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9 

 

 

 

Article 

Theoretical analysis of the performance and 

optimization of indirect flat evaporative coolers  
Hamidreza Asemi1, Rahim Zahedi2*, Sareh Daneshgar3 

1Faculty of Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran 
2Department of Renewable energy and Environmental Engineering, University of Tehran, Tehran, Iran  
3Faculty of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran 

A R T I C L E   I N F O 
 

Article history: 
Received 13 July 2022  
Received in revised form 
16 August 2022 
Accepted 22 August 2022 
 
Keywords: 
Indirect Evaporative Cooling,  
Exergy, Optimization 
 
*Corresponding author 
Email address:  
Rahimzahedi@ut.ac.ir 

 
DOI: 10.55670/fpll.fuen.2.1.2 

A B S T R A C T 
 

External-cooling indirect evaporative coolers with different configurations and 
working air sources are incomprehensively analyzed and compared so far. This 
paper investigates the mechanism and theory of operation of indirect flat-panel 
evaporative coolers based on X-analysis. Then, based on the second law of 
thermodynamics analysis, the entropy production rate of the flat-plate heat 
exchanger of the cooler is calculated. As a result of this analysis, the optimal 
energy efficiency-evaporation efficiency and cooling capacity values are 
presented in terms of effective parameters in the design. 

 

 

 

 

 

 
1. Introduction  

One of the most common methods used to cool 

environments is the passage of hot air over wet surfaces at 

low temperatures, called "evaporative cooling." This method 

is considered compared to mechanical (compression) cooling 

systems due to its cheaper and non-use harmful refrigerant in 

the ozone layer [1]. An essential drawback of this type of 

cooling is the lack of proper control of ambient humidity. This 

form is produced by the indirect evaporative cooling method 

to a large extent, and cooling with the desired quality is 

prepared. Mode of operation as shown in Figure 1, indirect 

evaporative coolers have two primary and secondary 

airflows. The initial air has no contact with moisture and 

enters the desired environment after cooling. Suppose the 

secondary air is in direct contact with the water film. In that 

case, this action causes the water film to evaporate and the 

transformer plates to cool down, eventually removing heat 

from the primary air. The mixture of these two breaths of air 

provides favorable conditions for the environment. 

Rawabawale and Sapali [2] analytically evaluated the 

influence of size variation of the cooling tower on exergy loss 

and found that the cooling tower with a larger length than 

height yields smaller exergy loss with higher thermal 

efficiency. Kiyaninia et al. [3] experimentally and theoretically 

investigated the exergoeconomic performance of a solar 

photovoltaic-based direct evaporative air-cooling system. 

The results showed that for an inlet air with a temperature of 

30oC and relative humidity of 30%, the maximum system 

exergy efficiency was obtained at about 20%. Martineza et al. 

[4] and Nada et al. [5] studied the energy and exergy 

performance of different wet pad materials of evaporative 

coolers through experiments and found that the exergy 

efficiency was between 70% and 94% under different pad 

thicknesses. This research comparatively analyzes the energy 

and exergy performance of indirect flat evaporative coolers 

on the basis of a verified numerical model and experimental 

correlation. 

2. Thermal analysis 

The control volume is considered for thermal analysis of 

the cooler according to Figure 2. It is worth mentioning that 

the analysis of the mass heat transfer process of this type of 

air conditioner is complex. Therefore, in the thermal study, it 

is necessary to consider hypotheses to do the calculations 

quickly. Hypotheses are: 

• Lewis number: 1 = Le 

• Mass heat transfer coefficients are assumed to be 

constant. 

 

 

Future Energy 

Open Access Journal 

https://doi.org/10.55670/fpll.fuen.2.1.2 

 

 

 

 

 

 

 

 

 

 

February 2023| Volume 02 | Issue 01 | Pages 09-14 

Journal homepage: https://fupubco.com/fuen 

 

 

ISSN 2832-0328 

mailto:Rahimzahedi@ut.ac.ir
https://doi.org/10.55670/fpll.fuen.2.1.2
https://fupubco.com/fuen


H Asemi et al. /Future Energy                                                                                                 February 2023| Volume 02 | Issue 01 | Pages 09-14 

10 

 

• The effect of mass heat transfer between secondary air 

droplets is considered to be negligible. 

• Saturation enthalpy changes with humid secondary air 

temperature are linear. 

• The plates are considered completely wet. 

• Heat transfer to the environment is negligible. 

 

 

 

 

 

 

 

 

 

 

 

Figure 1. Isometric view of indirect evaporation cooler 

 

 

Figure 2. Control volume of an indirect evaporative cooler 

 

2.1 Energy balance for primary air 

According to Figure 2, the primary air loses its heat 

significantly, so: 

𝑈0(𝑡 − 𝑡𝑤) = �̇�𝑃𝑐𝑃 ⅆ𝑡                                            (1) 

As a result, after integration, the number of primary air units 

using X-theory is as follows [6]. 

(𝑁𝑇𝑈)𝑝 =
𝑈0𝐴

𝑚𝑃𝑐𝑃 
=  −ln (

𝑡2−𝑡1

𝑡1−𝑡𝑤
)                                  (2) 

The initial air impact factor will be as follows. 

𝜀𝑃 = 1 − 𝑒𝑥𝑝(𝑁𝑇𝑈)𝜌 =
𝑡1−𝑡2

𝑡1−𝑡𝑤
             (3) 

2.2 Energy balance for secondary air 

In Figure 2, a differential element from the air contact 

surface shows a wet surface. Thermal balance for the earth 

element from [7]: 

ℎ𝐷(ℎ𝑤 − ℎ) ⅆ𝐴 = �̇�𝑠 ⅆℎ              (4) 

By integrating equation (5) 

(𝑁𝑇𝑈)𝑠 =
ℎ𝐷𝐴

�̇�𝑠
=  

ℎ𝐷𝑐𝑃𝐴

�̇�𝑠𝑐𝑃
=  −ln (

ℎ0−ℎ𝑤

ℎ0−ℎ𝑖
)             (5) 

After arranging the above equation, the coefficient of the 

effect of the secondary air is as follows: 

𝜀𝑠 = 1 − 𝑒𝑥𝑝(−𝑁𝑇𝑈)𝑠 =
ℎ𝑖−ℎ0

ℎ𝑖−ℎ𝑤
             (6) 

According to the linear proportion of changes in the enthalpy 

of humid air with the ratio of temperatures [8]: 

𝑐𝑤𝑏 =
ℎ0−ℎ𝑖

𝑡𝑤0− 𝑡𝑤𝑖
               (7) 

Cwb is the specific saturation temperature of equation (7) as 

follows. 

𝜀𝑠 = 1 − 𝑒𝑥𝑝(−𝑁𝑇𝑈)𝑠 =
𝑡𝑤0− 𝑡𝑤𝑖

𝑡𝑤𝑖− 𝑡𝑤 
             (8) 

Assuming 1 = Le (i.e., assuming that the rate of penetration of 

the rate of heat transfer and mass diffusion are the same at 

the contact surface of air-water): 

𝐿𝑒2∕3 =
ℎ𝐶

ℎ𝐷𝐶𝜌
→ ℎ𝐶 = 𝐶𝜌ℎ𝐷             (9) 

With replacement (9) in (5): 

(𝑁𝑇𝑈)𝑠 =
ℎ𝑐𝐴

�̇�𝑠𝑐𝑃
=  −ln (

𝑡𝑤0− 𝑡𝑤 

𝑡𝑤𝑖− 𝑡𝑤 
)           (10) 

2.3.    Energy balance between the primary and secondary 

air 

By defining the coefficient of cooling effect in indirect 

evaporative coolers [9]: 

𝜀𝑠 =  
𝑡1−𝑡2

𝑡1− 𝑡𝑤𝑖
              (11) 

The energy balance equation between the primary and 

secondary air will follow: 

�̇�𝑝𝑐𝑃(𝑡1 − 𝑡2) = �̇�𝑠 (ℎ𝑂 − ℎ𝑖)            (12) 

Incidentally, according to equation (7) 

𝑡2 = 𝑡1 −
𝑐𝑚𝑎𝑥

𝑐𝑚𝑖𝑛
(𝑡𝑤0 − 𝑡𝑤𝑖)           (13) 

While Cmin and Cmax are: 

𝑐𝑚𝑖𝑛 = �̇�𝑝𝑐𝑃               (14) 

𝑐𝑚𝑎𝑥 = �̇�𝑠𝑐𝑤𝑏             (15) 

By placing equation (13) in equation (2): 



H Asemi et al. /Future Energy                                                                                                 February 2023| Volume 02 | Issue 01 | Pages 09-14 

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𝜀𝑃 =  
𝑐𝑚𝑎𝑥

𝑐𝑚𝑖𝑛
(

𝑡𝑤0− 𝑡𝑤𝑖

𝑡1− 𝑡𝑤 
)             (16) 

 By placing equation (6) in equation (5), water temperature 

𝑡𝑤 =
𝜀𝑠(

𝑐𝑚𝑎𝑥
𝑐𝑚𝑖𝑛

)+𝜀𝑃+𝑡1

𝜀𝑠(
𝑐𝑚𝑎𝑥
𝑐𝑚𝑖𝑛

)+𝜀𝑃

            (17) 

By placing equation (17) in equation (2), the coefficient of the 

cooling effect of the cooler is: 

𝜀𝐶 =
1

1

𝜀𝑃
+

1

𝜀𝑆
(

�̇�𝑝𝑐𝑃

�̇�𝑠𝑐𝑤𝑏
)
            (18) 

In addition, to calculate the heat transfer coefficients of the 

primary and secondary air passages and the sticky surface, 

respectively [10] are: 

𝑁𝑢 = 0.023𝑅𝑒𝐷ℎ
4/5𝑃𝑟1/2    𝑅𝑒 > 2300          (19) 

𝑁𝑢 = 7.54     𝑅𝑒 ≤ 2300            (20) 

Where:  𝑅𝑒𝐷ℎ =
4�̇�

𝛱𝐷𝜇
  , 𝐷ℎ = 2𝑏 

2.4. Calculate the drop in air flux by multiplying the 

capacity of the air conditioner 

The pressure drop of the whole system due to local 

friction drops is calculated according to the following 

equations [11]. 

𝛥𝑃𝑓 = 𝑓
𝐿

𝐷ℎ

𝑣2𝜌

2
             (21) 

𝛥𝑃𝑙 = ∑ 𝐾
𝑣2𝜌

2
             (22) 

𝛥𝑃𝑡 =  𝛥𝑃𝑓 + 𝛥𝑃𝑙           (23) 

 As a result, the required power of the device blower, 

assuming ηm =1, is [12]: 

𝑤 =
�̇�𝑃𝛥𝑃𝑃

𝜂𝑃
+

�̇�𝑠𝛥𝑃𝑠

𝜂𝑠
           (24) 

2.5.    Calculating the energy efficiency ratio 

The energy efficiency ratio of EER is equal to 

𝐸𝐸𝑅 =
𝑄𝑐

𝑤
             (25) 

Where Qc is the cooling capacity of the indirect air conditioner 

can be calculated from the following equation. 

𝑄𝑐 = �̇�𝑝𝑐𝑃(𝑡1 − 𝑡2)           (26) 

2.6. Analysis of the second law for indirect plate 

evaporative coolers 

There are several evaluation criteria for the optimal 

operating conditions of thermal systems. One of the most 

reliable criteria is the second law analysis or exergy analysis. 

Figure 2 shows that an indirect plate air conditioner is like a 

plate heat exchanger with the opposite flow for secondary 

airflow and cross-flow for primary air. In order to obtain a 

statement to calculate the initial airflow exergy rate, we 

assume that the surface temperature of the plates is constant 

throughout the length of the plate, which is a reasonable 

assumption for this type of cooler. The rate of production of 

primary exergies of such plates is as follows, which includes 

two parts: output and output [13, 14]. 

(𝑒𝑡,𝑝)
𝑖𝑛

=  𝑅𝑎𝑡1ln (1 + 1.6)𝜔𝑜           (27) 

(𝑒𝑡,𝑝)
𝑜𝑢𝑡

= 𝑐𝑃𝑡0 (
𝑡2

𝑡0
− 1 − 𝑙𝑛

𝑡2

𝑡0
) + 𝑅𝑎𝑡0 ln (1 −

∆𝑃

𝑃0
) +

𝑅𝑎𝑡𝑜ln (1 + 1.6)𝜔𝑜            (28) 

Moreover, entropy production for secondary air is in 

equations (29) and (30). In addition to heat transfer and 

friction, it also has the mass transfer term due to evaporation, 

which consists of input and output. 

(𝑒𝑡,𝑆)
𝑖𝑛

= (𝑐𝑃 + 𝜔𝑐𝑃𝑣)𝑡0 (
𝑡𝑊𝑖

𝑡0
− 1 − 𝑙𝑛

𝑡𝑊𝑖

𝑡0
) + (1 +

1.6𝜔𝑜)𝑅𝑎𝑡0 ln (
𝑃

𝑃0
) + 𝑅𝑎𝑡0[(1 + 1.6𝜔𝑜)𝑙𝑛

1+1.6𝜔𝑜

1+1.6𝜔 
+

1.6𝜔𝑜 ln
𝜔

𝜔0
  ]            (29) 

(𝑒𝑡,𝑆)
𝑜𝑢𝑡

= (𝑐𝑃 + 𝜔𝑜𝑢𝑡𝑐𝑃,𝑣)𝑡0 (
𝑡𝑊𝑜

𝑡0
− 1 − 𝑙𝑛

𝑡𝑊𝑜

𝑡0
) +(1 +

1.6𝜔𝑜𝑢𝑡) 𝑅𝑎𝑡0 ln (1 −
∆𝑃

𝑃0
) + 𝑅𝑎𝑡0[(1 + 1.6𝜔𝑜𝑢𝑡)𝑙𝑛

1+1.6𝜔𝑜

1+1.6𝜔𝑜𝑢𝑡 

+

1.6𝜔𝑜𝑢𝑡 ln
𝜔𝑜𝑢𝑡

𝜔𝑜
  ]           (30) 

As a result, the total entropy production rate is calculated as 

follows: 

𝑠𝑔𝑒𝑛 =
1

𝑇0
{[�̇�𝑃𝑒𝑡𝑝 + �̇�𝑠𝑒𝑡𝑠]

𝑖𝑛
−[�̇�𝑃𝑒𝑡𝑝 + �̇�𝑠𝑒𝑡𝑠]

𝑜𝑢𝑡
}         (31) 

Based on the entropy production equations (27 to 31), the 

design method of indirect coolers mentioned in the first part 

of the article, the functional relationship of the total entropy 

production rate is as follows: 

�̇�𝑔𝑒𝑛 = 𝑓(�̇�𝑝, �̇�𝑠, 𝐿, 𝑤)          (32) 

3. Results and Discussion 

A computer code has been prepared to calculate the 

design values such as the temperature of the air leaving the 

air conditioner, the energy efficiency ratio, and to observe the 

effect of effective parameters such as plate length and mass 

flow rate on the EER according to Figure 3. Figures 3 and 4 

result from these calculations. As shown in Figure 4, the initial 

outlet air temperature of the T2 model was compared with 

the experimental reference results [4]. The difference 

between the modeling results and the experimental results is 

about 3%, and this difference is due to ignoring the effects of 

environmental parameters in modeling. Figure 5 compares 

the cooling effect of X with the experimental results [4]. After 

designing the system according to equations (27) to (32), the 

entropy produced by the system is investigated. It is evident 

that the system, based on the design conditions, is faced with 

limitations in selecting the values of W and L. Due to these 

limitations, the mentioned values must be considered in a 

specific range. The variables are examined by keeping the 

other parameters constant on the entropy production rate of 

the system, and finally, the optimal values of the system 

design are obtained. It is worth noting that the optimization 

of design variables by the exergy method (the second law of 

thermodynamics) is not economically optimal for systems. 

Therefore, in practice, in order to optimize the systems, both 

thermo economic studies should be done.  



H Asemi et al. /Future Energy                                                                                                 February 2023| Volume 02 | Issue 01 | Pages 09-14 

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Figure 6 shows the entropy production rate in terms of 

the secondary air flow rate changes at a constant value of the 

primary airflow rate. The design shows that the maximum 

and minimum point distance values are minimal. Figure 7 

shows the entropy production rate of initial airflow. As shown 

in this figure, the maximum and minimum entropy 

production points are not within the design range. This trend 

indicates the low sensitivity of the entropy production rate to 

the initial air flow rate.  

 

Figure 4. Comparison of the initial air outlet temperature of 

the model with experimental results 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5. Comparison of the cooling coefficient of the model 

air conditioner with the experimental results 

Figures 8 and Figure 9 show the entropy production rate 

in secondary air discharge in different primary discharges 

and the entropy production rate in secondary air discharge in 

different primary discharges, respectively. As can be seen, at 

a certa in amount of primary or secondary air flow rate, the 

minimum entropy production can be achieved by selecting 

another reasonable flow rate. Figure 10 also shows the 

entropy production rate in terms of plate length. It is 

observed that the longer the plate lengths are selected in this 

type of short coolers, the more the entropy production rate 

Figure 3. Flowchart design of direct evaporative coolers 



H Asemi et al. /Future Energy                                                                                                 February 2023| Volume 02 | Issue 01 | Pages 09-14 

13 

 

increases. As a result, the length of the pages should be 

selected by the primary and secondary discharges for 

maximum efficiency. 

    

Figure 6. Entropy production rate in terms of secondary air 

flow 

 

    Figure 7. Entropy production rate in terms of initial air 

flow 

   

Figure 8. Entropy production rate in terms of primary air 

discharge in secondary discharges 

 

 

 

 

 

  Figure 9. Entropy production rate in terms of secondary air 

flow in primary flows 

 

    Figure 10. Entropy production rate in terms of plate 

length in secondary discharges 

4. Conclusions 

In this study, the energy and exergy performance of 

cooling indirect flat evaporative coolers and direct 

evaporative coolers are compared and analyzed on the basis 

of the developed mathematical model. The main conclusions 

are given as follows: 

1) Indirect evaporative coolers have better performance 

than direct evaporative coolers in dry areas. 

2) Exergy analysis is an excellent tool to optimize the 

parameters affecting the performance of indirect 

evaporators. The increased fresh air flowrate reduces the 

cooling efficiency and exergy efficiency of all evaporative 

coolers. 

Ethical issue 

The authors are aware of and comply with best practices 
in publication ethics, specifically with regard to authorship 
(avoidance of guest authorship), dual submission, 
manipulation of figures, competing interests, and 
compliance with policies on research ethics. The authors 
adhere to publication requirements that the submitted work 
is original and has not been published elsewhere in any 
language. 

Data availability statement 
Data sharing is not applicable to this article as no 

datasets were generated or analyzed during the current 
study. 

Conflict of interest 
The authors declare no potential conflict of interest. 

 



H Asemi et al. /Future Energy                                                                                                 February 2023| Volume 02 | Issue 01 | Pages 09-14 

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Abbreviations and Greek symbols 

A Effective heat transfer surface (m2) 
B The distance between the two cooler plates (m) 
Cmax Maximum heat transfer capacity(W/oC) 
Cmin Minimum heat transfer capacity(W/oC) 
Cp Specific heat capacity to air (KJ/Kg oC) 
Cpv Specific heat capacity of water vapor (KJ/Kg oC) 
Dh Hydraulic diameter (m) 
et,p Primary air exergy (W/KgK) 
et,s Secondary air exergy (W/KgK) 
hi Enthalpy of secondary air inlet (KJ/Kg) 
ho Enthalpy of secondary air outlet (KJ/Kg) 
hw Enthalpy of air saturation at the common air and 

water level (KJ/Kg) 

K Local drop coefficient 
L The length of the plates of indirect evaporators 

(m) 
Le Louis Number 
�̇�𝑎 Air mass flow (Kg/s) 
�̇�𝑝 Primary air flow (Kg/s) 

�̇�𝑠 Secondary air flow (Kg/s) 
EER Energy efficiency ratio  
f Friction coefficient  
hc Conduction heat transfer coefficient (W/m2oC) 
hD Mass transfer coefficient (Kg/m2s) 
Nu Nusselt number  
𝛥𝑃𝑓 Pressure drop due to friction (Pa) 

𝛥𝑃𝑙 Local pressure drop (Pa) 
𝛥𝑃𝑝 Initial pressure drop 

𝛥𝑃𝑠 Secondary pressure drop 
Pr Prandt number 
Re Reynolds number 
Sgen Production entropy 
t1 Dry bubble temperature of the primary air inlet 
t2 Dry bubble temperature of the primary air outlet 
twi Wet bubble temperature of secondary air inlet 
two Wet bubble temperature of secondary air outlet 
tw Saturation temperature of the joint surface of air 

and water 
to Ambient temperature 
ṁw Mass flow of water 
NTUp Number of primary air transfer units 
NTUs Number of secondary air transmission units 
U0 The total heat transfer coefficient between the 

primary air and the common surface of the 
secondary air of water 

𝜀𝐶  Cooler effect coefficient 
𝜀𝑝 Primary air impact coefficient 

𝜀𝑠 Secondary air impact coefficient 
ρ Air density 

𝜂𝑝 Fan efficiency of the first part 

𝜂𝑠 Secondary fan efficiency 
𝜇 Adhesion coefficient 

ω𝑜𝑢𝑡  Humidity for secondary exhaust air 
ω0 Humidity for the environment 

 

 

 

 

 

 

 

 

 

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 This article is an open-access article 

distributed under the terms and conditions of the Creative 

Commons Attribution (CC BY) license 

(https://creativecommons.org/licenses/by/4.0/). 


