Aplicación Práctica / Practical Issues

Recibido: 31-10-2023, Aprobado tras revisión: 21-12-2023

Forma sugerida de citación: Simbaña, I.; Quitiaquez, W.; Cabezas, P.; Quitiaquez, P. (2024). “Comparative study of the efficiency

of rectangular and triangular flat plate solar collectors through finite element method”. Revista Técnica “energía”. No. 20, Issue

II, Pp. 81-89

ISSN On-line: 2602-8492 - ISSN Impreso: 1390-5074

Doi: https://doi.org/10.37116/revistaenergia.v20.n2.2024.593

Esta publicación es de acceso abierto bajo una licencia Creative Commons

Comparative Study of the Efficiency of Rectangular and Triangular Flat Plate

Solar Collectors through Finite Element Method

Estudio Comparativo de la Eficiencia de Colectores Solares de Placa Plana

Rectangular y Triangular mediante el Método de Elementos Finitos

I. Simbaña1

0000-0002-3324-3071

W. Quitiaquez2

0000-0001-9430-2082

P. Cabezas2

0000-0001-5114-6913

P. Quitiaquez2

0000-0003-0472-7154

1Instituto Superior Universitario Sucre, Grupo de Investigación en Ingeniería Mecánica y Pedagogía de la Carrera de

Electromecánica (GIIMPCEM), Quito, Ecuador

E-mail: isimbana@tecnologicosucre.edu.ec

2Universidad Politécnica Salesiana, Grupo de Investigación en Ingeniería, Productividad y Simulación Industrial

(GIIPSI), Quito, Ecuador

E-mail: wquitiaquez@ups.edu.ec, ecabezasg@est.ups.edu.ec, rquitiaquez@ups.edu.ec

Abstract

This investigation compared the efficiency of a

flat-plate solar collector with triangular and square

geometry, by using the finite element method (FEM).

The design of the geometries and the utilized

parameters for the simulation were obtained from

previous publications. SolidWorks was used to model

the two collectors, meanwhile, the Fluent module of the

ANSYS software was used for the simulation by the

FEM. Collectors integrated a pipe of 7 mm internal

diameter with a plate thickness of 11 mm; the defined

material was aluminum. The experiment was conducted

under ambient conditions at a temperature of 20 °C,

accompanied by a solar radiation intensity of

1000 W·m-2. The heat transfer surfaces employed were

61 250 mm2 for the triangular collector and

122 500 mm2 for the square collector. The quality of

the mesh was excellent, obtaining a skewness of 0.2486,

with which efficiencies of 62 and 39 % and maximum

temperatures of 27 and 25.5 °C were obtained for the

triangular and square collectors, respectively. Due to

the geometries performing as fins, the temperatures are

higher in the corners and, therefore, achieving higher

efficiency is impossible.

Resumen

En esta investigación, se comparó la eficiencia de un

colector solar de placa plana con geometría triangular y

cuadrada, a través del enfoque del método de elementos

finitos (FEM, por sus siglas en inglés), se llevó a cabo

la simulación. Las geometrías y los parámetros

empleados en la simulación fueron extraídos de

investigaciones previamente publicadas. Se utilizó el

software SolidWorks para el modelado de los dos

colectores, mientras que el módulo Fluent del software

ANSYS fue utilizado para la simulación por el método

FEM. Para los colectores se utilizó una tubería de 7 mm

de diámetro interno, con un espesor de placa de 11 mm

y de aluminio. Se consideró temperatura ambiente de

20 °C, con una radiación solar de 1000 W·m-2

y superficies de transferencia de calor 61 250 y

122 500 mm2 para el colector triangular y cuadrado,

respectivamente. La calidad del mallado fue excelente,

alcanzando skewness de 0.2486. De esta manera, luego

del proceso de simulación se obtuvo eficiencias de 62 y

39 % y temperaturas máximas de 27 y 25.5 °C para el

colector triangular y cuadrado, respectivamente. Debido

a que las geometrías actúan como aletas, las

temperaturas son más altas en las esquinas y por esto no

es posible alcanzar mayor eficiencia.

Index terms— Solar collector, flat-plate, efficiency,

FEM.

Palabras clave— Colector solar, placa plana,

eficiencia, FEM.

Edición No. 20, Issue II, Enero 2024

1. INTRODUCTION

The existing problems of environmental changes

worldwide have been derived from the consumption of

fossil fuels, which is the main dominant in the energy

matrix worldwide [1]. The elevated consumption has led

to a rapid surge in the levels of carbon dioxide (CO2).

This has led to a crisis of distribution, production, and

consumption of energy, causing the scarcity of this fossil

resource [2]. To mitigate the effects of environmental

changes, the International Atomic Energy Agency

(IAEA) and the Kyoto Protocol have proposed the use of

renewable sources, which is accepted by developed

countries [3]. This suggestion has spurred the

development of fresh concepts involving the

implementation of inventive prototypes aimed at

alleviating the repercussions of high-consumer demand

of fossil resources. [4].

Solar energy is the world's most abundant energy

resource, provided by nature providing clean energy [5].

The photovoltaic cell converts solar energy into

electricity through the use of doped semiconductors [6].

Extensive literature exists on the characterization of the

cell or solar cell, as outlined below. The application of

solar collectors has been evolving by proposing new

models, such as Ion et al. [7] presented, a new flat plate

thermal solar collector for the integration of facades. The

numerical analysis considered a triangular geometry

collector, modeling it in SolidWorks as design software

and ANSYS to identify deformation and stagnation

points. Three solar collectors with absorber plates in

black, green, and orange hues were produced, yielding

efficiencies of 55, 42, and 35 %, respectively.

The numerical and experimental investigation of a

triangular storage collector was carried out by Ahmed

[8]. As collectors are suitable devices for use in heating

water, they are applied for domestic and industrial

purposes. Andrade et al. [9] developed a systematic

investigation for hours in the summer and winter seasons

with and without a hot water outlet. Their experimental

results showed that the efficiency during winter and

summer at maximum temperatures was 48.7 % at 65 °C

and 62.2 % at 70 °C, respectively. Moss et al. [10] affirm

that collectors must be designed to be used in cold

conditions, on cloudy days, or with high supply

temperatures, to achieve maximum efficiency. The

authors have deduced that the enhanced efficiency of

solar collectors is attributed to the decrease in internal

pressure, necessitating values below 0.5 Pa.

When comparing the efficiency of a solar collector

based on ANSYS Fluent results with experimental data,

it becomes evident that the inflow into the storage

collector should occur at the bottom of the coldest

section. Simultaneously, the outflow from the storage

collector should be positioned at the top of the collector's

coldest section [11]. By considering the literature review

carried out and that has been taken as a reference, for the

development of this investigation, Table 1 presents a

brief description of the most relevant publications, which

used a flat-plate solar collector.

Table 1: Investigations carried out on flat collectors

Parameter

Efficiency

[%]

Water as

the

working

fluid

Coating

Max.

Temp.

[°C]

Ion et al. [7]

Ahmed [8]

62.2

Moss et al.

[10]

Fathabadi

[12]

74.9

Saffarian et

al. [13]

Müller et

al. [14]

16.5

Fan et al.

[15]

69.4

The aim of this work is the geometric comparison

between triangular and square solar collectors. For this,

the obtained data, the efficiencies, and the maximum

temperatures were contrasted, which can be reached with

the constant solar irradiation flow of 1 000 W·m-2. This

document is organized as described below, Materials and

Methods present the parameters of each collector, as well

as the mathematical modeling necessary to support the

investigation. In Results, the obtained graphs for each

geometry with the simulation software are compared.

Finally, in the Conclusions, the obtained results are

discussed and the presented information is summarized.

2. MATERIALS AND METHODS

2.1 Conceptualization of the design

Solar collectors are mechanisms used to gather,

impregnate, and transfer solar energy to a fluid, which

can be water or air. They can also be utilized to heat

water, for heating systems, or heating swimming pools

[16]. Among the types of renewable energy, solar is very

prominent, due to the use of photovoltaic cells and

collectors to produce electricity and water heating,

respectively, in industrial processes or integrated systems

of homes [17]. Solar collectors are used for various

applications and different utilities, trying to take better

advantage of the use of solar energy. Therefore, a

different geometry of a solar collector is proposed,

considering the design presented by Ion et al. [7]. Fig. 1a

shows a triangular solar collector, to compare the

efficiencies against a square collector, as shown in

Fig. 1b.

The central body consists of an aluminum cavity that

will be located under the absorbent plate of the collector

through which water, as working fluid, will circulate. Fig.

2 shows the cross-section used for both geometrics,

where the absorber plate contained in a box is displayed.

Simbaña et al. / Comparative study of the efficiency of rectangular and triangular flat plate solar collectors through FEM

Figure 1: Solar collector (a) triangular, (b) square

Figure 2: Collector cross-section

The central body consists of an aluminum cavity that

will be located under the absorbent plate of the collector

through which water, as working fluid, will circulate.

Being made of corrosion-resistant aluminum material, it

will allow hot water to be used for any need that the user

requires. When used for the facades of buildings or

houses, coatings must be placed to obtain an architectural

design. For the joining and sealing elements, rubber is

applied between the aluminum cavities and the absorbent

plate [18]. The design needs to use not very thick

absorber plates to take advantage of the heat transfer to

the fluid since thick plates absorb a lot of heat and not all

of this heat is transferred to the fluid [19].

2.2 Solar thermal collector model

The triangular and square collector models were

designed in SolidWorks, resembling the measurements in

the article by Ion et al. [7]. The simulation was carried

out in ANSYS Fluent, in which, the input conditions are

presented in Table 2.

Table 2: Parameters utilized in the simulation

Parameter

Symbol

Units

Value

Tilt angle

Ambient temperature

°C

Absorber plate thermal

conductivity

W·m-1·K-1

237

Specific flow rate

󰇗

kg·m-1·s-1

0.02

Absorber plate

emissivity

εp

0.115

Absorber plate

temperature

°C

20 - 100

Solar irradiation

W·m-2

1000

For the design of the collectors, both triangular and

square, only the geometry sketch was considered. It

allows to parameterize the simulation and obtain the

required results. The velocity was determined in relation

to a specific flow rate of 0.02 kg·m-2·s-1 and irradiance

ranging from 800 to 900 W·m-2 [7]. Under these

conditions, there is a favorable trend towards elevating

the temperature from 20 °C to approximately 60 °C.

In Fig. 3, the schematic of the triangular solar

collector is depicted, with the water inlet marked in blue

at a temperature of 20 °C. The central section of the

collector corresponds to the region where the working

fluid will circulate. At the top, there is the outlet of the

working fluid that has been heated. The proposed square

solar collector also complies with the same schematic

characteristics as the triangular solar collector.

Figure 3: Triangular solar collector scheme

Table 3 details the parameters that both triangular and

square solar collectors share, with similar characteristics.

Table 3: Characteristics of triangular and square solar collectors

Characteristics

Triangular

Square

Length [mm]

350

Height [mm]

350

Pipe diameter [mm]

Transfer area [mm2]

61 250

122 500

Distance of the inlet pipe [mm]

Width [mm]

Material

Aluminum

2.3 Mathematical modeling of the flat-plate solar

collector

In flat-plate solar collectors, regardless of geometry,

the amount of solar radiation absorbed by the collector is

an important factor [12]. The received solar heat (Qin) for

a flat plate collector is expressed with equation (1):

 

(1)

Where Ac is the collector area and G is the solar

irradiation. Equation (2) is employed to calculate the

Edición No. 20, Issue II, Enero 2024

useful heat (Qu) within the solar collector, considering

the circulation of the working fluid within the collector,

as detailed by Diez et al. [20]:

󰇗 󰇛󰇜

(2)

Where 󰇗 is the mass Flow rate, cp is the specific heat

for water, Tout and Tin are the water temperature at the

inlet and outlet, respectively. The global coefficient of

heat loss (UL) is a variable taken into account during the

design phase of solar collectors. Anca et al. [21] have

used equation (3), which expresses the sum of the loss

coefficients:



(3)

The higher loss coefficient (Ut) is obtained in the

glazing due to the convection of heat and radiation that

takes place in the absorber plate and as the radiation loss

from the external glass plate to the environment. Wang et

al. [22] use equation (4):

󰇧 

ℎℎ

ℎℎ󰇨

(4)

The convective heat coefficient (hc,p-c) from the cover

glass and absorber plate can be calculated by Deshmukh

et al. [23] from equation (5):

ℎ 



(5)

Where L is the characteristic length and k is the

thermal conductivity of the material. The Nusselt number

(Nu) can be determined based on the Rayleigh number

and the inclination angle, ranging from 0 to 75° taking

into account its operation in a natural convection [24],

through the application of equation (6):

󰇩󰇛󰇜

 󰇪 



󰇯

 

󰇰

(6)

Where β is the Coefficient of thermal expansion. The

Rayleigh number (Ra) is calculated by Robles-Campos et

al. [25] by applying equation (7):





(7)

Where g is gravity. The radiation coefficient from the

absorber plate to the cover (hc,p-c) is derived using the

equation (8):

ℎ 







(8)

Where ε is the emissivity. For the calculation of the

radiation coefficient of the outer cover (hr,p-c), Gunjo et

al. [26] applied equation (9):

ℎ 

(9)

The background loss coefficient caused by heat

conduction (Ub) is calculated with equation (10):





(10)

Mustafa et al. [27] calculated the border loss

coefficient (Uε) due to heat conduction by using

equation (11):

󰇛󰇜



(11)

To obtain the efficiency of the collectors, the

following relationships are applied [28]. Equation (12)

allows to obtain the thermal power (󰇗) of heat output

from the collector, which is expressed as follows:

󰇗󰇗 󰇛󰇜

(12)

Finally, the relation of the equation (13) has been

used by Xu et al. [29] for the direct calculation of the

efficiency of the solar collector:

󰇗



(13)

2.4 ANSYS Software Equations

2.4.1 Energy

Quitiaquez et al. [30] conceptualize that the most

important and powerful scientific laws are the

conservation laws, in which, energy is a constant that is

conserved in the universe. Therefore, it is declared that

energy is neither created nor destroyed, it is only

transformed. Equation (14) expresses the energy

conversation equation to carry out a steady-state thermal

analysis [31]:



󰇛󰇜

󰇍



󰇛󰇜

󰇟ℎ󰇠

(14)

2.4.2 Energy Radiation

Solar energy represents an essentially limitless source

of power, serving as a foundation for the creation of

projects centered around this renewable alternative. This

type of analysis considers radiative heat transfer, which

is the transfer of thermal energy via electromagnetic

waves. Mohamad et al. [32] articulated the representation

of solar radiation in equation (15):

󰇛󰇛󰇜󰇜󰇛󰇜󰇛󰇜





 󰇛󰇜



󰇛󰇜

(15)

Where  and  representes the position and direction

vectors, a and σs are the absorption scattering coefficients,

respectively. I is the solar intensity, n is the refractive index, T

is the ambient temperature and Ω is the solid angle. And for heat

transfer by radiation, the Stefan-Botlzmann constant (σ) is

5.669 x 10-8 W·m-2·K-4.

2.5 Meshing

The finite element method allows the structural model

to be perfected, creating an iteration process by applying

a level of precision that ensures adequate convergence of

the process. If the guarantee of this process is not

Simbaña et al. / Comparative study of the efficiency of rectangular and triangular flat plate solar collectors through FEM

adequate, the quality of the results in the analysis of the

design will be poor, granting a convergence of the non-

optimal solution [33].

The mesh metric presents the information of nodes

and elements, hence, the selection of a good mesh quality

provides a better accuracy in the convergence of the

solution in the simulation. For an excellent mesh, it is

found in the values of 0 to 0.25 and for a lousy mesh

surrounds the range of 0.98 to 1.00 [34]. Based on the

aforementioned parameters and selecting a method of

dominant tetrahedra, the metric of the triangular and

square solar collector designs is performed by a selection

of the mesh that is approximately in a range of 0.2486.

Fig. 4a and Fig. 4b presents the discretization for the

triangular and square collectors, respectively.

Figure 4: Mesh of solar collectors (a) triangular, (b) square

3. RESULTS

Fig. 5a and Fig. 5b present the temperature contours,

therefore, it is possible to predict the behavior that the

collectors will have throughout the increase in the flow

of solar irradiation. The fluid initiates with an inlet

temperature of approximately 20 °C, and the outlet

temperature is monitored as the fluid exits the collectors.

Figure 5: Heat distribution of solar collectors (a) triangular,

(b) square

In Fig. 6a, the temperature trend graph of the working

fluid originates at 20 °C, tracing the path of the triangular

collector until it reaches an outlet temperature of 25.5 °C.

Similarly, in the square solar collector illustrated in Fig.

6b, the identical trend line is apparent, albeit with the

distinction that the working fluid's outlet temperature is

27 °C. Consequently, when compared to the temperature

of the triangular collector, a notable difference is evident

between the two geometries.

Figure 6: Working fluid temperature along the solar collectors

(a) triangular, (b) square

The trend lines illustrate a proportional relationship

between temperature and distance. Consequently, the

forecast includes the maximum temperature that the

working fluid is anticipated to reach based on the solar

irradiation flux and the heat transfer area of each

collector. The linear correlation coefficient R² serves as

an indicator of the model's fit [35] and in the simulation,

the obtained data exhibits a value of 85 %, signifying the

proportion of variability in temperature increase relative

to the distance covered by the working fluid. This figure

suggests a reasonably acceptable reliability in the

correlation between the data.

The calculation of the efficiency, equations (13) and

(14) are applied, in which the thermal power of the

collectors is calculated. It describes a tendency of the

efficiency as a function of the temperature. Fig. 7a

illustrates the performance of the triangular collector,

depicting the efficiency trend. The highest efficiency is

achieved at a maximum temperature of 62 %, occurring

Edición No. 20, Issue II, Enero 2024

at 25.5 °C. The square collector had an efficiency of

39 % at 27 °C and a cause for this very high variation in

efficiency and temperature could be due to the geometry

of each collector, as shown in Fig. 7b. The heat transfer

areas of the absorber plate of both the square collector

and the triangular collector are wider.

Figure 7: Working fluid temperature along the solar collectors

(a) triangular, (b) square

The designs put forth in this research were developed

by considering the geometry of an actual rectangular

solar collector, which served as the basis for subsequent

modifications. Moldovan et al. [36] conducted CFD

simulations aimed at optimizing the efficiency of a

triangular solar collector, taking layer thickness into

account. They achieved an average efficiency of 55 %,

and with a mass flow of 0.01 kg·s-1 and a 5 mm thickness,

the maximum efficiency reached 66.12 %. Similarly,

Noghrehabadi et al. [37] explored the efficiency of a

square solar collector using both water and a nanofluid

with silicon oxide as working fluids. The incorporation

of nanofluid significantly enhanced the collector's

efficiency, exceeding 50 %, while the use of water

resulted in efficiency variations between 30 and 50 %,

considering variations in mass flow and incident solar

radiation. The efficiency outcomes for the proposed

geometries align with those found in the literature,

validating their credibility. However, further

investigation is warranted to optimize the design, taking

into consideration a more in-depth analysis of other

variables associated with efficiency.

Fig. 8 presents the comparison of the efficiencies

between both solar collectors that start from the same

temperature of 20 °C. The efficiency of the solar

collectors changes as the temperature increases, reaching

a great difference in efficiency of 23 % between the

collectors. The efficiency of the triangular solar collector

closely mirrors the efficiency of the model introduced by

Ion et al. [7], the first one to propose a collector with

these characteristics. A triangular solar collector may

potentially exhibit higher efficiency owing to its capacity

to capture solar radiation from broader angles throughout

the day, particularly at varying latitudes. Moreover, the

triangular geometry can provide a larger surface area for

absorption in a specific orientation when compared to a

square collector. Comprehensive evaluations through

computational simulations and experimental tests are

essential to assess and compare the efficiency of various

solar collector geometries in specific contexts.

Figure 8: Comparison of the efficiencies of the triangular and

square solar collectors

4. CONCLUSIONS

The efficiency comparison of the triangular and

square solar collectors was analyzed. Modeling was

designed in SolidWorks and simulations were carried out

in ANSYS Fluent, obtaining a mesh skewness of 0.2486.

The triangular collector demonstrates notable efficiency

in converting solar radiation into usable heat, with 62 %

of the incident radiation being effectively converted into

thermal energy. In contrast, the square solar collector

exhibits a lower efficiency, signifying a less effective

conversion of only 39% of the incident solar radiation

into thermal energy. Temperature reaches maximum

values for square and triangular solar collectors of 27 and

25.5 °C, respectively. The material utilized for the design

of the solar collectors is aluminum which has a high

thermal conductivity. Finally, the collector behavior was

analyzed with a color scale figure, which shows that

higher temperatures tend to concentrate in the corners of

the collectors. Therefore, a certain amount of heat will be

discarded and for this reason, it is not possible to have

higher levels of efficiency. Attaining an optimal design

that maximizes the capture of solar radiation while

minimizing thermal losses can pose challenges, and the

process of optimization frequently demands a thorough,

application-specific strategy.

20 21 22 23 24 25

Efficiency [%]

Temperature [°C]

Square

Triangle

Simbaña et al. / Comparative study of the efficiency of rectangular and triangular flat plate solar collectors through FEM

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Biografías

Isaac Simbaña.- He was born in

Quito, Ecuador, in 1990.

He received his Mechanical

Engineering degree from the

Universidad Politécnica Salesiana,

in 2018; his Master's Degree

in Mathematical Methods and

Numerical Simulation in

Engineering from the Universidad Politécnica Salesiana

in 2022; his Master's Degree in Education from the

Universidad Politécnica Salesiana, in 2023. He currently

works in the Electromechanical Career at the Instituto

Superior Universitario Sucre. His research fields are

related to Numerical and Statistical Analysis, as well as

Thermodynamics, Manufacturing Processes, Materials

Science, and Educational Innovation.

William Quitiaquez.- He was born

in Quito, Ecuador, in 1988.

He received his Mechanical

Engineering degree from the

Universidad Politécnica Salesiana

in 2011; a Master's in Energy

Management from the Universidad

Técnica de Cotopaxi, in 2015;

Master of Engineering from the Universidad Pontificia

Bolivariana de Medellín, in 2019; Ph.D. in Engineering

from the Universidad Pontificia Bolivariana de Medellín,

in 2022. He is the coordinator of the Productivity and

Industrial Simulation Research Group (GIIPSI) of the

Universidad Politécnica Salesiana. His research field is

related to Renewable Energy Sources, Thermodynamics,

Heat Transfer, and Simulation.

Patricio Cabezas.- He was born in

Quito, Ecuador and received his

degree in Mechanical Engineering

in 2021. He has worked in the

position of QA/QC technician, and

he has Certification as an Inspector

in visual methods and penetrating

inks Level II by the American

Association o Non-Destructive Testing (ASNT) in 2023.

He is a member of the Association for Materials

Protection and Performance (AMPP) and he is certified

under the coatings inspector program as Basic Coating

Inspector Certification Level I (CIP). His areas of interest

are welding and industrial coatings.

Patricio Quitiaquez.- He was born

in Quito in 1969. He received his

Mechanical Engineer degree from

the Escuela Politécnica Nacional in

2002; and his Master's Degree in

Production Management from the

Universidad Técnica de Cotopaxi,

in 2007. His research field is related

to Operations Management, Structural Design,

Manufacturing Processes, and Simulation.