Optical and thermal analysis of different cavity receiver designs for solar dish concentrators

Energy Conversion and Management: X 2 (2019) 100013

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Optical and thermal analysis of different cavity receiver designs for solar dish concentrators

T

Evangelos Bellosa, , Erion Bousia, Christos Tzivanidisa, Sasa Pavlovicb ⁎

a b

Department of Thermal Engineering, National Technical University of Athens, Zografou, Heroon Polytechniou 9, 15780 Athens, Greece Department of Energetics and Process Technique, Faculty of Mechanical Engineering, University in Nis, Serbia

ARTICLE INFO

ABSTRACT

Keywords: Solar dish Cavity receiver Conical cavity Cylindrical cavity Thermal analysis Optical analysis

Cavity receivers are the most usual design in solar dish concentrators in order to achieve high thermal performance. The objective of this work is the investigation of five different cavity receivers under different operating temperature levels and the selection of the most appropriate designs. More specifically, the examined cavities have the following shapes: cylindrical, rectangular, spherical, conical and cylindrical-conical. All the cavities are optimized in order to determine the best design which maximizes the thermal efficiency of the solar collector. The optimization variables are different for every design and they regard the cavity length, the cone angle and the distance from the concentrator base. According to the results, the best design is the novel one with cylindrical-conical shape, while the conical and the spherical are the next choices. The worst design is rectangular, while the cylindrical is the fourth design in the performance sequence. For operation at 300 °C, the cylindricalconical design is found to have 67.95% thermal efficiency, 35.73% exergy efficiency while the optical efficiency is 85.42%. The analysis is conducted with a developed model in SolidWorks Flow Simulation which is validated with literature experimental data.

1. Introduction 1.1. Solar energy and collectors Solar energy is an important renewable energy source for covering a part of the human energy needs with a sustainable way [1]. Solar thermal collectors are the devices which can utilize properly the solar irradiation for useful thermal production [2]. Concentrating solar systems are the most promising solar technologies because they can produce heat at low, medium or high-temperature levels with sufficient efficiency [3]. The most usual solar concentrating technologies are the parabolic trough collector (PTC), the linear Fresnel reflector (LFR), the solar towers and the solar dishes [4]. The PTC and the LFR belong to the linear concentrating technologies which can operate up to 500 °C with the conventional designs and they can be applied in a great variety of applications [5]. Solar towers and solar dishes are point focus technologies which can operate in extremely high temperatures with gas working fluids (air or CO2) because of the high concentration ratio [6]. The solar tower is used practically for only power generation applications, while solar dishes can be used for a great range of high-temperature applications [7]. This fact makes it a promising technology ⁎

which presents high interest by the scientific and commercial point of view. Solar dishes can be used for power production [8], solar cooling [9], desalination [10], industrial heat [11] and chemical processes [12]. Wang et al. [13] examined the use of a solar dish concentrator for assisting a gas turbine and practically to reduce fuel consumption. Meas and Bello-Ochende [14] examined different solar-driven gas turbine cycles and they found that the intercooling is more effective than reheating. Khan et al. [15] studied the use of a nanofluid-based solar dish collector for feeding a supercritical CO2 Brayton cycle with recompression and reheating. They found the maximum system energy efficiency to be 33.73% with Al2O3/Oil nanofluid as the working fluid in the solar collector. Barreto and Canhoto [16] investigated the use of a solar concentrator coupled to a Stirling engine for power production and they found a global system efficiency 10.4%. On the other hand, Sandoval et al. [17] found the global efficiency of a similar system to be 19%. Mohammadi and Mehrpooya [18] studied a system with a solar dish collector coupled to fuel cell for hydrogen and electricity production. They found that the use of solar energy reduces fuel consumption by about 74%. Hou and Zhang [19] studied a system with a solar dish collector which feeds a thermoelectric generator coupled to an absorption chiller. This system produces cooling and electricity,

Corresponding author. E-mail address: [email protected] (E. Bellos).

https://doi.org/10.1016/j.ecmx.2019.100013 Received 11 May 2019; Received in revised form 11 June 2019; Accepted 12 June 2019 Available online 22 June 2019 2590-1745/ © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

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Nomenclature Aa C cp drec D Ds E f fr Gb h hout k L L1 m Nu p Pr Q Re T t u Wp

ΔP ε η μ ρ Φ ω

aperture area, m2 concentration ratio, specific heat capacity under constant pressure, J/kg K receiver position, mm receive diameter, m spherical receive diameter, m exergy flow, W focal length, m friction factor, solar direct beam irradiation, W/m2 heat transfer coefficient, W/m2K convection coefficient with the ambient, W/m2K thermal conductivity, W/mK cavity length, m cylindrical part of the cavity length, m mass flow rate, kg/s Nusselt number, helical coil pitch, mm Prandtl, – heat rate, W Reynolds number, – temperature, °C thickness, mm fluid velocity, m/s pumping work, W

pressure drop, Pa emittance, – efficiency, – dynamic viscosity, Pa s reflectance, – cone angle, ° design angle for the cone (Appendix A),

o

Subscripts and superscripts am cav1 cav2 coil cone ex fm in iso max opt out rec s sph th u

ambient side cavity back cavity absorber coil conical geometry exergetic mean fluid inlet insulation maximum optical outlet receiver solar sphere thermal useful

Greek symbols

Abbreviations

α

PTC

absorber absorbance, –

parabolic trough collector

examined studies.

while its maximum efficiency is around 27%. Javidmehr et al. [20] studied a polygeneration system with gas turbine, organic Rankine cycle, solar dish and distillation unit for heating, electricity and fresh water production which presents maximum efficiency at 70%. The solar dish is a compact design which can produce useful heat in high temperatures and it is a promising technology. It mainly is manufactured by using a parabolic dish concentrator, a receiver and a supporting mechanism. The total configuration tracks the sun in two directions in order to achieve the proper reflection of the sun rays from the concentrator to the receiver [21]. There are numerous designs of solar dishes and there is not a well-established design, as in the case of PTC for example. Different ideas have been tested numerically and experimentally, as well as there are commercial solutions at this time. In any case, the field is open and the optimum design from the energetic and financial point of view is still under investigation.

1.2.1. Cylindrical cavity receivers The cylindrical cavity receiver is the most usual in the literature and many studies have used this design. Azzouri et al. [22] examined experimentally a cylindrical cavity receiver with the absorber tubes to be inside the insulation layer. They found that the maximum thermal efficiency of their design is 75%. Avargani et al. [23] studied a cylindrical cavity receiver and they found maximum exergy efficiency at 23% with a developed numerical model, while the maximum exergetic efficiency was 10% experimentally. Karimi et al. [24] found that thermal performance of the cylindrical receiver (without the optical losses) can reach up to 90% for a receiver aperture diameter at 0.2 m while the concentrator has a diameter of 2.88 m. Mawire and Taole [25] studied experimentally a homemade parabolic dish concentrator with a cylindrical receiver. They found that the thermal efficiency is up to 40% and the exergy efficiency by up to 10%. Zou et al. [26] found that the optimum length of the cavity is 0.39 m with 5 tube loops and in this case, the thermal efficiency can be 72%. Wang et al. [27] studied a cylindrical receiver with a modified back surface. Their work is optical and they found maximum optical efficiency at 72%. This value is 7.67% higher than the optical efficiency of the conventional cylindrical cavity receiver with 10% dead space. Loni et al. [28] optimized a cylindrical receiver for application in an Organic Rankine Cycle. They found that the optimum depth of the cavity is equal to the cavity diameter. They stated that the optimum concentration ratio is around 32 and the receiver efficiency reaches 90%. Soltani et al. [29] performed and optimization work about a cylindrical cavity receiver. They found that the receiver position has to be at the 90% of the focal distance, while its length has to be 1.5–2 times greater than the aperture diameter. The maximum thermal efficiency of the system was about 78%.

1.2. Different cavity receivers The most usual receiver in the solar dish concentrators are the cavity receivers. These receivers usually have not a cover in the front side and they try to achieve high optical efficiency by capturing the incident solar rays inside the cavity. Practically, the cavity receivers utilize the secondary reflections inside the cavity and so the effective absorbance of the receiver is increased. However, the lack of cover increases the thermal losses of the cavities. The insulation layer is used in the outside area of the cavity, expect the side to the concentrator. In the literature, there are numerous studies which are associated with the investigation of different cavities optically and thermally. However, there is not a well-established design and the comparative studies do not give always the same candidates as the most appropriate. Below a brief literature review about the studies of cavity receivers which are presented in terms of efficiency are given. Table 1 summarizes the 2

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are reduced with the inclination angle increase of the receiver, while the radiation losses do not change. Tan et al. [36] also found that the convection losses have a decreasing rate with the inclination angle increase and they suggested correlations for their determination which are based on experimental data.

Table 1 Summary of the literature studies about the cavity receivers. Cavity/receiver

Study

Main findings (maximum values)

Cylindrical

Azzouri et al. [22] Avargani et al. [23] Karimi et al. [24] Mawire and Taole [25] Zou et al. [26] Wang et al. [27] Loni et al. [28] Soltani et al. [29]

ηopt = 75% ηex = 23% ηrec = 90% ηth = 40% – ηex = 10% ηth = 72% ηopt = 72% ηrec = 90% ηth = 78%

Rectangular

Le Roux et al. [30] Loni et al. [31]

ηrec = 85% ηth = 75.4%

Spherical

Loni et al. [33] Zhou et al. [34]

ηth = 60% ηth = 84.4% – ηopt = 88.9%

Conical

Dun et al. [37] Yan et al. [39] Pavlovic et al. [40]

ηth = 35% ηopt = 90% ηth = 79.4% – ηopt = 85.4% ηrec = 80%

Turini et al. [41] Hetero-conical

Pye et al. [44] Thirunavukkarasu and Cheralathan [45]

ηrec = 98.7% ηth = 40%

Spiral

Pavlovic et al. [46] Pavlovic et al. [47]

ηth = 34% ηth = 65%

Flat

Senthil and Cheralathan [50]

ηth = 67.1%

Volumetric

Zhu et al. [52] Wang et al. [53]

ηth = 87% – ηopt = 36% ηth = 64%

Tapered tube bundle

Xu et al. [55]

ηth = 80%

Three-part receiver

Dahler et al. [12]

ηopt = 82%

1.2.4. Conical cavity receivers The conical design is an alternative choice which has been also examined by many researchers. The conventional conical design, as well as a double-conical or hetero-conical design, is found in the literature. Dunn et al. [37] studied a solar dish system for operation with ammonia which has a conical receiver. Their experiments proved the low performance of about 35%. Giovannelli and Bashir [38] studied the use of a receiver with a conical cavity and phase change materials. Yan et al. [39] studied a system with a conical receiver with an internal reflecting surface and they managed to achieve 90% optical efficiency. Pavlovic et al. [40] studied a classical cavity receiver with a conical shape which includes tubular absorber tubes. They found that this design is better than a flat one with a spiral shape. They found the thermal efficiency of up to 79.4% and the optical efficiency of 85.2%. A similar design has been studied experimentally by Turini et al. [41]. In this design, there is only a small parabolic part of the concentrator and the cavity has tubular conical absorber which presents receiver efficiency of around 80%. Recently, Khalil et al. [42] suggested a thermal plate conical receiver for a solar dish concentrator. In the other part of the literature studies, the hetero-conical exist. This design has been suggested in an old study by Williams [43] in 1980 for operation up to 750 °C with ammonia. Pye et al. [44] found numerically the maximum receiver efficiency at 98.7%. Lastly, Thirunavukkarasu and Cheralathan [45] found maximum thermal performance up to 40% with experiments with a hetero-conical design. 1.2.5. Flat receivers There are some studies in the literature which investigates flat cavities which have a small depth and practically the receiver close to the aperture. The most usual design is the spiral one which has been examined extensively by Pavlovic et al. [46,47]. In Refs. [46], the system was studied experimentally and the maximum experimental efficiency was found at 34%. In another numerical study [47], it is found that the efficiency can reach up to 65% by eliminating the optical errors. Yang et al. [48] tested a double-stage dish concentrator with a flat receiver. They found that the double-stage receiver is beneficial compared to the conventional dish design and it presents 19% higher peak optical efficiency. In another interesting work, a solar dish coupled to a still is examined by Bahrami et al. [49]. The solar still is placed in the focal region of the concentrator and its aperture is a flat area which can be assumed to be a flat receiver. Senthil and Cheralathan [50] studied a flat receiver with straight and curved paths in its back. They found thermal efficiency 63.3% with the straight paths and 67.1% with the curved paths. Furthermore, Toygar et al. [51] studied the manufacturing process of a solar dish collector design with a flat receiver.

1.2.2. Rectangular cavity receivers The rectangular cavity shape is not so usual to the literature but it has studied by some researchers. It includes the cubical design for the special case with an aspect ratio equal to 1 (cavity length to cavity diameter ratio). The reason for the lower interest on this cavity is the greater outer surface compared to the cylindrical cavity which can increase the thermal losses. Le Roux et al. [30] found the optimum aperture of a rectangular cavity to be about 0.35% of the reflector diameter, while the receiver efficiency can reach up to 85%. Loni et al. [31] found that the optimum design regards high cavity height in order to maximize the optical efficiency. The maximum thermal efficiency is 75.40% and it found for aspect ratio equal to 1 with the minimum examined tube diameter (10 mm). 1.2.3. Spherical cavity receivers The spherical receiver design is a usual choice for solar dishes and it also includes the hemispherical designs. The advantage of this design is the minimized external surface which leads to low thermal losses. The majority of the studies are focused on the convective thermal losses calculations and no to the performance of the solar system. So, a brief summary of the studies is given below. Kumar and Reddy [32] compared a cavity receiver, a semi-cavity and a modified cavity with spherical shapes. They condoled that the higher convection thermal losses exist in the case of the cavity design, while the modified cavity with a decreased aperture in the down part is the best choice. Loni et al. [33] found that wind speed increases the thermal losses of a receiver in a linear way. During their experimental study, they found the thermal efficiency to be up to 60%. In another work, Zhou et al. [34] studied a spherical cavity receiver with 88.9% optical efficiency. They found that the thermal conversion efficiency of this system is varied from 81.9% up to 84.4%. Reddy and Kumar [35] found that the convection losses

1.2.6. Other receivers Except for the previous classifications, there are also alternative designs which have been studied by researchers. The volumetric receivers have been examined in Refs. [52] and [53]. Zhu et al. [52] studied a cylindrical shape volumetric receiver with maximum thermal efficiency 87% and maximum exergy efficiency 36%. Wang et al. [53] investigated a windowed volumetric solar receiver with a thermal efficiency of around 64%. Yang et al. [54] suggested the use of an air path inside the volumetric receiver which can reduce the convective thermal losses by about 58%. Xu et al. [55] suggested a tapered tube bundle receiver which has a shape similar to spherical and it can achieve up to 80% thermal efficiency. Dahler et al. [12] examined a double stage receiver which has three different absorbers for fuel production via 3

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thermochemical redox cycles. This design achieves high concentration and optical efficiency is about 82%. 1.2.7. Comparative studies about the cavity receivers In the literature, there are many studies which compare some different designs. However, the comparisons are not always detailed which means in optical and thermal efficiency terms. Generally, the conical design has been found to be better than spherical and cylindrical by Daabo et al. [56–58]. Moreover, Jilt et al. [59] found the convective thermal losses of the conical cavity to be lower compared to other designs. Moreover, Xie et al. [60] found the conical design to be better than spherical and cylindrical. Seo et al. [61] stated that the conical design is better than the dome design for operation at 200 °C, while the dome is better at 100 °C. The next part of the literature studies highlights the use of spherical designs to be the optimum. Loni et al. [62] found experimentally the hemispherical design to have better thermal efficiency than cubical and cylindrical cavities. Yan et al. [63] found that the spherical design is better optically compared to conical and cylindrical respectively. In another work which is based on the heat flux uniformity, Harris and Lenz [64] performed one of the oldest studies on the field and they found the spherical design to have the most smoother profile, while the conical to be the worst case. Similar results have been found by Shuai et al. [65,66] with the spherical design to be the best case. The conical design presents a region with high heat flux concentration which creates temperature peaks (hot spots), something that reduces the thermal efficiency due to the local high radiation thermal losses.

Fig. 1. The examined solar dish collector (the case with the cylindrical cavity).

1.3. The present work The previous literature review indicates that there is a lot of interest in the design of different cavity receivers of solar dish concentrators. The reason for this situation is based on the need for creating highly efficient designs, especially in medium and high temperatures. In this direction, this paper comes to examine the most usual cavities which are the cylindrical, rectangular, spherical and conical, as well as a promising design with the cylindrical-conical shape. All the designs are optimized using the thermal efficiency maximization as the objective function. To our knowledge, there is not any other so detailed comparative work which compares and optimizes the cavities optically, thermally and exergetically under various operating conditions. Usually, many literature studies optimize the cavity receivers only optically, something that does not lead always to the maximization of the thermal efficiency and consequently of the useful heat production. The present analysis is conducted with a developed model in SolidWorks Flow Simulation [67] and this model is validated with experimental results from Ref. [22]. The importance of this work is based on a clear comparison of the most usual designs, as well as the suggestion of a new one. Moreover, the comparative study is able to make clear the performance deviations between the different designs in optical and thermal terms. It is important to state that in the present work, the working fluid is thermal oil which can operate up to 400 °C and thus the maximum selected inlet temperature is 350 °C. 2. Material and methods 2.1. The examined cavity receivers In this work, five different cavity receivers are investigated. Every design is optimized using the thermal efficiency maximization criterion and then the optimum cases are compared. The examined designs have the following shapes: cylindrical, rectangular, spherical, conical and cylinder-conical. All the designs have generally studied by many researchers but the last one has only studied optically in Refs. [27,59]. Moreover, the analysis is optical, thermal and exergetic for various operating temperatures and so it can be characterized as a detailed one.

Fig. 2. The examined cavities of the present study.

4

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Fig. 1 shows the examined solar collector with the concentrator and the cavity receiver. Fig. 2 illustrates the five examined cavities. The dimensions of the concentrator have been taken from the Ref. [22] which includes experimental results for a cylindrical cavity receiver. The optimization of all the examine cavities is conducted firstly with the cavity placed at the focal distance. In this case, the parameter (drec) is equal to 526.5 mm. The length of the cavity (L) is the main optimization parameter for the designs (a) to (d). For the design (e) is the ratio of the cylindrical part to the total part is the optimization parameter. The cone angle (Φ) is extra optimization parameters for the designs (d) and (e). The next step in the optimization is the investigation of the optimum location of the cavity by investigating the different values of the parameters (drec). The optimized designs for (drec = 526.mm or drec = f) are the optimum geometries which are studied for different (drec). Finally, the five optimized cases are compared for different operating temperatures with thermal oil. Therminol VP-1 is the used thermal oil which can operate up to 400 °C [68]. Table 2 includes the main parameters of this work about the designed geometry and the simulation parameters. During the optimization procedure, the length of the cylindrical and conical receivers has to be ranged from 0.5∙D up to 2.5∙D, while in the rectangular case it is ranged from 0.5∙a up to 2.5∙a. For the spherical case, the length changes by using the sphere diameter (Ds) as the equivalent parameter. More specifically, the parameter (Ds) takes the values D, 1.1∙D, 1.2∙D and 1.3∙Ds. The equation which correlates the length (L) with the parameters (Ds) and (D) is the following:

Lsph =

Ds +

Ds 2 2

2.2. Mathematical formulation In this section, the basic equations about the evaluation of the solar collector are given. The available solar irradiation on the collector aperture (Qs) is calculated as the product of the dish aperture (Aa) and the of the direct solar beam irradiation (Gb). The useful heat production (Qu) of the solar dish collector is calculated as below:

Qu = m · cp·(Tout

th

2 = 2.249·10 7·Trec + 1.039·10

=

opt

Trec + 5.599·10

2

(5)

=

Qabs Qs

(6)

The heat transfer coefficient (h) of the flow can be calculated according to the next formula:

(1)

4

Qu Qs

The optical efficiency (ηopt) is calculated as the ratio of the absorbed solar irradiation from the absorber coils (Qabs) to the incident solar irradiation (Qs):

Qu A coil ·(Trec

(7)

Tfm )

The mean fluid temperature (Tfm) is calculated as:

In the case of the cylinder-conical design, the length (L) is selected to be the same as the aperture diameter (D), while the ratio of the cylindrical part (L1) to the total (L) is ranged from L1/L = 0.3 up to L1/ L = 0.7. The cone angle (Φ) for the designs with conical and cylindricalconical shapes takes values 50%, 75% and 100% of the (Φmax). The receiver position (drec) is selected to be ranged from 496.5 mm up to 556.5 mm, with the initial position is at the focal point (drec = 526.5 mm). Practically, the displacements have been examined with a step of 10 mm. It has to be said that the aperture of the cavity is the same in all the cases and it is at 121 mm, a value in accordance with the Ref. [22] about the present concentrator. The length of the cavity is the variable parameter. The way that the conical shapes are created is given in Appendix A, as well as the values of the maximum cone angles are given in this Appendix. The system is examined for ambient temperature equal to 21 °C and solar irradiation at 1000 W/m2 which are reasonable choices for the evaluation of the different cases. The main goal of this work is the investigation of the optimum cavity geometry and so the emphasis is not given in the different operating condition. Only the inlet temperature is ranged from 50 °C to 350 °C, while the flow rate is kept at 0.03 kg/s, a value which leads to turbulent flow in order to achieve high heat transfer rates. More specifically, the Reynolds number is greater than 5000 in all the cases, so the flow can be easily adopted as turbulent. The convection thermal loss coefficient between the cavity and the ambient air is selected at 10 W/m2K which is a typical value according to the literature for air velocity close to 1 m/s [69]. This value is not depended on the working fluid kind indie the tube and it can be used for simulations with thermal oil, molten salt or gas working fluid. The absorber tube is selected to be made of copper, the cavity to be made of steel and the insulation is glasswool with thermal conductivity of 0.04 W/mK. All the used parameters in the developed model are summarized in table 2. About the emittance of the receiver for the absorber tubes and metallic cavity (εrec), the following formula has been used. The temperature of the receiver (Trec) is in (°C) [70]: rec

(4)

Tin)

The thermal efficiency of the collector (ηth) is the ratio of the useful heat production to the available solar irradiation:

h=

D2

(3)

Qs = Aa · Gb

Tfm =

Tin + Tout 2

(8)

The Nusselt number (Nu) of the fluid flow is calculated as below:

Nu =

h· Dcoil, in k

(9)

The theoretical value of the Nusselt, which is used only for validation reasons, is estimated using the model of Dittus-Boelter [71]: (10)

Nu = 0.023·Re 0.8·Pr 0.4

The Reynolds number (Re) for the tubular coil is calculated as:

Re =

4· m ·Dcoi, in · µ

(11)

Table 2 Basic data of the present work (geometry and simulation parameters).

(2) 5

Parameter

Symbol

Value

Aperture of the collector Concentrator diameter Focal distance Concentrator rim angle Concentrator reflectance Concentration ratio Receiver diameter Metallic cavity thickness (side) Metallic cavity thickness (back) Insulation thickness Tube inner diameter Tube outer diameter Helical coil pitch Receiver absorbance Insulation absorbance Insulation emittance Ambient temperature Solar direct beam irradiation Mass flow rate Heat transfer coefficient for convection

Aa Dcon f Φrim ρ C D tcav1 tcav2 tiso Dcoil,in Dcoil,out p αrec αiso εiso Tamb Gb m hout

1.676 m2 1461 mm 526.5 mm 69.5o 0.93 146 121 mm 1.5 mm 3 mm 28 mm 7 mm 9 mm 10 mm 0.9 0.8 0.8 21 °C 1000 W/m2 0.03 kg/s 10 W/m2K

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The friction factor of the flow (fr) can be calculated using the pressure drop along with the coil (ΔP), as below:

fr =

P· 1 · 2

· u2

·

Dcoil, in Lcoil

sun in two directions. – The mass flow rate (m) is selected at 0.03 kg/s. – The inlet temperature (Tin) is ranged from 50 °C up to 350 °C. – The heat convection coefficient of the cavity (hout) is selected at 10 W/m2K [69]. – The pressure at the outlet is selected at 15 bar in order to keep the fluid at the liquid state in all the cases [68]. – The concentrator reflectance (ρ) is selected at 93%. – The absorbance of the cavity is selected at 90%, while its emittance is calculated by Eq. (2). – The insulation material has absorbance 80%, emittance 80% and thermal conductivity 0.04 W/mK.

(12)

The theoretical value of the friction factor, which is used only for validation reasons, is estimated using the formula [72]:

fr =

0.3164 + 0.03 Re 0.25

Dcoil, in D

0.5

µcoil

0.27

µ

(13)

The velocity (u) inside the coil is calculated as:

m

u= ·

2 ·Dcoil , in

Moreover, the proper materials of the system have been selected as below:

(14)

4

The pumping work demand is calculated as:

Wp =

m

· P

– – – – –

(15)

The useful exergy production of the solar collector (Eu) is calculated as below [69]:

Eu = Qu

m · cp·Tam·ln

Tout Tin

m ·Tam· P fm · Tfm

4 Tam 1 T · + · am 3 Tsun 3 Tsun

4

(17)

The sun temperature is selected at 5770 K, while it has to be said that the temperature levels in Eqs. (16) and (17) have to be in Kelvin units. The exergetic efficiency of the solar collector (ηex) is the ratio of the useful exergy production to the solar exergy input, as below [69]: ex

=

Eu Es

working fluid is thermal oil (Therminol VP-1) [67]. material of the insulation is glasswool. absorber tubes are made of copper. absorber cavity is made of steel. reflector is a proper mirror material.

About the calculation procedure, it has to be said that the SolidWorks Flow Simulation [67] solves the governing equations for turbulent flow conditions. The turbulent model in this work is the k-ε which has been used in many similar studies [75,76]. Lastly, it has to be said that the fluid is assumed to be fully developed in the coil inlet. During the optical analysis, the proper number of solar rays has to be selected in order to have correct results. Table 3 gives the results of this simple sensitivity process for the cylindrical design. According to the results, the use of 2∙106 solar rays is selected as the minimum value which leads to converged results. The last step in the model development is the selection of a proper mesh. An independence procedure is followed and the results are given in Table 4. The final mesh has about 2.28 million cells. It is important to state that proper treatment for the fluid cells has been done in order to have denser mesh in the fluid and the partial cells. The use of a local mesh was the main way of achieving independent results. This procedure is performed for the cylindrical cavity and for inlet temperature equal to 21 °C.

(16)

The exergy flow of the solar irradiation (Es) is estimated using the Petela model [73]:

Es = Qs· 1

The The The The The

(18)

The exergy efficiency is an index that shows the maximum work production that can be produced by solar irradiation. This index evaluates the useful heat production, the operating temperature, as well as the pumping work demand through the pressure drop. Therefore, it is an ideal index for the overall evaluation of solar collector performance.

2.4. Model validation

2.3. The developed model

The developed model is SolidWorks Flow Simulation is validated using literature experimental results for the cylindrical design. The experimental results of the Ref. [22] are used and the comparison is given in Fig. 3. The validation regards the temperature different (Tout − Tin) and this quantity is used because this was given in Ref. [22]. It is useful to state that the validation has been performed for water as the working fluid and for inlet temperature equal to 21 °C. Practically this temperature difference is equivalent to the thermal efficiency validation. The mean deviation is found to be 4.7% which is an acceptable value. The next part of the validation procedure regards the comparison of

In this work, the analysis is conducted with SolidWorks Flow Simulation [67] which is a computational fluid dynamics program. The advantage of this software is the simultaneous optical, thermal and flow analysis, as well as the possibility for investigation of different geometric designs. This software has been used for a great number of solar thermal studies and so it can be assumed as reliable. More specifically, the present software has been used for the simulation of solar dish collectors in Refs. [47,74]. All the examined designs have been created in SolidWorks environment and they automatically imported in the Flow Simulation studio for further analysis. The proper boundary conditions have been selected in order to study the problem properly. Below, the boundary conditions of the present work are presented. For more details, the previously referenced papers can be used. It is important to state that the absorbed solar irradiation by the coils is calculated by the program and it is given as an output of the simulation.

Table 3 Selection of the solar rays for the case with the cylindrical receiver. Number of solar rays 3

5∙10 5∙104 5∙105 106 2∙106 3∙106 4∙106

– The ambient temperature is equal to be 21 °C in all the simulations. – The solar direct beam irradiation (Gb) is equal to 1000 W/m2 in all the cases. – The solar incident angle is zero because the solar dish follows the 6

Absorbed energy by the coils (W) 934.6 950.1 947.0 948.9 948.5 948.5 948.5

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88.11% but in this case, the thermal efficiency is only 34.44%. The reason for the thermal efficiency reduction with the cavity length increase is based on the higher outer surface of the cavity. Moreover, it has to be said that higher cavity length increases the absorption of the incoming solar rays into the cavity and thus the optical efficiency increases with the cavity length increases. Practically, higher cavity length captures more secondary reflections inside the cavity and so it is beneficial for the cavity by the optical point of view. Fig. 7 shows the optimization procedure of the cavity location with L = D. It is obvious that the optimum location is for (drec = 536.5 mm) and in this case, the thermal efficiency is 60.00% and the optical efficiency 81.34%. Practically, the optimum location is very close to the focal point position (initial analysis of Fig. 6) and it is about 10 mm higher than this location. This is a reasonable result because the solar rays are concentrated close to the focal point region. The next step in this analysis regards the temperature distribution and the heat flux distribution over the examined designs at the focal point which is exhibited in Figs. 8 and 9 respectively. Fig. 8 shows that the maximum outlet temperature is found for the case L = D and this fact is in accordance with the thermal efficiency optimization. Higher cavity lengths reduce the outlet temperature and it is obvious that the coil turns close to the outlet (upper part) are cooled, something nonbeneficial. Fig. 9 gives an image of the heat flux distribution of the various designs. It can be said that the optimum case (L = D) has high heat flux in all the internal surface, while the other designs with greater length have not high heat flux at the upper coils. So, the upper coils cool the thermal oils and the performance is reduced. This fact makes the outlet temperature to be reduced in the cases with lower heat flux in the upper coils. Lastly, it has to be said that the main reason for the lower efficiency at L = 0.5D is the reduced optical efficiency of this scenario.

Table 4 Mesh independence procedure for cylindrical cavity and Tin = 21 °C. Number of cells

71,746

229,314

543,689

1,050,645

1,802,921

2,281,728

Tout (°C) Tcoil (°C) ηth fr

27.3 27.7 50.5% 0.71

27.5 26.3 52.2% 0.14

27.5 26.1 52.2% 0.08

27.5 26.1 52.3% 0.05

27.5 26.2 52.6% 0.04

27.5 26.3 52.5% 0.04

Fig. 3. Validation of the developed model for the cylindrical case using experimental results of Ref. [22] about the temperature difference of the fluid.

the Nusselt number and the friction factor with theoretical values. More specifically, for the cases of Fig. 3, the Nusselt number is calculated with SolidWorks and it is compared with the results from Eq. (10). Fig. 4 illustrates the comparison for two different inlet temperatures (21 °C and 81 °C) and the mean deviation is found close to 7%. Fig. 5 shows the comparison of the friction factor of the SolidWorks with the theoretical results of Eq. (13). The mean deviation of the friction factor is found to be 6.4%. These results indicate that the developed model leads to reasonable and acceptable results. So, the thermal analysis has been validated by the results of Fig. 3 and the flow by the results of Figs. 4 and 5.

3.1.2. Optimization of the rectangular cavity receiver The optimization of the rectangular cavity receiver is depicted in Figs. 10 and 11. Fig. 10 illustrates the thermal efficiency and the optical efficiency for different lengths of the rectangular cavity. The optimum cavity is found for L = a and in this case, the thermal efficiency is 53.77%, while the optical is 79.01%. The maximum optical efficiency is found for L = 2.5∙a and it is 88.00% but in this case, the thermal efficiency is only 23.76%. As in the analysis of the cylindrical cavity, the thermal efficiency maximization exists for a length equal to the opening aperture and for greater lengths the thermal efficiency is reduced while the optical is enhanced. Fig. 11 indicates that the optimization procedure of the cavity location with L = a leads to an optimum location for (drec = 536.5 mm). In this case, the thermal efficiency is 54.58% and the optical efficiency of 80.11%. Practically, the optimum location is about 10 mm higher than the focal point. Figs. 12 and 13 show the fluid temperature distribution and the heat flux distribution for the examined cases. The results are similar to the

3. Results and discussion 3.1. Optimization of cavity receivers The first part of this work regards the optimization of the five different cavity receivers. For every receiver, the distance from the concentrator (drec) and the cavity length (L) are the optimization parameters Moreover, for the designs with conical parts, the cone angle (Φ) is an extra optimization parameter. In every case, the optimization is conducted by examined different designs and evaluating them according to the thermal efficiency for inlet temperature at 300 °C. This is a high-temperature level which is close to the real applications of solar dishes and thus it is selected to be used as the reference once for the optimization of the different scenarios. It has also to be said that firstly the cavity length and the cone angle are optimized with the cavity located at the focal point. The next step is the optimization of the cavity position. This methodology has been found to lead to the global optimum solution, after conducting a more detailed procedure for the cylindrical cavity, and so it is selected to be followed in this work. 3.1.1. Optimization of the cylindrical cavity receiver The optimization of the cylindrical cavity receiver is given in Figs. 6 and 7. Fig. 6 shows the thermal efficiency and the optical efficiency for different lengths of the cavity. The optimum cavity is found for L = D and in this case, the thermal efficiency is 59.59%, while the optical is 80.67%. The maximum optical efficiency is found for L = 2.5∙D and it is

Fig. 4. Nusselt number validation with the theoretical results of Eq. (10). 7

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Fig. 5. Friction factor validation with the theoretical results of Eq. (13).

Fig. 7. Thermal and optical efficiency for different locations of the cylindrical cavity (L = 1∙D).

previous about the cylindrical cavity. The case with L = a leads to a heat flux distribution in all the geometry, while the greater lengths present regions with low heat flux. This fact makes the working fluid in the high length cases to be cooled in the last coil turns close to the outlet. In the end, it has to be said that the optimum design is the cubical one because this is the most compact rectangular shape. In other words, the compact design leads to lower outer area and so the thermal losses are reduced.

the optical efficiency of 78.78%. Practically, the optimum location is about 10 mm higher than the focal point. Fig. 16 shows that the maximum outlet temperature is found for Ds = 1.1∙D, while only the case Ds = 1.3∙D gives an important decrease in the outlet temperature. These results indicate that the spherical design has to be with a reasonable sphere diameter and close to the hemispherical one (Ds = 1.1∙D). The optimum design has a sphere diameter 10% greater than the aperture and this rule is a useful one for the design of spherical cavities. In Fig. 17, it is obvious that for the design of Ds = 1.1∙D the entire upper region is red, while for the Ds = 1.3∙D there is a yellow part. This is the reason for the decrease in the thermal efficiency for Ds over 1.1∙D. Practically, the lower heat flux in the upper coils leads to reduced outlet temperature due to the local cooling of the working fluid.

3.1.3. Optimization of the spherical cavity receiver The optimization of the spherical cavity receiver is displayed in Figs. 14 and 15. Fig. 14 shows the thermal efficiency and the optical efficiency for different lengths of the spherical cavity. The optimum cavity is found for sphere diameter Ds = 1.1∙D which means that the length is L = 0.729∙D, according to equation (1). In this case, the thermal efficiency is 62.68%, while the optical is 74.13%. The maximum optical efficiency is found for Ds = 1.3∙D and it is 82.75% but in this case, the thermal efficiency is only 40.49%. As in the previous subsections (3.1.1 and 3.1.2), the thermal efficiency maximization is not achieved for the same geometry as the optical efficiency maximization. Fig. 15 shows that the optimization procedure of the cavity location with Ds = 1.1∙D leads to an optimum location for (drec = 536.5 mm). In this case, the thermal efficiency is 63.98% and

3.1.4. Optimization of the conical cavity receiver The optimization of the conical cavity receiver is given in Figs. 18–20. Figs. 18 and 19 show the optical and the thermal efficiency respectively for the different combinations of the cavity length (L) and the cone angle (Φ). Fig. 18 indicates that the optical efficiency is higher for the cavities with greater length and lower cone angle. On the other hand, Fig. 19 shows that the thermal efficiency is maximized for length

Fig. 6. Thermal and optical efficiency for different cylindrical cavity lengths (cavity at the focal point – drec = 526.5 mm). 8

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Fig. 8. Fluid temperature distribution for the cylindrical cavity cases.

Fig. 10. Thermal and optical efficiency for different rectangular cavity lengths (cavity at the focal point – drec = 526.5 mm).

Fig. 9. Heat flux distribution of the cylindrical cavity cases.

(L = D) for all the examined cone angles. Moreover, the cone angle which gives the maximum thermal efficiency is equal to the 75% of the maximum possible and more specifically it is 18.53° (see also the Appendix A for more details about the maximum angle calculation). In the optimum case (L = D and Φ = 0.75Φmax), the thermal efficiency is 66.56% and the optical efficiency of 80.69%. Fig. 20 shows that the optimum location of the previous design is at drec = 516.5 mm and in this case, the thermal efficiency is 66.85% and the optical efficiency 80.96%. More specifically, the optimum location is about 10 mm lower than the focal point. Fig. 21 makes clear that the maximum fluid temperature is found for the case L = D, while Fig. 22 shows that this optimum case has the most uniform heat flux distribution. These figures depict results for (Φ = 0.75Φmax) which is the optimal cone angle. Moreover, it has to be said that for the cases for L > D, the upper coils do not receive so high

Fig. 11. Thermal and optical efficiency for different locations of the rectangular cavity (L = 1∙a).

amounts of solar irradiation and so they cool the working fluid. Practically, the last coil turns in these cases are useless and thus the case of L = D is the optimum one. About the case L = 0.5D, the existing coils are not enough to capture the solar irradiation and this fact leads to a decreased optical performance. 3.1.5. Optimization of the cylindrical-conical cavity receiver The optimization of the cylindrical-conical cavity receiver is given in Figs. 23–25. Figs. 23 and 24 show the optical and the thermal efficiency respectively for the different combinations of the cavity length 9

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Fig. 12. Fluid temperature distribution for the rectangular cavity cases.

Fig. 14. Thermal and optical efficiency for different sphere diameters (cavity at the focal point – drec = 526.5 mm and constant opening at 112 mm).

Fig. 13. Heat flux distribution for the rectangular cavity cases.

ratio (L1/L) and the cone angle (Φ). For this cavity, the optimum length has been selected to be L = D, as in the conical and the cylindrical design. However, the cylindrical part is the optimization parameter through the ratio (L1/L). Fig. 23 indicate that the optical efficiency is maximized for ratio values around 50%-60% and the optical efficiency is higher for lower cone angles. Fig. 24 proves that the thermal efficiency is maximized for length ratio at 40% (L1/L = 40%) for all the examined cone angles. Furthermore, the cone angle which gives the maximum thermal efficiency is equal to the 75% of the maximum possible and more specifically it is 26.03° (see also the appendix A for more details about the maximum angle calculation). In the optimum case (L1/L = 40%) and Φ = 0.75Φmax), the thermal efficiency is 66.96% and the optical efficiency of 84.22%. Fig. 25 shows that the optimum location of the previous design is at drec = 536.5 mm and in this case, the thermal efficiency is 67.95% and the optical efficiency

Fig. 15. Thermal and optical efficiency for different locations of the spherical cavity (Ds = 1.1∙D).

85.42%. More specifically, the optimum location is about 10 mm higher than the focal point. Figs. 26 and 27 shows the temperature distribution and the heat flux distribution for the different length ratios and for (Φ = 0.75Φmax). It can be said that the found results have not great variations because the ratio of the cylindrical part does not affect the results in a high way. Generally, values in the range of 40%–60% give similar results and close to the optimum design of 40%. So, it can be concluded that the ratio (L1/L) has a small impact on the results and a soft optimization of 10

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Fig. 16. Fluid temperature distribution for the spherical cavity cases.

Fig. 19. Thermal efficiency for different conical cavity lengths and cone angles (cavity at the focal point – drec = 526.5 mm). Fig. 17. Heat flux distribution for the spherical cavity cases.

Fig. 20. Thermal and optical efficiency for different locations of the conical cavity (L = 1∙D and Φ = 0.75∙Φmax). Fig. 18. Optical efficiency for different conical cavity lengths and cone angles (cavity at the focal point – drec = 526.5 mm).

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Fig. 21. Fluid temperature distribution for the conical cavity cases.

this parameter is required. 3.2. Comparison of the different designs Section 2 is devoted to comparing the optimum designs which have been found in Section 3.1. Table 5 summarizes the optimum geometries for the five different cavity receivers. The real values of the parameter and not the relative values are given in this table. Fig. 28 illustrates the thermal efficiency of the five different collectors for different inlet temperatures from 50 °C up to 350 °C. The cylindrical-conical design is found to be the most efficient choice for all the examined temperature levels. Conical and spherical designs follow with the spherical to be better up to 100 °C and conical to be more efficient than the spherical for inlet temperature over 100 °C. The fourth design is the cylindrical one and the less efficient choice is the rectangular one. The rectangular one presents greater outer surface due to the edges. On the other hand, all the other designs have smoother outer

Fig. 23. Optical efficiency for different cylinder-conical length ratios and cone (cavity at the focal point – drec = 526.5 mm).

Fig. 22. Heat flux distribution for the conical cavity cases.

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losses of this design which does not make suitable the high thermal efficiency in great temperatures. Also, the exergy efficiency takes into consideration the pressure drop of every cavity which is depicted in Fig. 30. The spherical design has the lowest pressure drop, while the rectangular the highest. Generally, there are not huge differences in the pressure drop among the examined designs, and also the pressure drop is similar for all the temperature levels. The pumping work demand, which is given in Fig. 31, shows that there is no great need for fluid circulation in the cavities. More specifically, the pumping work demand is up to 0.6 W which is a very low value, compared to the useful heat production which is about 1000 W. Also, the high values of the exergy efficiency (up to 37%) prove that the pumping work demand is not something which reduces the overall system performance. The next presented parameter is the heat transfer coefficient inside the coil absorber. Fig. 32 depicts this parameter and it can be said that this coefficient takes high values up to 2100 W/m2K. This coefficient is higher at greater temperature levels because the flow becomes more turbulent in these cases. More specifically, higher temperature leads to lower dynamic viscosity and consequently to higher Reynolds number, the fact that increases the Nusselt number and the heat transfer coefficient. The greater values of the heat transfer coefficient are found for the conical cavity, while spherical, cylindrical-conical, cylindrical and rectangular cavities follow respectively. It can be said that the rectangular cavity is the worst design by all the points of view because it has the lowest thermal efficiency, the greatest pumping work demand and the lowest heat transfer coefficient. A summary of all the results is given in Table 6, while Fig. 33 shows a detailed comparison of the optimum designs. Fig. 33 shows the results for inlet temperature equal to 300 °C. The thermal, optical and exergy efficiencies are given in this figure. It is remarkable to state that the spherical design is the one with the lowest optical efficiency with 78.78%, while it is the third in the thermal efficiency sequence. Practically the spherical geometry increases the optical losses and reduces the thermal losses. The highest optical efficiency is found for the cylindrical-conical design with 85.42%, while this design has also the maximum thermal efficiency which is 67.95%. The conical comes to the second place according to the thermal efficiency criterion, while in the third place according to the optical efficiency criterion. The cylindrical design is the fourth in the thermal efficiency sequence and the third in the optical efficiency sequence. The less efficient thermally design is rectangular but it has higher optical efficiency than the spherical one. Lastly, it has to be said that the exergy efficiency trends follow the thermal efficiency trends in this work. Finally, it has to be said that the main reason for the highest thermal efficiency of the cylindrical-conical design is its higher optical efficiency compared to the other examined designs. Moreover, it is important to state that this design is the optimum in all the examined temperature levels which is an important result.

Fig. 24. Thermal efficiency for different cylinder-conical length ratios and cone (cavity at the focal point – drec = 526.5 mm).

Fig. 25. Thermal and optical efficiency for different locations of the cylindricalconical cavity (L1/L = 40% and Φ = 0.75∙Φmax).

surfaces and they are more efficient. The spherical design is the one which has the smallest outer surface but it is not suitable for capturing the solar rays inside it. On the other hand, the conical design is able to capture properly the solar rays, but the conical design presents a region with high heat flux which creates hot spots. So, the design with the cylinder and conical parts has been suggested in order to capture properly the solar rays and not to have hot spots. In this way, the maximum thermal efficiency can be achieved, as Fig. 28 proves. Fig. 29 shows the exergy efficiency of the examined cases. The performance sequence is the same energetically and exergetically, with the cylindrical-conical design to have the highest exergetic efficiency. It is remarkable to state that the exergy efficiency is higher when the inlet temperature increases and this fact proves that solar dish collectors can be used in medium and high temperatures. The rectangular design, which is the less efficient thermally, shows to have the minimum exergy efficiency and its curve tends towards to horizontal for high inlet temperatures. The reason for this result is based on the high thermal

3.3. Discussion of the results and future work The present work compared five different optimized designs in

Fig. 26. Fluid temperature distribution for the cylinder-conical cavity cases. 13

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Fig. 27. Heat flux distribution for the Cylinder-conical cavity cases. Table 5 Summary of the optimum designs. Shape

Optimum design parameters

Distance from the concentrator

Cylindrical Rectangular Sphere Conical

L = 121 mm L = 121 mm L = 94.3 mm L = 121 mm Φ = 18.53o L = 121 mm L1 = 48.4 mm Φ = 26.03o

drec = 536.5 mm drec = 536.5 mm drec = 536.5 mm drec = 516.5 mm

Cylindrical – Conical

drec = 536.5 mm

Fig. 29. Exergy efficiency of the examined designs for different inlet temperatures.

relatively novel one with cylindrical-conical geometry. It is important to state that the present work compares the design thermally, while an optical comparison is also conducted. However, many previous comparative literature studies [56–58,63–66] have only evaluated the cavity designs optically which introduces important questions about the final results. The thermal comparison is the best way for the evaluation because it is directly connected with the final useful output. The thermal efficiency includes both optical and thermal losses and thus it is the most suitable criterion for the cavity evaluation. It is important to state that in this work, the optimum length of the cavity has been found to be the same as the aperture for the cylindrical, rectangular and conical designs. This result is in accordance with the existing literature about the cylindrical design [28] and the rectangular design [31]. Moreover, the Refs. [27] and [29] give similar data about the aspect ratio of the cylindrical cavity receiver. So, it can be said that the found results are reasonable in accordance with the existing studies. Moreover, the present results can be extended and to state that the

Fig. 28. Thermal efficiency of the examined designs for different inlet temperatures.

various operating temperature levels with thermal oil. This work is a detailed comparative study which comes to add something new in the literature and to give clear results about the optimum design of the cavity receivers for solar dishes. Practically, all the usual cavities are examined (cylindrical, rectangular, spherical and conical), as well as a 14

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upper limit of this work. However, the operation in higher temperature levels has to be conducted with different working fluids. Up to 650 °C, the use of molten salts is suggested as an alternative and effective choice. For temperature levels up to 800 °C or 1000 °C, the use of gas working fluids such as air or supercritical CO2 are recommended. The next step in this discussion section regards the suggestions for future studies on the field. Firstly, an experimental comparison of the different cavities is something useful. At this time, only the Ref. [62] has results about the comparison of different cavity receivers in low temperatures levels. In this work, the spherical design has been found more efficient than the rectangular and the cylindrical designs, a result which is in accordance with the present work. Except for the experimental works with more cavities, there is a need for both experimental and numerical studies in higher temperature levels with other working fluids such as air and supercritical carbon dioxide. Moreover, the cavity receivers can be optimized and design using alternative methods such as free-form optimization procedures. An interesting idea about the use of Bezier polynomials for optimization the secondary reflector of a linear Fresnel reflector [77] can be extended in the solar dished for the cavity optimization. Lastly, the cavity receivers can be used instead of the evacuated tubes in the linear parabolic trough concentrators, something that has highlighted also in the recent review paper of Ref. [5]. Another important future project regards the investigation of the dish capacity for different dish and cavity dimensions.

Fig. 30. Pressure drop of the examined designs for different inlet temperatures.

4. Conclusions The objective of this work is the investigation and the optimization of five different cavity receivers for the solar dish concentrators. More specifically, the examined cavity designs are cylindrical, rectangular, spherical, conical and cylindrical-conical. Every design is optimized in order to achieve maximum thermal efficiency for operating at 300 °C with thermal oil. The most important conclusions of this work are summarized below:

Fig. 31. Pumping work demand of the examined designs for different inlet temperatures.

– The best design optically is found to be the cylindrical-conical with an optical efficiency of 85.42%. The next design is cylindrical with 81.34%, while the conical, rectangular and spherical follow with 80.96%, 80.11% and 78.78% respectively. – According to the thermal efficiency criterion at 300 °C, the performance sequence is the following: cylindrical-conical, conical, spherical, cylindrical and rectangular with 67.95%, 66.85%, 63.98%, 60.00% and 54.59% respectively. – According to the exergy efficiency criterion at 300 °C, the performance sequence is the following: cylindrical-conical, conical, spherical, cylindrical and rectangular with 35.73%, 35.14%, 33.61%, 31.49% and 28.62% respectively. – It is important to state that the thermal efficiency sequence is generally the same for all the examined temperatures except than the spherical design which is better than the conical one for temperatures up to 100 °C. – It is found that the optimum length is equal to the aperture for the majority of the examined designs and more specifically for the cylindrical, the rectangular and the conical. Furthermore, in the designs with great cavity lengths, it is found that there is not adequate heat flux in the absorber coils close to the upper part and so the fluid is cooled when it flows through these coils. This is the fact that explains the existence of an optimum length for the examined cavity receivers. – The highest-pressure drop has been found for the rectangular design, while the lowest for the spherical design.

Fig. 32. Mean heat transfer coefficient in the flow of the examined designs for different inlet temperatures.

optimum shape has to be compact in order to have a small external surface for the minimization of the thermal losses and an adequate length for achieving high optical efficiency. Furthermore, it has to be said that the solar dishes can operate in higher temperature levels that the 350 °C which is approximately the

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Table 6 Summary of the results about the examined inlet temperatures. Parameters

Shapes

Inlet temperature – Tin (°C) 50

100

150

200

250

300

350

ηth (%)

Cylindrical-Conical Conical Spherical Cylindrical Rectangular

79.77 77.25 78.29 75.44 73.97

79.26 76.85 77.20 74.12 71.67

76.87 74.77 74.50 71.04 67.93

74.11 72.32 71.20 67.61 63.77

71.16 69.68 67.70 63.94 59.34

67.95 66.85 63.98 60.00 54.59

64.45 63.73 60.01 55.74 49.45

ηex (%)

Cylindrical-Conical Conical Spherical Cylindrical Rectangular

10.68 10.25 10.43 9.95 9.69

20.21 19.54 19.63 18.77 18.09

26.64 25.88 25.78 24.52 23.39

31.02 30.25 29.77 28.23 26.58

33.95 33.22 32.26 30.43 28.20

35.73 35.14 33.61 31.49 28.62

36.67 36.25 34.11 31.66 28.05

h (W/m2K)

Cylindrical-Conical Conical Spherical Cylindrical Rectangular

277 338 388 300 321

731 925 849 721 703

997 1375 1307 940 933

1147 1633 1531 1059 1043

1263 1840 1709 1146 1120

1348 1999 1841 1203 1164

1400 2101 1923 1233 1178

ΔΡ (Pa)

Cylindrical-Conical Conical Spherical Cylindrical Rectangular

5706 5855 5376 7088 10,157

5982 5973 5064 7293 9913

5717 5771 4756 6935 9177

5641 5760 4684 6839 9060

5687 5855 4716 6889 9136

5823 6035 4822 7049 9356

6043 6305 4998 7312 9712

Fig. 33. Final comparison of the optimum cases (Tin = 300 °C).

Declaration of Competing Interest

Acknowledgment

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Dr. Evangelos Bellos would like to thank “Bodossaki Foundation” for its financial support.

Appendix A. Design of the conical geometry In this appendix shows the way that the conical geometry has been created. Fig. A1 shows the design procedure in the SolidWorks. The angle (ω) can be calculated as below, from the triangle (AOC):

= tan

1

Hcoil Dcone 2

(A1)

where (Hcoil) and (Dcone) are the height and the aperture of the cone respectively. It has to be said that the (Dcone) is smaller than the aperture diameter (D) due to the tube thickness. The length (BD) can be calculated as: 16

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Fig. A1. Design of the conical absorber tube.

Table A1 Values of the (Φmax) for the conical design. Cases

L = 0.5∙D

L = 1∙D

L = 1.5∙D

L = 2∙D

L = 2.5∙D

Φmax (°)

43.8

24.5

16.6

12.4

9.9

Table A2 Values of the (Φmax) for the Cylindrical-conical design. Cases

L1 = 0.3∙L

L1 = 0.4∙L

L1 = 0.5∙L

L1 = 0.6∙L

L1 = 0.7∙L

Φmax (°)

30.9

34.7

39.5

45.4

52.7

(BD) =

Dcone 2

(AD)

(A2)

The length (AD) can be calculated as below, using the triangle (ADE):

AD =

Dcoil, in 2· sin ( )

(A3)

Finally, the maximum angle of the cone (Φmax) can be calculated from the triangle (BDC) as below: max

= tan

1

(BD ) H coil

(A4)

So, the values of the angle (Φmax) can be calculated for the different cases of the conical design (Table A1) and of the cylindrical-conical design (Table A2).

[4] Wu S-Y, Xiao L, Cao Y, Li Y-R. Convection heat loss from cavity receiver in parabolic dish solar thermal power system: a review. Sol Energy 2010;84(8):1342–55. [5] Wang F, Cheng Z, Tan J, Yuan Y, Shuai Y, Liu L. Progress in concentrated solar power technology with parabolic trough collector system: a comprehensive review. Renewable Sustainable Energy Rev 2017;79:1314–28. [6] Jafrancesco D, Cardoso JP, Mutuberria A, Leonardi E, Les I, Sansoni P, et al. Optical simulation of a central receiver system: comparison of different software tools. Renewable Sustainable Energy Rev 2018;94:792–803. [7] Hafez AZ, Soliman A, El-Metwally KA, Ismail IM. Design analysis factors and specifications of solar dish technologies for different systems and applications. Renewable Sustainable Energy Rev 2017;67:1019–36. [8] Gavagnin G, Sánchez D, Martínez GS, Rodríguez JM, Muñoz A. Cost analysis of solar

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