lumped parameter models seems to be more appropriate. Theaim of this work is to simulate the performance of counter-flow cooling towers anddirect evaporative coolers through a simple effectiveness–NTU method, proposedby Jaber and Webb [9].

Mathematical model

The mathematical model was obtained from the mass, energy and heat transferequations by considering the properties of water and air uniform in a given cross-sectional area of the equipment. The following working hypotheses were employed:steady-state operation; negligible variation of kinetic and potential energy; adiabaticequipment.

Mass balance equationsThe global mass balance is obtained when the conservation of mass equation isapplied to the control volume of the direct evaporative cooler (schematically shownin Fig. 2), yielding:

m· aW1 + m· w1 = m· aW2 + m· w2 (1)

At the water basin (schematically shown in Fig. 3):

m· w1 = m· w0 + m· w2 (2)

From equations 1 and 2, one obtains:

m· w0 = m· a(W2 − W1) (3)

The dry air mass flow rate can be expressed in terms of the moist air mass flowrate as follows:

(4)

From equations 3 and 4, one has:

(5)˙˙

mm

WW Ww0

m1=+( )

−( )1 1

2 1

˙˙

mm

Wa

m1=+( )1 1





wm

2

a

mw1

ma mw2

mw3

mw0

w1

m

mh

a mw2

mw0

m1

mh

a

m2

w1hm1

Air Water

hw1

hw0mw1hw3

Q

Fig. 2 Global mass balance control volume at the direct evaporative cooler.

Fig. 3 Energy and water balance control volumes at the water basin, the water side and air sides.

Energy balance equationsThe control volume of the heat and mass transfer processes at the air–water interface, illustrated in Fig. 3, is described in terms of energy balance equations as follows.

At the water side:

Q· = m· w1hw1 − m· w2hw2 (6)

where hw1 is the specific enthalpy of the water as saturated liquid, at temperatureTw1.

At the saturated liquid state, one may consider with good approximation that:

hw = cpwTw (7)

Combining equations 6 and 7, one obtains:

Q· = cpw(m· w1Tw1 − m· w2Tw2) (8)

On the moist air side (Fig. 3), the energy balance provides:

Q· = m· a(hm2 − hm1) (9)

Combining equations 9 and 4, one obtains:

(10)

The energy balance equation at the (cold) water basin (Fig. 3) is:

m· w2hw2 + m· w0hw0 = m· w1hw3 (11)

From equation 7:

m· w2Tw2 + m· w0Tw0 = m· w1Tw3 (12)

The thermodynamic properties of moist air and water were found in Threlkeld [5]and Hill and MacMillan [19].

Heat and mass transfer by the effectiveness–NTU methodFollowing Jaber and Webb [9], the heat and mass transfer equation by theeffectiveness method is given by:

Q· = e m· min(hs1 − d − hm1) (13)

where hs1 is the enthalpy of the saturated air at temperature Tw1. The enthalpycorrection factor, d, accounts for the non-linearity of the enthalpy–temperature curve and can be calculated as follows [9]:

(14)

where hs is the enthalpy of the saturated air at temperature Tw, which is the arithmeticmean between Tw1 and Tw2.

d =+ −( )h h hs1 s2 s2

4

˙ ˙Q

m

Wh h=

( )−( )m1

m2 m11+ 1



According to Jaber and Webb [9], the e-NTU equation for a counter-flow coolingtower and direct evaporative cooler is given by:

(15)

where the fluid capacity rate ratio, CR, is defined by:

(16)

and the number of transfer units, NTU, from:

(17)

Solution

The resulting algebraic system of non-linear equations has been solved using aFORTRAN-coded computer program by a sequential algorithm whose input datawere given by the incoming conditions of moist air, water and make-up water (Tm1,inlet dry-bulb temperature of moist air, W1, m· m1, Tw1, m· m2 and Two) and equipmentcharacteristics (K, a and V).

The sequence of calculation was as follows:

(1) input data;(2) calculate Ws1, hm1 and hs1;(3) estimate Tn

w2 and Wn2;

(4) calculate f ′ and m· min;(5) calculate CR, NTU, e, m· w0, d, Q

·, hm2, Tm2;

(6) calculate Tw2n+1 and W2

n+1;(7) calculate tolerances zW and zT;(8) if zW and zT are within accepted values, then calculate results;(9) otherwise proceed to new estimate of Tw2 and W2 and return to (4).

Stopping criteria were defined as follows:

(18)

(19)

Updated values of Tw2 and W2 were obtained with a relaxation coefficient of 0.2.

Comparison with experimental data

The validity of the model has been tested by comparing predicted results withexperimental data, from Gomes [20], obtained at identical operating conditions. Thelaboratory data were collected from a counter-flow cooling tower rig 400mm wide,

zW

n n

n

W W

W=

−≤2

+12

2

0 01.

zT

n n

n

T T

T=

−≤w2

+1w2

w2

0 001.

NTU = =KA

m

KaV

m˙ ˙min min

Cm

mR =

˙

˙min

max

e = − − −( )[ ]− − −( )[ ]1

1

exp

exp

NTU 1

NTU 1R

R R

C

C C



900mm long and 1530mm high, with a high-impact polystyrene packing, corru-gated and moulded in a honeycomb structure. The cooling tower had a water dis-tribution manifold with 10 spray nozzles, a volume of the cold basin of 0.050m3

and a mass transfer and packing volumes of 0.320m3 and 0.230m3, respectively.The laboratory data selected consisted of 34 runs in the ‘cooling tower mode’ and

32 runs for the direct evaporative cooler. The exit air temperature (Tm3), the exitwater temperature (Tw3) and the exit air humidity ratio (W2), as predicted by the sim-ulation model, were the thermodynamic parameters selected for comparison. Theexperimental mass transfer coefficient, an input parameter for the model, wasdetermined from measured values of temperature and humidity ratio. The calcula-tion procedure is outlined in the Appendix.

Figs 4–6 show the comparison between predicted and experimental values of theoutlet variables for the cooling tower (and the direct evaporative cooler in Fig. 6).Agreement was very good, as the temperature and humidity measurements, takenfrom the test rig, were used to evaluate the product UA at each run. The consistencyof the heat and mass transfer equations, either in the form of a logarithmic temper-ature difference or in the form of the exponential of the number of transfer units,guaranteed the validation of the model adopted.

Applications

The model was applied to predict the performance of a typical direct evaporativecooler. Direct evaporative coolers and wet cooling towers are thermodynamicallyequivalent devices, so that the influence of operating parameters, such as the inlet air conditions, on their performance follows the same pattern. A typical example



290 295 300 305 310 315

Predicted Temperature (K)

290

295

300

305

310

315

Mea

sure

d T

empe

ratu

re (

K)

Fig. 4 Comparison between simulated and measured values of the water exit temperatureof a cooling tower.

simulation of an evaporative cooler (T1 = 300.55K; m· 1 = 0.0161kg/s; Tw1 = 298.95K;m· w1 = 2.8kg/s; KaV = 0.65kga/s) is presented. Figs 7 and 8 show the dependence of theevaporative cooler on the inlet air relative humidity. Fig. 7 shows that evaporativecoolers do present, as expected, higher capacity in drier climates. On the other hand,



290 295 300 305 310 315

Predicted Temperature (K)

290

295

300

305

310

315

Mea

sure

d T

empe

ratu

re (

K)

0.016

0.017

0.018

0.019

0.020

0.021

0.016 0.017 0.018 0.019 0.020 0.021

Measured Humidity Ratio (kg/kg)

Pre

dic

ted

Hu

mid

ity

Rat

io (

kg/k

g)

cooling tower

direct evap cooler

Fig. 5 Comparison between simulated and measured values of the air exit temperature ofa cooling tower.

Fig. 6 Comparison between simulated and measured values of the exit humidity ratio of acooling tower and a direct evaporative cooler.

Fig. 8 shows that the outlet relative humidity is always high, and not substantiallyaffected by the inlet conditions.

Concluding remarks

Most mechanical engineering students are introduced to the effectiveness–NTUmethod in a second undergraduate heat transfer course, but its application is usuallylimited to common heat exchangers. Here we have applied this important heat trans-fer technique to other heat transfer devices. The simulation of counter-flow coolingtowers and direct evaporative coolers has demonstrated, once more, the usefulnessof the effectiveness–NTU method. Accurate predictions were obtained except whereconstant properties could not be assumed throughout the cross-section of the towerand evaporative cooler (e.g. low water mass flow rate).

Appendix. Determination of the mass transfer coefficient from experimental data

The definition of the mass transfer coefficient should be taken into account, that is:

m· a dW = KadV (Ws − W) (A1)



0

2

4

6

8

10

12

0 20 40 60 80 100

Inlet Air Relative Humidity (%)

Hea

t Tra

nsfe

r R

ate

(kW

)

40

50

60

70

80

90

100

0Inlet Air relative Humidity (%)

Out

let A

ir R

elat

ive

Hum

idity

(%

)

20 40 60 80 100

Fig. 7 Influence of the air inlet relative humidity on the capacity of a direct evaporativecooler.

Fig. 8 Influence of the air inlet relative humidity on the exit relative humidity of a directevaporative cooler.

where Ws is the saturated air absolute humidity at the temperature Tw at the sectionconsidered.

Integrating equation A1 one has:

(A2)

The integral in equation A2 may be evaluated by the log-mean difference betweenthe inlet and exit air humidity, assuming that the function (Ws − W) varies linearlywith absolute humidity. Thus:

(A3)

and

(A4)

whose integration yields:

(A5)

where WS1 is the saturated air absolute humidity at TW1, and WS2 at Tw2.Equation A5 provides a convenient analytical expression for the student to obtain

the mass transfer coefficient from data obtained in the heat transfer laboratory.

Acknowledgements

The financial support provided by Brazilian Research Agencies (CNPq, CAPES andFAPERJ) is gratefully acknowledged.

References

[1] W. Walker, H. Lewis and W. K. McAdams, Principles of Chemical Engineering (McGraw-Hill BookCompany, 1923).

[2] C. S. Robinson, Mech. Eng., 45 (1923), 99.[3] F. Merkel, ‘Evaporative cooling’, Zeits Verein Deutscher Ingenieure, 70 (1926), 123–128.[4] J. Lichtenstein, ‘Performance and selection of mechanical-draft cooling towers’, Transactions of the

ASME (October 1943).[5] J. L. Threlkeld, Thermal Environmental Engineering, 2nd edition (Prentice-Hall International,

1970).[6] S. M. Zivi and B. B. Brand, ‘An analysis of the cross-flow cooling tower’, Refrigeration

Engineering, 64 (1956), 31.

Km

V

W WW W W W

W W

W W

a = a

s s

s

s

˙

ln

2 1

2 1

2

1

--( ) - -( )

-( )-( )

mW W

W W W W

W W

W WK Va

s s

s

s

dad2 1

2 1

--( ) - -( )

-( )-( ) =

d ds s

sWW W

W W W WW W= −

−( ) − −( )−( )2 1

2 1

Km

V

W

W WW

W

a =da

s

˙

-( )Ú1

2



[7] H. J. Lowe and D. G. Christie, ‘Heat transfer and pressure drop in cooling tower packings and modelstudies of the resistance of natural-draft towers to airflow’, Transactions of the ASME (1961),933–950.

[8] D. J. Benton and W. R. Waldrop, ‘Computer simulation of transport phenomena in evaporativecooling towers’, Transactions of the ASME, Journal of Engineering for Gas Turbines and Power,110 (April 1988).

[9] H. Jaber and R. L. Webb, ‘Design of cooling towers by the effectiveness–NTU method’, Transac-tions of the ASME, Journal of Heat Transfer, 111 (November 1989), 837–843.

[10] L. D. Berman, Evaporative Cooling of Circulating Water, 2nd edition (Henrych Stawistowshi,Pergamon Press, 1961), ch. 2, pp. 94–99.

[11] J. Verdu-Estevez and E. Torrella-Alcarez, ‘Theoretical and experimental study of a cooling tower’,Instalador Refrig. FRIAL, 9(32) (1994), 46–54, in Spanish.

[12] S. P. M. Gomes, F. A. L. Azevedo and F. E. M. Saboya, ‘Experimental and theoretical investigationon the performance of a cooling tower’, Proceedings of 6th Brazilian Congress of Engineering andThermal Sciences, (November 11–November 14 1996), pp. 741–746.

[13] M. A. Bernier, ‘Thermal performance of cooling towers’, ASHRAE Journal, 37(4) (1995), 56–61.[14] A. K. M. Mohiuddin and K. Kant, ‘Knowledge base for the systematic design of wet cooling towers.

Part I: Selection and tower characteristics’, International Journal of Refrigeration, 19(1) (1996),43–51.

[15] A. K. M. Mohiuddin and K. Kant, ‘Knowledge base for the systematic design of wet cooling towers.Part II: Fill and other design parameters’, International Journal of Refrigeration, 19(1) (1996),52–60.

[16] G. Gan and S. B. Riffat, ‘Numerical simulation of closed wet cooling towers for chilled ceilingsystems’, Applied Thermal Engineering, 19 (1999), 1279–1296.

[17] L. Lu and W. Cai, ‘A universal engineering model for cooling towers’, Proceedings of the NinthInternational Refrigeration and Air Conditioning Conference at Purdue, paper R20-2 (WestLafayette, 2002).

[18] J. E. Braun, S. A. Klein and J. W. Mitchell, ‘Effectiveness models for cooling towers and coolingcoils’, ASHRAE Transactions, 95(2) (1989), 164–174.

[19] P. G. Hill and R. D. C. MacMillan, ‘The properties of steam: current status’, Transactions of theASME, Journal of Heat Transfer, 119 (August 1988), 763–777.

[20] M. S. P. Gomes, Theoretical–Experimental Analysis of Cooling Towers and Air Washers, MSc thesis,in Portuguese, Department of Mechanical Engineering, Pontifical Catholic University of Rio deJaneiro (September 1988).



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