Accepted Manuscript
Parametric investigation of thermal characteristic in trapezoidal cavity receiver for a
linear Fresnel solar collector concentrator
Soroush Dabiri, Erfan Khodabandeh, Alireza Khoeini Poorfar, Ramin Mashayekhi,
Davood Toghraie, Seyed Ali Abadian Zade
PII:
S0360-5442(18)30629-7
DOI:
10.1016/j.energy.2018.04.025
Reference:
EGY 12664
To appear in:
Energy
Received Date: 28 November 2017
Revised Date:
16 March 2018
Accepted Date: 6 April 2018
Please cite this article as: Dabiri S, Khodabandeh E, Poorfar AK, Mashayekhi R, Toghraie D, Abadian
Zade SA, Parametric investigation of thermal characteristic in trapezoidal cavity receiver for a linear
Fresnel solar collector concentrator, Energy (2018), doi: 10.1016/j.energy.2018.04.025.
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to
our customers we are providing this early version of the manuscript. The manuscript will undergo
copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please
note that during the production process errors may be discovered which could affect the content, and all
legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Parametric investigation of thermal characteristic in
2
trapezoidal cavity receiver for a linear Fresnel solar
3
collector concentrator
RI
PT
1
SC
4
Soroush Dabiria, Erfan Khodabandehb, Alireza Khoeini Poorfarc, Ramin Mashayekhid, Davood
6
Toghraiee, *, Seyed Ali Abadian Zadef
M
AN
U
5
7
a
8
Environmental Engineering Dept., University of Tehran, 16th Azar St., Enghelab Sq., P.O. Box 14155-6619,
9
Tehran, Iran
b
10
Mechanical Engineering Dept., Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Avenue,
11
TE
D
c
Combustion Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology
13
(IUST), Narmak, 16846-13114 Tehran, Iran
d
14
15
e*
Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran
Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran,
16
18
19
20
f
Toghraee@iaukhsh.ac.ir
Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran
AC
C
17
EP
12
P.O. Box 15875-4413, Tehran, Iran,
21
22
23
1
ACCEPTED MANUSCRIPT
Abstract
2
Solar collectors—especially Linear Fresnel Reflectors—are one of the main implements to
3
utilize the energy of sun light. The receiver cavity of the collector is recognized as an important
4
component of each concentrating solar power plant. In this study, the heat transfer rate and heat
5
loss in a trapezoidal cavity of the linear Fresnel reflector are analyzed. The calculations are
6
performed for steady-state, laminar model in which temperature-dependent density is assumed
7
for the air inside the cavity. The effects of the cavity angle and the effect of the tube size are
8
evaluated, in various models. DTRM radiation model is employed for the simulation, while
9
considering radiation, conduction and convection heat transfers are as the boundary conditions.
10
Finally, it will be observed that by increasing the cavity angle, the total value for heat transfer
11
rate is increased, but the heat absorbed by each tube is decreased. The results also show that the
12
tube size is less effective on heat transfer rate compared to the cavity angle.
13
Keywords: Trapezoidal cavity receiver, Linear Fresnel reflector, Computational fluid dynamics,
14
Discrete transfer radiation model.
15
TE
D
M
AN
U
SC
RI
PT
1
1. Introduction
17
Today, the extravagant consumption of the limited fossil fuel resources has resulted in the global
18
warming through emission of greenhouse gases. Therefore, studies are conducted and laws are
19
legislated so as to substitute renewable energy sources—especially solar energy—for coal, oil
20
and natural gas resources during the last decades [1, 2]. There are different ways to benefit from
21
the sun energy, and convert it to either thermal or electrical energy, such as photovoltaic (PV)
22
Panels, concentrating solar power (CSP) plants, solar cookers, etc. [3].
AC
C
EP
16
2
ACCEPTED MANUSCRIPT
Due to considerable ongoing enhance in energy consumption, renewable energy technologies
2
should be developed to play a more important role in the expanding energy market. In this
3
regard, CSPs are expected to supply up to 10% of the demand until 2050 [4]. The most
4
developed types of CSPs consist of linear Fresnel reflectors (LFRs), parabolic trough collectors
5
(PTCs), parabolic dish reflectors (PDRs) and heliostat field collectors (HFCs) [5].
6
Linear Fresnel collectors are one of two viable line-focus CSP technologies, along with the
7
parabolic trough [6]. Although the PTC technology can be considered as the most mature type of
8
CSPs [7], it requires high investment cost. Hence, in order to reduce manufacturing costs, the
9
LFR technology is supposed to be a promising application for utilizing the sun energy, which is
10
due to using flat or slightly curved reflectors [8]. Additionally, linear Fresnel is mounted close to
11
the ground, that cause the structural requirements will be minimized. Also, in some cases, using
12
conventional glass on the bottom side of the cavity receiver, and minimal operation and
13
maintenance costs are the other advantages of LFRs. However, at the same conditions, the LFRs
14
have a lower efficiency compared with PTCs [9]; therefore, various aspects affecting the
15
efficiency of the LFRs should be considered in order to improve the efficiency of these plants.
16
The linear Fresnel reflector is composed of many mirrors that focus beams of light to the central
17
receiver, where the absorbing tubes surrounded by secondary reflector are located. The receiver
18
is the crucial part of plant [10]. It absorbs the concentrated solar radiation and transfers it into a
19
heat transfer fluid, which depending on whose type can be used either directly, or indirectly to
20
run a thermodynamic power cycle [11].
21
There are different designs for LFRs. In the basic design, there is just one absorber tube;
22
however, there are new designs, which hold on two or more tubes in the receiver [12]. The
23
simple form of Fresnel reflector is illustrated in Fig. 1.
AC
C
EP
TE
D
M
AN
U
SC
RI
PT
1
3
ACCEPTED MANUSCRIPT
Totally, the heat transferred to the medium depends on receiver geometry, ambient conditions,
2
optical and material properties of the receiver, etc. Therefore, in order to increase the efficiency
3
and performance of LFRs, studies are conducted, investigating the different aspects of LFR
4
plants which affect the efficiency. The first Linear Fresnel collectors was designed by Giorgio
5
Francia in Genoa, Italy in the 60s [13]. Then, it was developed by other research laboratories and
6
companies.
7
Various sketches of collector receiver configurations are designed and evaluated by the
8
researches such as; one-horizontal receiver in the middle of each array [14], two-tilted receivers
9
in the middle of array [15] and receivers in both sides of the array [16].
M
AN
U
SC
RI
PT
1
In some cases, researchers applied secondary reflector to the receiver and mentioned its effects
11
on efficiency [17]. However, Abbas et al. [18] analyzed the different types of receiver
12
configurations and stated that a multi-pipe cavity receiver without any secondary reflectors is
13
appropriate to be used in an LFR plant.
14
Furthermore, there are studies investigating the dimensions of the trapezoidal receivers in LFRs.
15
Lin et al. [19] based on Monte Carlo ray tracing method applied a V-shaped receiver cavity in a
16
LFR system. Then, they analyzed and optimized different variables such as; time, the width of
17
the cavity, and the temperatures of the fluid and ambient to enhance the efficiency. Lai et al. [20]
18
by using CFD method, analyzed ambient temperature, absorber temperature, cavity depth,
19
insulation thickness, glass window and the emissivity of the selective absorption coating,
20
affecting the performance of the receiver. Facão and Oliveira [21] with simplified ray-trace
21
simulation investigated the number of the receiver absorber tubes and the inclination of the
22
lateral walls. They employed CFD to achieve the optimum depth for the receiver cavity and the
23
optimum thickness for the insulation layer. Qiu et al. [22] analyzed thermal and optical
AC
C
EP
TE
D
10
4
ACCEPTED MANUSCRIPT
characteristics of a LFR via both Monte Carlo Ray Tracing and CFD approaches. By the realistic
2
LFR application, their validation indicated that their model is proper for simulation of the LFR.
3
They evaluated the circumferential temperature, and the temperature of the fluid. Reddy and
4
Kumar [23] based on non-Boussinesq approximations probed into different geometric parameters
5
in order to analyze heat transfer via radiation, convection and conduction at the receiver cavity.
6
The obtained results showed that the total convective heat transfer decreased by 12.76% and total
7
radiative heat transfer increased by 54%, once the receiver width varied from 300 mm to 800
8
mm at a certain receiver temperature and wind velocity. They also described how the heat
9
transfer through the different parts of the receiver occurs and showed that the optimum
10
magnitudes for the insulation thickness, cavity depth and aspect ratio are 300 mm, 300 mm and
11
2, respectively.
12
Manikumar et al. [24] modeled a two-dimensional receiver cavity of the LFR, utilizing steady
13
state, laminar heat transfer model in which Boussinesq density model was employed. The CFD
14
codes were performed by ANSYS Fluent. Quantifying heat transfer rate and heat loss in the
15
cavity, they evaluated the location of Copper absorber tubes and showed the significance of the
16
surface coating. Moreover, the cavity angle has been analyzed by Saxena et al. [25] who carried
17
out a CFD method for two cavity angles, and presented some correlations related to Nusselt and
18
Grashof numbers. Also Natarjan et al. [26] studied on the effects of cavity angle, as well as
19
Grashof number, surface emissivities and temperature ratio, and presented a correlation for both
20
the natural convection and surface Nusselt number.
21
However, a review on the relevant literature shows that the authors investigating on the cavity
22
angle and other geometrical aspects of the trapezoidal receiver cavities, including Saxena et al.
AC
C
EP
TE
D
M
AN
U
SC
RI
PT
1
5
ACCEPTED MANUSCRIPT
[25] and Natarjan et al. [26], didn’t deal with the number of tubes and tube sizes. They assumed
2
absorber plates rather than absorber tubes, while analyzing cavity angle.
3
The novelty of this paper is that the number of tubes is evaluated, changing the angles, while the
4
downer edge of the trapezoidal cavity is constant. In this study, the aim is to analyze the effects
5
of the cavity angle and the tube size on the total and radiation heat transfer rate through the
6
absorber tubes, by employing CFD method. In the considered case-study, the heat transfer rate
7
absorbed by the tubes of a prototype cavity receiver is evaluated for different angles of the
8
cavity. Additionally, the simulations are repeated for different tube sizes to observe the effect of
9
tube size on the heat transfer rate and heat transfer, and Finally, correlations for heat transfer
10
coefficient to the tubes, from within the cavity, as a function of the cavity angle and tube
11
dimension, is presented, respectively.
12
2. Material and Method
SC
M
AN
U
2.1. Procedure and assumptions
TE
D
13
RI
PT
1
In the present research, the heat transfer from within the cavity domain to the absorber tubes, and
15
the heat loss, as the heat dissipation through the glass cover and insulation walls is calculated,
16
while changing the cavity angle and the tube size in the receiver.
17
Since the process of heat transfer in solar receivers is a complicated simulation including
18
convection, radiation and conduction, the simplifying assumptions are applied, as follows:
AC
C
EP
14
•
Gravity acceleration is considered as 9.81 ms-2.
•
Heat transfer and flow in modeled as laminar, steady state flow.
21
•
All tubes are assumed isothermal.
22
•
Boussinesq approximation is employed for the air within the cavity, since the
19
20
23
variation of temperature, as well as the variations of density, is not high, in the model.
6
ACCEPTED MANUSCRIPT
1
•
The absorber surface is covered by ordinary black coating according to [24].
2
•
The effect of the heat absorbed by the glass aperture is neglected.
3
2.2. Geometry
The cavity receiver is modeled trapezoidal due to proper insulation causing to reduce heat
5
dissipation [26]. The 3D receiver trapezoidal cavity of this research is illustrated by Fig. 2. The
6
rays reflected from a 40-mirror field located along north-south direction. Each mirror is single-
7
axis sun tracking, and according to [24] 0.1 m long and 0.4 m wide. The medium fluid crosses
8
the copper absorber tubes situated at the upper surface of the receiver. However, upper section of
9
the tubes, where the fluid is flown, is disregarded to reduce the number of the calculations. Also,
10
there is a glass cover situated at the aperture of the receiver to reduce heat dissipation due to
11
external wind conditions, by enhancing the greenhouse effect to trap more heat in the receiver
12
cavity. Indeed, the beams are radiated to the inside of the receiver, crossing the covering glass of
13
the cavity aperture. As shown in Fig. 3, the cavity geometry is designed in 2D shape, due to
14
remarkable computation time and grid numbers. Imposing the two-dimensional geometry to the
15
simulation provides almost the same results as the three-dimensional analysis, because the cavity
16
absorber covers the full length of the reflector.
17
In this article, two parameters are analyzed to see the effects of each one on the heat transfer; the
18
cavity angle and the tube size. While analyzing the effect of the cavity angle (ɸ), the outer
19
diameter of each tube (D) is considered to be 1/2. In addition, while analyzing the effect of tube
20
size (D), the angle of the cavity (ɸ) is fixed at 50°. First, the cavity angle is fixed as 40°, 45°, 50°,
21
55° or 60°, and the flow and energy equations are solved to evaluate the effect of the angle of
22
cavity on both the heat transfer to the absorber tubes, and heat loss through the glass cover and
23
wall insulation. Similarly, to observe the effect of tube sizes on the heat transfer and heat loss,
AC
C
EP
TE
D
M
AN
U
SC
RI
PT
4
7
ACCEPTED MANUSCRIPT
the tube size is considered as 1/4 in, 3/8 in, 1/2 in, 5/8 in or 3/4 in. As the tube size increases, the
2
tubes number decreases, while fixing the cavity angle at 50°. Table. 1 depicts the dimensional
3
properties of the designed models according the parameters illustrated by Fig. 2. In addition,
4
material properties of the models are summarized by Table. 2. The employed materials are
5
selected depending on a previous research by Manikumar et al. [24].
6
RI
PT
1
2.3. CFD properties and Governing Equations
Because of the high cost of the experimental platforms, the computation fluid dynamics (CFD)
8
has emerged as an effective approach to predict the thermodynamic behavior of the power plants
9
components. The present CFD model, is carried out according to flow and heat transfer equations
M
AN
U
SC
7
10
describing continuity, momentum and energy in the system. The formulations are as follows:
11
The Continuity equation:
is the velocity of the fluid in the control volume.
Where is the density, 13
The momentum equation:
EP
12
= −∇p + ∇. (̿) + ∇. (2)
Where p is the static pressure and ̿ is the stress tensor, given by:
AC
C
14
(1)
TE
D
) = 0
∇(
2
+ ∇
− ∇. ]
̿ = [∇
3
(3)
15
Where denotes the unit tensor and is the viscosity.
16
The energy equation:
= ∇. (∇)
∇. (4)
8
ACCEPTED MANUSCRIPT
Where and k represent thermal conductivity and specific heat, respectively.
2
The utilized model to simulate radiation heat transfer is the Discrete Transfer Radiation Model,
3
also named as DTRM Method. It is employed due to its simplicity so as it estimates the heat
4
transfer inside the cavity by tracing rays [27]. In the DTRM model, the equation for the change
5
of radiant intensity, , along a path, , written as below:
(5)
SC
!" #
+ ! =
$
RI
PT
1
Where ! is the gas absorption coefficient, is intensity, is the local temperature of the gas and
7
" is the Stefan-Boltzmann constant (5.67 × 10*+ W/m/ K# ). The Boussinesq model is employed
8
to estimate the density of the trapped air. According to this method, the density is intended
9
constant in all equations apart from the buoyancy term in the momentum equation. Moreover, the
10
non-isothermal face zone of the receiver cavity is divided to smaller isothermal zones, and then
11
the energy equation is solved for all of the mesh elements of the zones [28]. The Bousinesq
12
equation is shown in Eq. (6).
TE
D
M
AN
U
6
= 1 (1 − 2( − 1 ))
(6)
Where 1 and 1 are the reference temperature and the corresponding density, respectively. 2 is
14
the volumetric thermal expansion coefficient of the fluid.
AC
C
15
EP
13
2.4. Boundary Condition
16
The input parameters inserted into the heat transfer equations depend on the situation of Tehran,
17
Iran. Tehran is located at the GPS coordinates of 35° 42' 55.0728'' N and 51° 24' 15.6348'' E.
18
According to latitude and longitude of Tehran, specific boundary conditions are applied to all
19
surfaces, and utilized for solving the energy equation.
9
ACCEPTED MANUSCRIPT
1
2
3
4
5
There is heat transfer crossing the absorber tubes from within the cavity. As depicted in Fig. 3,
the wall side of the receiver is insulated with a 0.25 m thick glass wool layer, however a small
amount of heat loss is occurred via the insulation wall, as well as the glass cover. Radiation and
convection are the methods via which heat loss occures. The related values to the boundary
conditions are detailed in
RI
PT
6
Table. 3.
8
It should be mentioned that isothermal boundary conditions might seem to be a very simple
9
assumption. However, Manikumar et al. [24], Natarjan et al. [26] and Reddy and Kumar [29],
10
have applied isothermal boundary conditions to both the absorbing and cover surfaces, validating
11
their results by experimental data.
2.5. Mesh Study
M
AN
U
12
SC
7
In the present simulation, the two-dimensional geometries of the receiver cavities are gridded by
14
face elements in order that the results will be independent from the gridding network, and
15
divergence error will be avoided. Also, the inflations option at all of the boundaries is imposed in
16
order to implement proper boundary meshes. In terms of mesh quality, the average aspect ratio
17
and the average skewness are 1.0606 and 0.0637, respectively. The determination of an optimum
18
value for the number of elements is described as following.
EP
2.5.1. Mesh independency
AC
C
19
TE
D
13
20
Analyzing mesh independence is necessary to assess the accuracy of the results of the CFD
21
calculations. In this paper, mesh independence results of the model with six 1/2-in tubes and the
22
cavity angle of 50° is described. However, the same procedure is carried out, for the other
23
models. In the calculations of the initial model, the number of elements is 10812. Then, to make
24
sure that the results are independent from the mesh network, the equations are solved again for
25
another mesh networks, including 19543 and 38026 elements. Also, the convergence criterion is
10
ACCEPTED MANUSCRIPT
considered we 10-6 for the relative residuals. The heat fluxes crossing the absorber tubes surface
2
are plotted for each meshing network in the Fig. 4. As shown, the number of 10812 elements was
3
not qualified to the aim of the research. Because, by increasing it to 19543 elements, the heat
4
transfer rate is changing. Whereas, increasing the number of elements to the values greater than
5
19543 does not lead to significant change in the results. Therefore, it is concluded that, the
6
meshing network with the 19543 elements is acceptable and adequate for this analysis.
7
3. Numerical method and Validation
8
Discretization of the nonlinear differential equations is carried out by the finite volume approach
9
using the second-order upwind method. To couple the velocity and pressure, the SIMPLE [31]
10
scheme is employed. Additionally, the convergence criterion is considered 10-6 for the relative
11
residuals related to all parameters.
12
The results are validated according to two previous studies, the numerical study by Manikumar
13
et al. [24] and the experimental research by Larsen et al. [14]. Indeed, according to Manikumar et
14
al. [24], considering tubes and plate as absorber of the cavity, two specific geometries are
15
modeled in which DTRM radiation model was employed for a trapezoidal cavity receiver. Also,
16
based on experiments of Larsen et al. [14], the heat transfer from a trapezoidal cavity with five
17
absorber tubes is modeled, exploiting the procedure utilized in the current study. In this regard,
18
the geometric and input parameters of the two studies [14, 24] are depicted by Table 4.
19
According to Manikumar et al. [24], the heat transfer rate crossing the tubes surface is calculated
20
in the models for various boundary conditions. Subsequently, the results of the present
21
simulation are contrasted with the results of Manikumar et al. [24], in Table. 5. The average
22
results indicate that, the obtained values for the heat transfer rate in the present simulation do not
23
differ with the Manikumar’s findings, significantly. Only 0.13 % and 1.04 % of differences are
AC
C
EP
TE
D
M
AN
U
SC
RI
PT
1
11
ACCEPTED MANUSCRIPT
observed for the tube-less cavity model and six-tube cavity model, respectively, which can be
2
due to differences in mesh arrangement of Manikumar’s research, which is not described
3
properly. However, this indicates that the results of the employed procedure in the present study,
4
have a good agreement with the results of Manikumar et al. [24].
5
In addition, the heat transfer rate from the cavity to absorber tubes is analyzed for six various
6
boundary conditions depicted by Table. 6 which includes the obtained overall heat transfer rate
7
values for both the present study and the research of Larsen et al. [14]. The average difference
8
between the heat loss obtained in this study, and that of Larsen is 2.020%. This shows that the
9
employed procedure, in the present study, conforms to the experimental results of Larsen et al.
10
[14], properly. This difference can be because of using average temperature for all tubes by this
11
simulation, while the temperatures of tubes are slightly different from each other.
12
4. Results and discussion
13
In order to realize the effect of cavity angle and tube size on the heat transfer and heat loss rates,
14
the total and radiation heat transfer rate to the tubes wall of the models with various angles are
15
calculated. Subsequently, the simulations are repeated while applying tubes with different
16
dimensions to the trapezoidal receiver cavity.
SC
M
AN
U
TE
D
EP
4.1. Temperature contours
AC
C
17
RI
PT
1
18
The calculations are performed for each of the five two-dimensional geometries designated for
19
evaluating cavity angle, while keeping the aperture length constant and changing the absorber
20
walls. The results as temperature contours are illustrated in Fig. 5, where it is shown that the
21
range of temperature is from 308.15 K in the glass aperture of the receiver to 408.15 K at the
22
walls of the copper tubes located at the top of the model. It is clearly seen that how the
23
temperature of the air is varied from the glass to the tubes.
12
ACCEPTED MANUSCRIPT
In order to increase the efficiency of the receiver, tube size can be assessed as an important
2
parameter, which affects the value of the heat transfer to the absorber tubes. Indeed, while
3
changing the tube size, there are external parameters affecting the total efficiency, such as
4
pumping power which will be increased, by decreasing the diameter of the tubes. However, so as
5
to focus on heat transfer, which occurs within the cavity, the pumping power effect is neglected.
6
In this regard, the model with the angle of 50° is selected, and subsequently, according to the
7
details depicted by Table. 1, the calculations are carried out, while considering the diameters of
8
the tubes as 1/4 in, 3/8 in, 1/2 in, 5/8 in and 3/4 in. Fig. 6 demonstrates the temperature contours
9
of the models with different tubes sizes. It is depicted that, the range of temperature is between
10
about 308.15 K in the glass aperture and 408.15 K at the walls of the tubes. The variation of
11
temperature from top surface to the bottom, depicted by Fig. 6 is due to considering both the
12
hotter wall for the upper surface, and buoyancy term.
13
In Fig. 5, as well as Fig. 6, it is indicated that the air at high temperature is aggregated closed to
14
the tubes. This is attributed to not only setting high temperature at the absorber wall, but also
15
considering buoyancy term for the air. Moreover, it is assumed that the variation of density is
16
low contrasted with the variation of temperature. By contrast, if the variation of density is high,
17
the Boussinesq approximation will be unable to prepare reliable results, according to [31].
18
However, validation of this research by previous experimental data of Larsen [14] shows that
19
this approximation for density is appropriate for modeling cavities in this range of temperature.
SC
M
AN
U
TE
D
EP
AC
C
20
RI
PT
1
4.2. Heat transfer rate analysis
21
In order to evaluate the cavity angle effect in a quantitative manner, two parameters as the results
22
of each model simulation are calculated. The first one is the value of the total and radiation heat
23
transfer crossing the absorber tubes of the five models, which are plotted in Fig. 7. The latter one
13
ACCEPTED MANUSCRIPT
is the average value of the total and radiation heat transfer, as the heat transfer to the tubes
2
divided by the absorber section of the tubes wall (in this two-dimensional model, it is the length
3
of the boundary condition assigned for the absorber tubes). This quantity is illustrated in Fig. 8.
4
To observe the effect of variation in sizes of the tubes, the heat flux crossing the tubes is
5
calculated for the five models with different tubes diameters. Then, the total and radiation heat
6
transfer rates from within the cavity to the absorber tubes are plotted by Fig. 9; and also the
7
average total and radiation heat transfer rates as the total and radiation heat transfer rate to the
8
absorber tubes divided by the exposed part of the tubes wall to the beams are depicted in and Fig.
9
10.
M
AN
U
SC
RI
PT
1
The values for heat transfer are captured by surface integral option in the software. By Fig. 7, it
11
is illustrated that the heat transfer rate crossing the tubes surface is increased from 74.63W for
12
the cavity with the angle of 40° to 143.13W for the cavity with the angle of 60°. Thus, it shows
13
an enhancement by 91.7% as the angle is changing from 40° to 60°. Also, the results show that
14
the radiation heat transfer at the tubes surface is increased from 63.56W to 129.77W, as the angle
15
changes from 40° to 60°. This indicates an increase by 104.2% in the radiation heat transfer (Fig.
16
7).
17
In addition, the heat transfer rate per the exposed length of the tubes wall, as the average heat
18
transfer rate, is evaluated. As demonstrated by Fig. 8, the average heat transfer rate is decreased
19
from 752.14 W/m to 641.11 W/m for the cavities with angles of 40° to 60°, respectively. It can
20
be seen that the average heat transfer rate is decreased by about 14.7%. Similarly, the radiation
21
heat transfer rate per the exposed length of the tubes wall, as the average radiation heat transfer
22
rate is decreased from 640.57 W/m to 581.26 W/m, which is equal to a reduction of about 9.2%.
23
Totally, as shown by the Fig. 7 and Fig. 8, on the one hand, the heat transfer is enhanced by
AC
C
EP
TE
D
10
14
ACCEPTED MANUSCRIPT
increasing the angle of the cavity, but on the other hand the average heat transfer rate is reduced.
2
This is because of increment in the number of the tubes as the angle is increased, which leads to
3
decrease in the absorbing wall. Hence, it is concluded that, the cavity with the more number of
4
tubes is more appropriate for transferring more amount of energy, neglecting the heat received
5
by each tube. However, it should be noted that although the cavity with angle of 60° transfers
6
more value of the heat, the heat received by the medium inside the tubes is not as great as the
7
heat received by the medium in the tubes of the cavity with angle of 40°. Thus, for transferring
8
more amount of energy through each tube, the model with smaller angle is more suitable.
9
It is demonstrated that the minimum heat transfer rate is 109.99W for the model in which 3/8 in-
10
diameter tubes are situated. In addition, the maximum heat transfer rate, 119.86W, occurs in the
11
model with 3/4-in-diameter tubes. The range of the total heat transfer rates for the other models
12
is between the two minimum and maximum values. Similarly, the simulated results indicate that
13
the minimum and the maximum value of the radiation heat transfer rate at the tubes surface are
14
99.14 W for the model with 3/8-in-diameter tubes and 107.07 W for the model with 3/4-in-
15
diameter tubes, respectively. Moreover, it is shown that, the radiation heat transfer rate is
16
significantly affected by the tube size compared to the total heat transfer rate. In other words, the
17
variation of the radiation heat transfer rate is greater than the variation of total heat transfer rate.
18
This is why the radiation heat transfer rate decreases, while the total heat transfer rate increases.
19
(Fig. 9).
20
Fig. 10 illustrates that unlike the total heat transfer rate, the average heat transfer rate has the
21
minimum value for the model with 5/8-in-diameter tubes while it has the maximum value for
22
3/8-in-diameter. Respectively, the minimum and maximum average total heat transfer rates are
23
691.76 W/m and 666.20 W/m. Similarly, it is shown that the minimum and maximum values for
AC
C
EP
TE
D
M
AN
U
SC
RI
PT
1
15
ACCEPTED MANUSCRIPT
the average radiation heat transfer are 598.15 W/m for the model with 5/8-in tubes and 623.22
2
W/m for the model with 3/8-in tubes, respectively. Consequently, considering Fig. 9 and Fig. 10,
3
as the diameter of the inserted tubes in the cavity whether increases or decreases, the total heat
4
transfer and average heat transfer values do not show a regular trend. In other words, it is found
5
out that changing the diameter of the tubes does not lead to increase or decrease in rate of heat
6
transfer, necessarily. It is concluded that the cavity with 3/4-in tubes is able to transfer more
7
amount of heat to each tubes, neglecting the total amount of heat transfer occurred at all of the
8
tubes. Indeed, based on Fig. 10, it can be realized that the model with 3/4-in tubes does not
9
transfer the most amount of heat in each tube, so for transferring more amount of energy in each
11
SC
M
AN
U
10
RI
PT
1
tube, the cavity with 3/8-in tubes is the most appropriate model.
4.3. Correlations for heat transfer rates
According to Larsen et al. [14], it is common in the literature to propose a correlation between
13
the heat transfer coefficient and any other parameter affecting the heat transfer rate to the
14
absorber surface, such as the receiver length, the mirror area, the aperture area, etc. Furthermore,
15
he presented a correlation to measure heat transfer coefficient based on the receiver length,456 ,
16
as following:
EP
TE
D
12
456 = 0.186(89:; − <=:>;?8 )1.@A+#
18
AC
C
17
(7)
Where denotes the temperature.
19
For each model, the heat transfer rate coefficient to the tubes is achieved, utilizing the analytical
20
approach presented by Pye et al. [32] in which the heat transfer coefficient of the cavity
21
absorber,45 , is calculated with the equation given below:
16
ACCEPTED MANUSCRIPT
45 =
B
(8)
C89:;D (89:;D − <=:>;?8 )
Where B is the heat transfer to the absorber tubes wall, C is the absorber area and is the
2
temperature.
3
Based on the obtained data for the effect of the cavity angle on the total heat transfer to the tubes
4
wall (Fig. 8), the following correlation, as a function of cavity angle, is presented for heat
5
transfer coefficient to the tubes per cavity length, which is the exposed length of tubes wall to the
6
cavity receiver, 4E (F/G):
SC
RI
PT
1
/
K = 0.9365 for 40° ≤ I ≤ 60°
M
AN
U
4E = 0.08087I / − 0.71742I + 7.7313
(9)
Where I is the angle of the cavity.
8
Considering the obtained data for the effect of the tube size on the average heat transfer to the
9
tubes wall (Fig. 10), the following correlation, as a function of the tube dimension, is presented
10
TE
D
7
for average heat transfer coefficient, exposed to the cavity, 4N (F/G):
4N = 0.00146OP − 0.51502O/ + 1.32752O + 5.48152
/
(10)
12
Where O is the outer diameter of absorber tubes.
AC
C
11
EP
K = 0.873 for 1/4QR ≤ O ≤ 3/4QR
4.4. Heat dissipation
13
The heat loss rate is calculated as the heat dissipation flux at the insulation walls and the glass,
14
from within the cavity domain to the outside. Heat loss is captured by post processing option in
15
the software. The heat loss occurs in the trapezoidal cavity through wall surface, which is due to
16
conduction and also convection and radiation, although the side walls are insulated (with a 0.25
17
m thick glass wool layer) to reduce the heat dissipation. Fig. 11 demonstrates the total heat loss
17
ACCEPTED MANUSCRIPT
in the five models. It is observed that, as the angle is increased from 40° to 60°, the heat loss is
2
increased from 6.92 W to 13.41 W. This shows an enhancement by 93.7 %.
3
As illustrated in Fig. 11, the differences between the values of the heat losses for the different
4
models do not exceed 6W for the maximum difference. Thus, the heat loss value is not an
5
important issue for selecting the angle of the cavity. It should be mentioned that, these values are
6
achieved due to the specific boundary condition applied to the models. Therefore, in this case,
7
heat loss can be neglected, evaluating the effect of cavity angle on the heat transfer.
8
The heat loss rates, as the heat dissipation through the glass cover and wall insulation, are
9
evaluated in the designed models with the tubes with different sizes. Fig. 12 demonstrates the
10
values for heat loss rates in the five models. It is observed that, as the size of the tubes is
11
incremented from 1/4 in to 3/4 in, the heat loss is increased from 8.17 W to 10.78 W. This shows
12
an enhancement of about 31.9 %.
13
As illustrated by Fig. 12 the differences between the values of heat losses for the different
14
models do not exceed 3W. Thus, the heat loss rate value is not an important issue for selecting
15
the kind of cavity.
16
It should be noted that the obtained values for total and average heat transfer and heat loss are
17
attributed to the specific boundary condition applying to the models. Therefore, it is not a general
18
conclusion that the heat dissipation value can be assumed as negligible for selecting cavities.
SC
M
AN
U
TE
D
EP
AC
C
19
RI
PT
1
4.5. Discussion
20
It is claimed that the main part of heat transfer in the cavity receivers is the heat loss due to
21
radiation. Dey et al. [33] has shown that up to 80% of heat losses in cavity receivers take place
22
by radiation. Also, Pye [30] found that radiation makes up for approximately 90% of the heat
23
transfer to the absorbing surface, applying an analytical model for a trapezoidal cavity. This is
18
ACCEPTED MANUSCRIPT
obviously depicted by Fig. 7, and Fig. 9 that most of the heats transfer is due to radiation heat
2
transfer. Based on the achieved data, the calculated ratio of radiation to total heat transfer is
3
between 85.2% and 91.3% for all models, either changing the cavity angle, or changing the tube
4
dimension. This has a good agreement by mentioned references [30, 33].
5
Additionally, it should be considered that in the current study the correlation for the heat transfer
6
rate coefficient are calculated according to the research by Pye et al. [32], where the absorber
7
tubes were simplified as absorber plates, but on the other hand this study considers tubes
8
geometry rather than plate geometry as absorber surface. Hence, it might cause some deviation
9
from the exact value, while calculating the heat transfer rate coefficient.
M
AN
U
SC
RI
PT
1
Moreover, the results showing the cavity angle effect in this research are contrasted with those of
11
Saxena et al. [25] in which simplifying the absorber tubes as absorber plate, the cavity angle
12
effect has been evaluated in the range between 30° and 60°, and subsequently it has been
13
concluded that by increasing the cavity angle from 40° to 60°, the average total Nusselt number
14
increases by about 89.3%, which conforms to the increase by 91.7% in the total heat transfer
15
rate, in this study. The difference is due to considering absorber plates instead of absorber tubes
16
by Saxena et al. [25]. Also, in both studies, the effect of cavity angle on the radiation heat
17
transfer rate is greater compared to the effect of cavity angle on the total heat transfer rate.
18
5. Conclusion
19
In this research, the aim was to analyze the effects of the cavity angle and tube size in a
20
prototype trapezoidal cavity utilized in a linear Fresnel reflector. The cavity angle changed in the
21
range between 40° and 60°, and the tube size was considered as 1/4 in, 3/8 in, 1/2 in, 5/8 in and
22
3/4 in. The following conclusions are concluded from the numerical simulation:
AC
C
EP
TE
D
10
19
ACCEPTED MANUSCRIPT
•
1
With the rise of the angle of the cavity, the heat transfer rate to the absorber tubes
2
increases, while the average heat transfer rate (the heat transfer rate divided by length
3
of the tubes wall exposed to cavity receiver) is decreased.
As the angle of the cavity increases, the heat dissipation through the glass cover and
RI
PT
•
4
wall insulation is increased about 93.7 %.
5
•
6
Necessarily, the increase in the sizes of tubes does not lead to increase or decrease in
heat transfer rate to the tube walls. Various values of heat transfer and average heat
8
transfer are obtained for various values of tubes sizes.
As the sizes of tubes are incremented, the value of heat loss to glass cover and wall
M
AN
U
•
9
SC
7
insulation is enhanced about 31.9 %.
10
•
11
From 85.2% to 91.3% of the total heat transfer rate to the absorber is made up of
radiation heat transfer rate, for the different designs of the cavity.
12
The extension of this paper according our previous works [34-58] affords engineers a good
14
option for CFD and power plant simulations.
15
16
6. Reference
17
18
19
20
21
22
23
24
25
26
27
1.
3.
4.
5.
Höök, M. and X. Tang, Depletion of fossil fuels and anthropogenic climate change—A
review. Energy Policy, 2013. 52: p. 797-809.
Kasaeian, A., et al., Experimental Studies on the Applications of PCMs and Nano-PCMs
in Buildings: A Critical Review. Energy and Buildings, 2017.
Mohanty, S., P.K. Patra, and S.S. Sahoo, Prediction and application of solar radiation
with soft computing over traditional and conventional approach–A comprehensive
review. Renewable and Sustainable Energy Reviews, 2016. 56: p. 778-796.
Espargilliere, H., et al., Applicability of CSP solar fields to the dry cooling of related
thermodynamic cycles. Applied Thermal Engineering, 2017. 127: p. 319-329.
Zhang, H., et al., Concentrated solar power plants: Review and design methodology.
Renewable and Sustainable Energy Reviews, 2013. 22: p. 466-481.
AC
C
2.
EP
TE
D
13
20
ACCEPTED MANUSCRIPT
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
RI
PT
10.
SC
9.
M
AN
U
8.
TE
D
7.
Price, H., et al., Advances in parabolic trough solar power technology. Journal of solar
energy engineering, 2002. 124(2): p. 109-125.
Kalogirou, S.A., Solar thermal collectors and applications. Progress in energy and
combustion science, 2004. 30(3): p. 231-295.
Moghimi, M., K. Craig, and J. Meyer, Optimization of a trapezoidal cavity absorber for
the Linear Fresnel Reflector. Solar Energy, 2015. 119: p. 343-361.
Giostri, A., et al., Comparison of two linear collectors in solar thermal plants: parabolic
trough versus Fresnel. Journal of Solar Energy Engineering, 2013. 135(1): p. 011001.
Zhu, G., et al., History, current state, and future of linear Fresnel concentrating solar
collectors. Solar Energy, 2014. 103: p. 639-652.
Bellos, E., C. Tzivanidis, and A. Papadopoulos, Optical and thermal analysis of a linear
Fresnel reflector operating with thermal oil, molten salt and liquid sodium. Applied
Thermal Engineering, 2018.
Montes, M.J., et al., Performance model and thermal comparison of different alternatives
for the Fresnel single-tube receiver. Applied Thermal Engineering, 2016. 104: p. 162175.
Francia, G., Pilot plants of solar steam generating stations. Solar Energy, 1968. 12(1): p.
51IN359IN763-58IN562IN1364.
Larsen, S.F., M. Altamirano, and A. Hernández, Heat loss of a trapezoidal cavity
absorber for a linear Fresnel reflecting solar concentrator. Renewable Energy, 2012.
39(1): p. 198-206.
Abbas, R., et al., Solar radiation concentration features in Linear Fresnel Reflector arrays.
Energy Conversion and Management, 2012. 54(1): p. 133-144.
Mills, D.R. and G.L. Morrison, Compact linear Fresnel reflector solar thermal
powerplants. Solar energy, 2000. 68(3): p. 263-283.
Qiu, Y., et al., Study on optical and thermal performance of a linear Fresnel solar
reflector using molten salt as HTF with MCRT and FVM methods. Applied Energy,
2015. 146: p. 162-173.
Abbas, R., et al., High concentration linear Fresnel reflectors. Energy conversion and
management, 2013. 72: p. 60-68.
Lin, M., et al., Experimental and theoretical analysis on a linear Fresnel reflector solar
collector prototype with V-shaped cavity receiver. Applied Thermal Engineering, 2013.
51(1): p. 963-972.
Lai, Y., et al., Thermal Performance Prediction of a Trapezoidal Cavity Absorber for a
Linear Fresnel Reflector. Advances in Mechanical Engineering, 2013. 5: p. 615742.
Facão, J. and A.C. Oliveira, Numerical simulation of a trapezoidal cavity receiver for a
linear Fresnel solar collector concentrator. Renewable Energy, 2011. 36(1): p. 90-96.
Qiu, Y., et al., A comprehensive model for optical and thermal characterization of a
linear Fresnel solar reflector with a trapezoidal cavity receiver. Renewable Energy, 2016.
97: p. 129-144.
EP
6.
AC
C
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
21
ACCEPTED MANUSCRIPT
28.
29.
30.
31.
32.
33.
34.
RI
PT
36.
27.
SC
35
36
37
38
26.
M
AN
U
35.
25.
TE
D
31
32
33
34
24.
Reddy, K. and K.R. Kumar, Estimation of convective and radiative heat losses from an
inverted trapezoidal cavity receiver of solar linear Fresnel reflector system. International
Journal of Thermal Sciences, 2014. 80: p. 48-57.
Manikumar, R., A. Valan Arasu, and S. Jayaraj, Computational fluid dynamics analysis
of a trapezoidal cavity absorber used for the linear Fresnel reflector solar concentrator
system. Journal of Renewable and Sustainable Energy, 2012. 4(6): p. 063145.
Saxena, A., et al., Numerical analysis of convective and radiative heat losses from
trapezoidal cavity receiver in LFR systems. Solar Energy, 2016. 137: p. 308-316.
Natarajan, S.K., K. Reddy, and T.K. Mallick, Heat loss characteristics of trapezoidal
cavity receiver for solar linear concentrating system. Applied energy, 2012. 93: p. 523531.
Karanth, K.V., M. Manjunath, and N.Y. Sharma. Numerical simulation of a solar flat
plate collector using discrete transfer radiation model (DTRM)–a CFD approach. in
Proceedings of the World Congress on Engineering. 2011.
Denys, S., J.G. Pieters, and K. Dewettinck, Computational fluid dynamics analysis of
combined conductive and convective heat transfer in model eggs. Journal of Food
Engineering, 2004. 63(3): p. 281-290.
Reddy, K. and N.S. Kumar, Combined laminar natural convection and surface radiation
heat transfer in a modified cavity receiver of solar parabolic dish. International Journal of
Thermal Sciences, 2008. 47(12): p. 1647-1657.
Pye, J.D., System modelling of the compact linear Fresnel reflector. PhD, University of
New South Wales, Sydney, Australia, 2008.
Fluent, A., Fluent 15 Users Guide. Lebanon, USA, 2016.
Pye, J.D., et al. Modelling of cavity receiver heat transfer for the compact linear fresnel
reflector. in ISES World Congress. 2003.
Dey, C., Heat transfer aspects of an elevated linear absorber. Solar Energy, 2004. 76(1):
p. 243-249.
S. Shareghi, D. Toghraie, Numerical Simulation of Blood Flow in Healthy Arteries by
Use of the Sisko Model, Computational Thermal Sciences: An International Journal 8 (4),
2016
EP
23.
AC
C
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Karimipour A. Esfe MH. Safaei MR. Semiromi DT. Jafari S. and Kazi SN. Mixed
convection of Copper–Water nanofluid in a shallow inclined lid driven cavity using the
lattice Boltzmann method. Physica A: Statistical Mechanics and its Applications. 2014,
402: 150-168 http://dx.doi.org/10.1007/s10973-016-5436-4.
Karimipour A. Alipour H. Akbari OA. Semiromi DT. Esfe MH., Studying the Effect of
Indentation on Flow Parameters and Slow Heat Transfer of Water-Silver Nano-Fluid with
Varying Volume Fraction in a Rectangular Two-Dimensional Micro Channel, Indian
Journal of Science and Technology, 2016, 8, 2015
22
ACCEPTED MANUSCRIPT
37.
Akbari OA. Karimipour A. Toghraie D. Karimipour A, Impact of ribs on flow parameters
and laminar heat transfer of Water/Alumina nanofluid with different nanoparticle volume
fractions in a three-dimensional rectangular microchannel, Adv Mech Eng, 2016; 7: 1–11
4
5
6
7
38.
Akbari OA. Karimipour A. Toghraie D. Safaei MR. Alipour M. Goodarzi H. and Dahari
M. Investigation of Rib's Height Effect on Heat Transfer and Flow Parameters of
Laminar Water- Al2O3 Nanofluid in a Two Dimensional Rib-Microchannel. Appl Math
Comp, 2016, 290, 135–153
8
9
10
39.
Akbari OA. Toghraie D. Karimipour A. Numerical simulation of heat transfer and
turbulent flow of Water nanofluids CuO in rectangular microchannel with semi attached
rib. Adv Mech Eng. 2016; 8: 1–25
11
12
13
14
40.
Alipour H. Karimipour A. Safaei MR. Semiromi DT. Akbari OA., Influence of T-semi
attached rib on turbulent flow and heat transfer parameters of a silver-Water nanofluid
with different volume fractions in a three-dimensional trapezoidal microchannel. Physica
E, 2016; 88: 60-76
15
16
41.
Nazari S. Toghraie D. Numerical simulation of heat transfer and fluid flow of Water-CuO
Nanofluid in a sinusoidal channel with a porous medium. Physica E, 123; 87: 134-140
17
18
19
42.
Sajadifar SA. Karimipour A. Toghraie D. Fluid flow and heat transfer of non-Newtonian
nanofluid in a microtube considering slip velocity and temperature jump boundary
conditions,European Journal of Mechanics-B/Fluids, 2017; 61: 25-32
20
21
22
43.
Aghanajafi A, Toghraie D. Mehmandoust B., Numerical simulation of laminar forced
convection of Water-CuO nanofluid inside a triangular duct, Physica E, 2017: 85: 103108
23
24
25
44.
Afrand M. Toghraie D. Karimipour A. Wongwises SA. Numerical Study of Natural
Convection in a Vertical Annulus Filled with Gallium in the Presence of Magnetic Field,
Journal of Magnetism and Magnetic Materials, 2017, 430: 22–28
26
27
28
29
30
31
32
33
34
35
36
45.
Faridzadeh MR. Semiromi DT. Niroomand A. Analysis of laminar mixed convection in
an inclined square lid-driven cavity with a nanofluid by using an artificial neural network.
Heat Transfer Research. 2014; 45
D Toghraie, Numerical thermal analysis of Water's boiling heat transfer based on a
turbulent jet impingement on heated surface, Physica E, 84, 454-465, 2016
F. Pourfattah, M. Motamedian, Gh. Sheikhzadeh, D. Toghraie, O.A. Akbari, The
numerical investigation of angle of attack of inclined rectangular rib on the turbulent heat
transfer of Water-Al 2 O 3 nanofluid in a tube, International Journal of Mechanical
Sciences 000–132 (2017) 1–11.
R. Mashayekhi, E. Khodabandeh, M. Bahiraei, L. Bahrami, D. Toghraiee, O.A. Akbari,
Application of a novel conical strip insert to improve the efficacy of water–Ag nanofluid
47.
48.
SC
M
AN
U
TE
D
EP
AC
C
46.
RI
PT
1
2
3
23
ACCEPTED MANUSCRIPT
53.
54.
55.
56.
57.
58.
RI
PT
SC
52.
M
AN
U
51.
TE
D
50.
EP
49.
for utilization in thermal systems: A two-phase simulation, Energy Conversion and
Management 151 (2017) 573–586.
M.R. Gholami, O.A. Akbari, Ali Marzban, D. Toghraie, Gh. Ahmadi Sheikh Shabani, M.
Zarringhalam, The effect of rib shape on the behavior of laminar flow of oil/MWCNT
nanofluid in a rectangular microchannel, J Therm Anal Calorim,
https://doi.org/10.1007/s10973-017-69023
R. Sarlak , Sh. Yousefzadeh, O.A. Akbari, D. Toghraie, S. Sarlak, F. assadi, The
investigation of simultaneous heat transfer of water/Al 2 O 3 nanofluid in a close
enclosure by applying homogeneous magnetic field, International Journal of Mechanical
Sciences 133 (2017) 674–688.
M. Heydari, D. Toghraie, O.A. Akbari, The effect of semi-attached and offset midtruncated ribs and Water/TiO2 nanofluid on flow and heat transfer properties in a
triangular microchannel, Therm Sci Eng Prog, 2 (2017) 140–150.
O.A. Akbari, H. Hassanzadeh Afrouzi, A. Marzban, D. Toghraie, H. Malekzade & A.
Arabpour, Investigation of volume fraction of nanoparticles effect and aspect ratio of the
twisted tape in the tube, J Therm Anal Calorim, DOI 10.1007/s10973-017-6372-7.
D. Toghraie, M.M. Davood Abdollah, F. Pourfattah, O.A. Akbari, B. Ruhani, Numerical
investigation of flow and heat transfer characteristics in smooth, sinusoidal and zigzagshaped microchannel with and without nanofluid, J Therm Anal Calorim, DOI
10.1007/s10973-017-6624-6.
Ahmadi, G., Toghraie, D., Akbari, O.A., 2017c. Solar parallel feed water heating
repowering of a steam power plant: A case study in Iran, Renew. and. Sus. En. Rev, 77,
474-485.
Ahmadi, Gh.R, Akbari, O.A., Zarringhalam, M., 2017b. Energy and Exergy Analyses of
Partial Repowering of a Natural Gas-Fired Steam Power Plant, Int. J. Exergy, 23 (2).
Ahmadi, Gh.R., Toghraie, D., 2015. Parallel feed water heating repowering of a 200 MW
steam power plant, J. Power. Techno. 95 (4), 288–301.
Ahmadi, Gh.R., Toghraie, D., 2016. Energy and exergy analysis of Montazeri Steam
Power Plant in Iran, Renew. and Sus. En. Rev, 56, 454–463.
Ahmadi, Gh.R., Toghraie, D., Azimian, A., Akbari, O., 2017a. Evaluation of
Synchronous Execution of Full Repowering and Solar Assisting in a 200 MW Steam
Power Plant, a Case Study, Appl. Therm. Eng, 112, 111–123.
AC
C
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
24
M
AN
U
SC
RI
PT
ACCEPTED MANUSCRIPT
1
2
Fig. 1 Position of cavity receiver in Linear Fresnel Reflector
4
5
AC
C
EP
TE
D
3
Fig. 2 A side view of the modeled receiver
6
Table. 1 Dimensional properties of the designed models
7
Aperture
Height
Angle
Tubes No.
25
Tube Size
ACCEPTED MANUSCRIPT
0.19666
(°)
(m)
0.0608
3
1/2
0.0127
45
5
1/2
0.0127
10
1/4
0.00635
7
3/8
0.009525
1/2
0.0127
5/8
0.015875
3/4
0.01905
1/2
0.0127
50
6
55
7
60
8
1/2
M
AN
U
1
SC
4
0.0127
Table. 2. Characteristics of the applied materials
2
Parameter
Insulation Material
Tubes Material
Air density
Air viscosity
Unit
TE
D
Cover Material
EP
Air thermal Expansion coefficient
Value
-
Glass-wool
-
Copper
-
Glass
kg/m3
1.225
kg/ms
1.7894e-05
1/K
0.002857
W/mK
0.071
Thermal conductivity of glass
W/mK
0.8
W/mK
387.6
AC
C
Thermal conductivity of glass-wool
Thermal conductivity of tubes
4
(m)
40
5
3
(in)
RI
PT
(m)
5
Table. 3. Applied boundary conditions
Domain
Quantity
Unit
Glass cover
Iso-thermal Temperature
K
26
Value
308
ACCEPTED MANUSCRIPT
W/m2K
16
Internal Emissivity
-
1
thickness
m
0.01
Iso-thermal Temperature
K
408
Convective heat transfer rate coefficient
Tubes
Internal Emissivity
thickness
Ambient Temperature
Insulation
External Radiation Temperature
M
AN
U
wall
Internal Emissivity [30]
Thickness
1
3
4
AC
C
EP
TE
D
2
Fig. 3 Boundary conditions of the model
27
W/m2K
16
-
1
m
0.00155
W/m2K
16
K
290
K
285
-
0.1
m
0.25
SC
Convective heat transfer rate coefficient
RI
PT
Convective heat transfer rate coefficient
M
AN
U
SC
RI
PT
ACCEPTED MANUSCRIPT
1
4
5
TE
D
Fig. 4. The plot of heat transfer rate to absorber tubes, according to tubes temperature for various grid sizes, in the
model with six 1/2-in tubes and the cavity angle of 50°
Table 4 Geometric and input parameters employed in the Manikumar [24] and Larsen [14] model
Parameter
Unit
EP
2
3
Value
Manikumar et al. [24]
Larsen et al. [14]
-
0/6
5
Cavity tilt angle
°
50
45
Aperture length
m
0.196
0.685
Absorber tubes length
m
1
1.4
Absorber emissivity
-
0.95
0.88
Ambient temperature
K
308.15
300.75
Convective heat transfer coefficient
W/m2K
16
0.96
AC
C
Tubes number
6
28
ACCEPTED MANUSCRIPT
Table. 5. Results of heat transfer for various tubes temperatures in the present simulation and those of Manikumar et
al. [24] (for both the tube-less and six-tube cavities)
For tube-less cavity
For six-tube cavity
Heat Transfer Rate to Plate
Heat Transfer Rate to Tubes
Absorber
Absorber
Manikumar et
Temperature
Present
(K)
study (W)
Difference
Difference
Temperature
Present study
Manikumar et
(K)
(W)
al. [24] (W)
(%)
al. [24]
(W)
216.883
0.053
451.15
348.825
346.970
0.532
436.15
182.072
181.818
0.140
436.15
303.407
300.001
1.122
422.15
152.176
151.948
0.150
422.15
261.963
259.091
1.094
408.15
126.203
125.974
0.181
408.15
224.441
221.212
1.440
M
AN
U
Table. 6. Results of heat transfer for various tubes temperatures in the present numerical simulation and the
experimental data of Larsen et al. [14]
557.95
510.65
471.65
445.35
AC
C
429.15
383.85
Difference
Present study (W)
Larsen et al. [14] (W)
(%)
1104.168
1130
2.286
804.760
825
2.453
574.761
580
0.903
439.116
430
2.076
367.242
360
1.972
175.629
180
2.428
EP
(K)
Heat Transfer Rate to Absorber Tubes
TE
D
Absorber temperature
8
SC
216.998
4
7
(%)
451.15
3
5
6
RI
PT
1
2
9
29
1
2
M
AN
U
SC
Angle: 40°
RI
PT
ACCEPTED MANUSCRIPT
3
4
7
8
TE
D
EP
6
Angle: 50°
AC
C
5
Angle: 45°
Angle: 55°
30
1
2
SC
Angle: 60°
3
Fig. 5. Temperature contours of the designed models with various angles
M
AN
U
4
Tube size: 1/4 in
8
9
AC
C
EP
7
TE
D
5
6
RI
PT
ACCEPTED MANUSCRIPT
Tube size: 3/8 in
31
1
2
M
AN
U
SC
Tube size: 1/2 in
RI
PT
ACCEPTED MANUSCRIPT
3
4
7
8
TE
D
EP
6
Tube size: 3/4 in
AC
C
5
Tube size: 5/8 in
Fig. 6. Temperature contours of the designed models with various tube sizes
32
ACCEPTED MANUSCRIPT
Total heat transfer rate
Radiation heat transfer rate
RI
PT
130
120
110
100
SC
90
80
70
60
50
35
40
45
50
55
60
Angle (degree)
1
Fig. 7. Total and radiation heat transfer rate from within the cavity, to the absorber tubes in the models with different
cavity angles
740
EP
720
Total heat transfer per unit of tubes length
Radiation heat transfer per unit of tubes length
700
680
660
AC
C
Heat transfer rate to absorber tube per
unit of length of tubes wall (W/m)
760
TE
D
2
3
M
AN
U
Heat transfer rate to absorber tubes (W)
140
640
620
600
580
560
35
4
5
6
40
45
50
55
60
Angle (degree)
Fig. 8. Total and radiation heat transfer rate from within the cavity, to the absorber tubes divided by length of the
tubes wall, for the models with different angles
33
Total heat transfer rate
Radiation heat transfer rate
RI
PT
120
115
110
SC
105
100
95
1/4 in
3/8 in
1/2 in
5/8 in
3/4 in
Tube Size (in)
1
680
670
660
650
640
630
AC
C
Heat transfer rate to absorber tubes per
unit of length of tubes wall (W/m)
690
TE
D
Fig. 9 Total and radiation heat transfer rate from within the cavity, to the absorber tubes in the models with different
tube sizes
Total heat transfer rate per meter
Radiation heat transfer rate per meter
EP
2
3
M
AN
U
Heat transfer rate to absorber tubes (W)
ACCEPTED MANUSCRIPT
620
610
600
590
4
5
6
1/4 in
3/8 in
1/2 in
5/8 in
3/4 in
Tube Size (in)
Fig. 10 Total and radiation heat transfer rate from within the cavity, to the absorber tubes divided by length of the
tubes wall, for the models with different tube sizes
34
RI
PT
12
SC
10
8
6
35
40
45
50
55
60
Angle (degree)
1
2
M
AN
U
Heat loss to glass and insulation wall (W)
ACCEPTED MANUSCRIPT
Fig. 11. The values of heat losses through the glass cover and insulation wall, for the models with different angles
TE
D
11
EP
10
9
AC
C
Heat loss to glass and insulation wall (W)
12
8
7
1/4 in
3
4
3/8 in
1/2 in
5/8 in
3/4 in
Tube Size (in)
Fig. 12 The values of heat losses through the glass cover and insulation wall, for the models with different tube sizes
35
ACCEPTED MANUSCRIPT
Research highlights
•
•
•
Heat transfer rate and heat loss in a trapezoidal cavity of a Linear Fresnel Reflector
are analyzed.
The calculations are performed for steady state laminar model.
Increasing the angle of cavity, the total value for heat transfer rate is increased.
AC
C
EP
TE
D
M
AN
U
SC
RI
PT
•