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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 [email protected] 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 )[email protected]+# 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. 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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 •