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Engenharia Térmica, nº 4, 2003 p. 30-34

A MATHEMATICAL MODEL FOR DIRECT EVAPORATIVE

COOLING AIR CONDITIONING SYSTEM

J. R. Camargo,

Universidade de Taubaté

Departamento de Engenharia Mecânica

Rua Daniel Danelli, s/n

12060-440; Taubaté; São Paulo; Brasil

rui@mec.unitau.br

C. D. Ebinuma,

Universidade Estadual Paulista – UNESP, FEG

Departamento de Energia

Rua Ariberto Pereira da Cunha, 333

12500-000; Guaratinguetá; São Paulo; Brasil

S. Cardoso,

Universidade de Taubaté

Departamento de Engenharia Mecânica

Rua Daniel Danelli, s/n

12060-440; Taubaté; São Paulo; Brasil

ABSTRACT

Air conditioning systems are responsible for increasing men's work

efficiency as well for his comfort, mainly in the warm periods of the year.

Currently, the most used system is the mechanical vapor compression

system. However, in many cases, evaporative cooling system can be an

economical alternative to replace the conventional system, under several

conditions, or as a pre-cooler in the conventional systems. It leads to a

reduction in the operational cost, comparing with systems using only

mechanical refrigeration. Evaporative cooling operates using induced

processes of heat and mass transfer, where water and air are the working

fluids. It consists in water evaporation, induced by the passage of an air

flow, thus decreasing the air temperature. This paper presents the basic

principles of the evaporative cooling process for human thermal comfort,

the principles of operation for the direct evaporative cooling system and

the mathematical development of the equations of thermal exchanges,

allowing the determination of the effectiveness of saturation. It also

presents some results of experimental tests in a direct evaporative cooler

that take place in the Air Conditioning Laboratory at the University of

Taubaté Mechanical Engineering Department, and the experimental results

are used to determinate the convective heat transfer coefficient and to

compare with the mathematical model.

INTRODUCTION

Evaporative cooling operates using induced

processes of heat and mass transfer, where water and air are

the working fluids. It consists, specifically, in water

evaporation, induced by the passage of an air flow, thus

decreasing the air temperature. When water evaporates into

the air to be cooled, simultaneously humidifying it, that is

called direct evaporative cooling (DEC) and the thermal

process is the adiabatic saturation. When the air to be cooled

is kept separated from the evaporation process, and therefore

is not humidified while it is cooled, it is called indirect

evaporative cooling (IEC). The main characteristic of this

process is the fact that it is more efficient when the

temperatures are higher, that means, when more cooling is

necessary for thermal comfort. It has the additional

attractiveness of low energy consumption and easy

maintenance. Due to use total airflow renewal, it eliminates

the recirculation flow and proliferation of fungi and bacteria,

a constant problem in conventional air conditioning systems.

Several authors dedicated their researches to the

development of direct evaporative cooling systems. Watt

(1963) developed the first serious analyses of direct and

indirect evaporative systems, Leung (1995) presents an

experimental research of the forced convection between an

air flow and an inner surface of a horizontal isosceles

triangular duct, Maclaine-Cross and Banks (1983) developed

equations to model evaporative regenerative heat exchanger,

Halasz (1998) presented

a general dimensionless

mathematical model to describe all evaporative cooling

devices used today (cooling water towers, evaporative

condensers of fluid, air washes, dehumidification coils, etc);

Cardoso, Camargo and Travelho (2000) developed a

research where a thermal balance study for direct and

indirect cooling systems was developed; Camargo and

Ebinuma (2002) presented the principles of operation for

direct and indirect evaporative cooling systems and the

mathematical development of thermal exchanges equations,

Dai and Sumathy (2002) researched a cross-flow direct

evaporative cooler, in which the wet durable honeycomb

paper constitutes the packing material, Liao and Chiu (2002)

developed a compact wind tunnel to simulate evaporative

cooling pad-fan systems and tested two alternative materials.

Al-Sulaiman (2002) evaluated the performance of three

natural fibers (palm fiber, jute and luffa) used as wetted pads

in direct evaporative coolers. Hasan and Sirén (2003)

investigated the performance of two evaporatively heat

exchangers operating under similar conditions of air flow

and inlet water temperatures

This paper develops a mathematical model for direct

evaporative cooling system and presents some experimental

tests results in a direct evaporative cooler that take place in

the Air Conditioning Laboratory at the University of Taubaté

Mechanical Engineering Department, located in the city of

Taubaté, State of São Paulo, Brazil.

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Engenharia Térmica, nº 4, 2003 p. 30-34

J. R. Camargo et al. A Mathematical Model...

DIRECT EVAPORATIVE COOLING

The principle underlying direct evaporative cooling is

the conversion of sensible heat to latent heat. Non-saturated air

is cooled by heat and mass transfer increases by forcing the

movement of air through an enlarged liquid water surface area

for evaporation by utilizing blowers or fans. Some of the

sensible heat of the air is transferred to the water and becomes

latent heat by evaporating some of the water. The latent heat

follows the water vapor and diffuses into the air (Watt and

Brown, 1997).

Figure 1 shows a schematic direct evaporative

cooling system.

Figure 1 – Direct evaporative cooler (DEC)

water decreases the air dry bulb temperature (DBT) and

increases its humidity, keeping the enthalpy constant (adiabatic

cooling) in an ideal process. The minimum temperature that can

be reached is the wet bulb temperature (WBT) of the incoming

air. The effectiveness of this system is defined as the rate

between the real decrease of the DBT and the maximum

theoretical decrease that the DBT could have if the cooling

were 100% efficient and the outlet air were saturated.

Practically, wet porous materials or pads provide a large

water surface in which the air moisture contact is achieved and

the pad is wetted by dripping water onto the upper edge of

vertically mounted pads.

MATHEMATICAL MODEL

In the study of the psychrometric process dry air

is considered as a single gas characterized by an average

molecular mass equal to 28.9645. In this work the humid

air is considered as a mixture of two gases: the dry air and

water vapor.

Considering the flow of humid air close to a wet

surface, according to Fig. (2), the heat transfer will occur if

the surface temperature Ts is different from the draft

temperature T. If the absolute humidity (concentration) of

the air close the surface ws is different from the humidity of

the draft w a mass transfer will also occur.

The elementary sensible heat is

In a DEC, the heat and mass transferred between air and

)TT(dAhQ

scs

???

(1)

where hc is the convective heat transfer coefficient, A is the

area of the heat transfer surface, Ts is the surface

temperature and T is the bulk temperature.

Figure 2 – Schematic direct evaporative cooler

The hc coefficient is determined from the Nusselt

number (Nu) expressed as a function of the Reynolds

number (Re) and Prandtl number (Pr).

In a similar way the rate of water vapour transfer

dmV between the draft and the air close to the surface will

be

)ww( dAh dm

samv

???

(2)

where hm is the mass transfer coefficient by convection and

? is the density of the water.

a

By analysis of the interface air-liquid, the latent

is determined by the energy conservation law.

heat

L

Q

?

v LvssL

dmhQQQ

??????

(3)

where ?Q is the flow of total heat and hLVS is the specific

enthalpy of vaporization of the water at surface

temperature. Rearranging Eqs. (1), (2) and (3), the total

differential heat flow is

hh)TT(h [ Q

Lvsasc

?????

dA] )ww(

sm

?

(4)

Equation (4) indicates that the total heat transfer is

the result of a combination of a portion originating from

temperature difference and other portion originating from the

difference of the absolute humidities. The total heat is caused

by two potentials and these potentials can be combined by the

Lewis relationship so that the total heat flow will be

expressed by a single potential that is the enthalpy difference

between the air close to the wet surface and the air free

current.

Using the specific enthalpy of the mixture as the

sum of the individual enthalpies (Moreira, 1999, p.99) gives

hw()h h (hh

vssa sas

?????

where hVS is the vapor enthalpy at surface temperature, hsa

is the enthalpy of the leaving air, ha is the air enthalpy and

hv is the vapor enthalpy. With the hypothesis that air and

vapor are perfect gases it follows that

w(h)TT(chh

s vss pus

????

)hw

v

(5)

)w

?

(6)

where the humid specific heat is

pvpa pu

cwcc

??

.

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J. R. Camargo et al. A Mathematical Model...

Cpa is the constant pressure specific heat of the air

and Cpv is the constant pressure specific heat of the vapor.

In the standard environmental conditions Cpa = 1,006 kJ/kg

K and Cpv = 1,805 kJ/kg K.

Therefore

ww(h ) hh (

TT

??

pu

s vss

s

c

)

???

(7)

Combining Eq. (4) and Eq. (7) gives

?

?

?

?

?

?

?

?

????

)hR h (

R

)ww(

) hh (

c

dAh

Q

vs Le Lvs

Le

s

s

pu

c

(8)

where RLe is the Lewis relationship, a dimensionless

number expressed as

h

R

?

?

pum

c

Le

Ch

(9)

In the above deduction the density of the humid

air was approximated by the density of the dry air. Taking

the Lewis relationship as

??

Lsvs Lvs

hhh

??

. It is also verified that the term

??

Ls

ww ?

is usually negligible in the presence of

difference of the specific enthalpies ?

the first term inside brackets is significant. In the same

way, the total heat flow is caused by the difference of

specific enthalpies of the air and of the saturated air close

to the wet surface and is given by

dAh

Q

s

pu

The sensible heat transferred is

being unitary, gives

sh

?

hhs?

, so that only

) h

?

h (

c

c

??

(10)

dTcmQ

puas??

(11)

where ma is the air mass flow. Therefore by combining Eq.

(11) with Eq. (1) gives

dTcm)TT( dAh

pu asc

??

(12)

which can be integrated, resulting in

A

T

c

dA

?

0

?

T

?

?

s

pua

2

1

)TT(

dT

cm

h

(13)

The integration yields

?

?

?

?

?

?

?

?

?

?

??

?

?

?

pua

c

s1

21

cm

Ah

exp

TT

TT

1

(14)

The effectiveness of a direct evaporative cooling

equipment is defined as

TT

?

s1

21

TT

?

??

(15)

then

??

?

?

??

?

??

???

puacm

Ah

exp1

(16)

Analysing the Eq. (15) it is verified that an

effectiveness of 100% corresponds to air leaving the

equipment at the wet bulb temperature of entrance. This

requires a combination of large area of heat transfer and a

high heat transfer coefficient and low mass flow.

It is also observed that the effectiveness is

constant if the mass flow is constant since it controls

directly and indirectly the value of the parameters on the

Eq. (16).

EXPERIMENTAL WORK

Experiments were developed during the months of

December/2002, January/2003 and February/2003 in the

Air Conditioning Laboratory at the University of Taubaté

Mechanical Engineering Department, located in the city of

Taubaté, State of São Paulo, Brazil. Performance tests

were carried on an air conditioning device by direct

evaporative cooling, model ECOBRISA20 manufactured

by VIVA Equipamentos Ltda. The evaporative device was

instaled in a room of 6.50m of length, 5.30m of width and

2.90m of height and the following parameters were

monitored: outdoor air humidity and temperature,

evaporative cooler inlet air humidity and temperature,

evaporative cooler outlet air humidity and temperature,

water temperature inside the supplying tubes, water

temperature at the evaporative cooling reservoir,

evaporative pad surface temperature, evaporative cooler

outlet air speed.

Air temperatures and humidity were monitored

using thermal-hygrometers HT-208 model from Gubintec,

with precision of 0.1ºC and 0.1% RH. Evaporative pad

surface and water temperature were monitored by digital

thermometers from SUMMIT, SDT20 CE model and the

market trades SALCAS, SALV TERM 700C model, with

precision of 0.1ºC. Air speed was monitored by a 9 point

measure matrix using portable anemometers type turbine,

PWM model from Dwyer Instruments Inc, USA with

precision of 0.1 m/s.

The ECOBRISA20 utilizes an evaporative pad

with 610x335x152 mm that is composed of a cellulose

material impregnated with a thermosetting resin to prevent

deterioration, shrinkage or

incorporates an internal geometry of transverse alternating

flutes and the arrangement increases cooling efficiency by

causing turbulence while the air is travelling through the

media (GLACIER-COR, 1999) and provides about 400 m2

of evaporative surface area per m3 of media.

sagging. The media

Engenharia Térmica, nº 4, 2003 p. 30-34

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J. R. Camargo et al. A Mathematical Model...

Engenharia Térmica, nº 4, 2003 p. 30-34

correlation to determinate the convective heat transfer

coefficients in a rigid cellulose evaporative media:

?

??

?

where le is the characteristic length and l is the pad

thickness.

?

?

Dowdy and Karabash (1987) presents a

3/1 8 . 0

12 . 0

e

PrRe

l

l

10. 0Nu

??

?

?

?

(17)

A

le

(18)

where ? is the volume occupied by the cellulose media

and A is the total wetted surface area.

The following air properties are used: k=0.0263

W/moC; Pr=0.708; Cpu=1033 J/kgoC and ? =15.7x10-6

m2/s.

The evaporative cooler provides air flow variation

by the of fan rotation. Figure 3 presents a variation of the

cooling effectiveness as a function of air speed at the outlet

face of the evaporative device. This figure shows the curve

plotted from the data presented at the evaporative pads

manufacturer catalogue (Glacier-cor) and the data obtained

from tests.

65

70

75

80

85

90

0,81 1,21,41,61,82 2,2 2,42,62,83 3,2

Air Speed (m/s)

Effectiveness (%)

Glacier-cor

Tests

Figure 3 – Effectiveness x Air Speed

Table 1 shows the resulting convective heat

transfer coefficient for several air velocities calculated

from Eq.(16).

Table 1 - Convective heat transfer coefficient for several

air speeds.

V

[m/s]

[kg/s]

0.96 0.203

1.12 0.233

1.42 0.297

2.02 0.419

2.21 0.458

2.32 0.480

m ?

Re

[1]

hc

[W/m2oC]

153

178

226

322

353

370

35.28

35.49

43.05

57.12

611.32

63.81

60

70

80

90

100

100 200 300400

Re

Effectiveness (%)

Equation 16

Equation 15

Figure 4 – Effectiveness x Reynolds number

Figure 4 shows the comparison between the

effectiveness calculated from the Eq. (15) and Eq. (16) as

function of the Reynolds number.

CONCLUSION

This paper presents a mathematical model for a

direct evaporative cooling air conditioning system that is

obtained by writing the energy conservation equation for

an elementary control volume and analyzing the heat and

mass transfer between the humid air and the water. The

resulting equation allows to determinate the DEC

effectiveness and compares it with the experimental

results, according to Fig. 4 which the curve concerning to

the Eq. 15 was determined by the temperature values

measured in the inlet and outlet air flow and the curve

referring to the Eq. 16 was determined using the mass air

flow and the convective transfer coefficient obtained from

the Eq. 17.

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REFERENCES

Al-Sulaiman, F., 2002, “Evaluation of the Performance of

Local Fibers in Evaporative Cooling”, Energy Conversion &

Management, Elsevier Science Ltd, p. 2267-2273.

Camargo, J. R.; Cardoso, S.; Travelho J. S., 2000,

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Congreso Nacional de Energia, COCIM-CONAE 2000,

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Camargo, J. R.; Ebinuma, C. D., 2002, “A

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J. R. Camargo et al. A Mathematical Model...