Performance Simulation of Solar Trough Concentrators: Optical and Thermal Comparisons

: The solar trough concentrator is used to increase the solar radiation intensity on absorbers for water heating, desalination, or power generation purposes. In this study, optical performances of four solar trough concentrators, viz. the parabolic trough concentrator (PTC), the compound parabolic concentrator (CPC), the surface uniform concentrator (SUC), and the trapezoid trough concentrator (TTC), are simulated using the Monte Carlo Ray Tracing method. Mathematical models for the solar trough concentrators are ﬁrst established. The solar radiation distributions on their receivers are then simulated. The solar water heating performances using the solar trough concentrators are ﬁnally compared. The results show that, as a high-concentration ratio concentrator, the PTC can achieve the highest heat ﬂux, but suffers from the worst uniformity on the absorber, which is only 0.32%. The CPC can generate the highest heat ﬂux among the rest three low-concentration ratio solar trough concentrators. Compared with the PTC and the CPC, the TTC has better uniformity, but its light-receiving ratio is only 70%. The SUC is beneﬁcial for its highest uniformity of 87.38%. Thermal analysis results show that the water temperatures inside the solar trough concentrators are directly proportional to their wall temperature, with the highest temperature rise in the PTC and the smallest temperature rise in the TTC. The solar trough concentrators’ thermal deformations are positively correlated to their wall temperatures. The radial deformation of the SUC is much larger than those of other solar trough concentrators. The smallest equivalent stress is found in the SUC, which is beneﬁcial to the long-term operation of the solar water heating system.


Introduction
Due to the low flux intensity, fluctuant and periodicity characteristics of solar energy, concentrators are of high importance to solar energy applications.The solar trough concentrator (STC) is one of the most developed solar concentrators.The STC has a trough shape reflector, which can focus sunlight to a line.There are many kinds of STCs reported in the literature, for example, the parabolic trough concentrator (PTC), the compound parabolic concentrator (CPC), the V-shaped trough concentrator (VTC) and the trapezoidal-shaped trough concentrator (TTC), etc. Extensive simulation and experimental studies have been reported in the literature for different STC types and their applications.
Padilla et al. conducted a one-dimensional numerical heat transfer analysis on PTC and found that the PTC has better performance for reducing heat loss [1].Wang et al. proposed a new concept of negative heat flux region, which provides a new optimization strategy for reducing the radiant heat loss of PTC [2].Avargani and Divband evaluated the performance of a solar water heating (SWH) system with glass-covered PTCs under different system tracking modes [3].Bellos et al. found that the absorber geometry was one main factor influencing the PTC performance and designed a dimpled absorber tube with sine geometry which increased the heat transfer surface and the flow turbulence [4].Freeman et al. proposed and evaluated a PTC-powered organic Rankine cycle (ORC) system for combined heating and power applications in the UK [5].Nafey and Sharaf designed and analyzed an ORC system with a reverse osmosis desalination process driven by solar PTC and CPC respectively [6].Zou et al. experimentally investigated a small-sized solar PTC for SWH in cold areas [7].Calise established a dynamic model of a solar PTC-based heating and cooling system with a double-stage LiBr-H 2 O absorption chiller [8].Coccia et al. manufactured a low-cost PTC for industrial process heat applications whose working temperature ranged from 70 to 250 • C [9].El Fadar et al. studied an adsorption refrigeration system powered by PTC with a heat pipe [10].Sanda et al. reviewed the modelling and simulation tools for direct steam generation (DSG) with PTCs [11].Temiz and Dincer proposed a solar PTC-driven thermochemical hydrogen production plant with thermal energy storage and geothermal systems [12].Sharaf et al. carried out a thermo-economic analysis of PTC-assisted MED-VC desalination processes [13].Kabeel et al. analyzed a passive solar water desalination system with a PTC-incorporated latent heat storage medium and concluded that the daily freshwater yield with the PTCs was nearly 50% higher than that with usual solar-based still [14].
Su et al. analyzed the optical and thermal performances of a CPC and found that the absorber received solar radiation was related to the sunlight incident angle [15].Abdullahi et al. compared the optical efficiencies of horizontal and vertical CPCs by using Monte Carlo Ray Tracing (MCRT) method and found that the performance of horizontal CPC was 15% higher than that of the vertical CPC [16].Ustaoglu et al. evaluated the optical, thermal and radiation distribution properties of a truncated CPC with the MCRT method, and found that the optimized incident angle was 20 • [17].Waghmare and Gulhane evaluated the CPC optical performance at low reception angles to improve the overall collector performance by minimizing optical loss [18].Xu et al. combined the CPC with a closed-end pulsating heat pipe and carried out an experimental investigation [19].Mohan et al. proposed a solar CPC-driven system for sustainable production of cooling, clean water and domestic hot water in the United Arab Emirates [20].Sonsaree et al. used CPC to drive a small-scale solar ORC power plant in Thailand [21].Dai et al. analyzed a hybrid solar hot water and Bi 2 Te 3 -based thermoelectric generator unit using a heat pipe with mini-CPCs [22].Brogren et al. discussed the optical efficiency of a water-cooled PV-thermal (PVT) system with low-concentration CPCs for high latitudes [23].Tiwari et al. analyzed the influence of water and condensing cover temperatures on the yield and electrical power output of a PVT-CPC active solar distillation system [24].Sharma et al. carried out energy, exergy, environmental impact, and economic analyses of solar CPC-powered thermal domestic SWH systems [25].Arunkumar et al. analyzed the performance of CPC-assisted tubular/concentric solar stills with different augmentation systems [26].
The V-shaped and trapezoidal-shaped cavities are usually used as the secondary reflector of the STCs.Zheng et al. proposed a new multi-cavity receiver STC, and their study showed that compared with the traditional STC, the multi-cavity receiver STC can achieve more sunlight absorption, higher heat transfer rate, and better temperature distribution at the receiver outer surface [27].Wang et al. analyzed the relationship between the concentration ratio and the gap efficiency of a V-shaped cavity CPC using the MCRT method [28].Reddy and Satyanarayana developed a 3-D numerical model to evaluate the performance of the receiver with square, triangular, trapezoidal and circular shapes [29].Wang et al. designed an inverted trapezoidal cavity receiver with an absorber of a bundle of tubes [30].Venegas-Reyes et al. simulated the optical performance of a circular trough solar concentrator with a trapezoidal secondary reflector [31].
The trough concentrators will generate uneven energy flow distribution on the absorber's outer surfaces.That will lead to large thermal stress and damage of absorbers in solar thermal applications; and also cause hot spots, current mismatch and efficiency reduction in the PV systems.To solve this problem, Yang et al. proposed a surfaceuniform-concentrator (SUC) and evaluated its optical performance in a solar photocatalytic hydrogen production system by using the MCRT method.They found that the solar energy conversion efficiency of the SUC is 8.57% higher than that of the CPC [32].Wei et al. experi-mentally used the SUC for solar photocatalytic hydrogen production, which validated the advantages of the SUC [33].With respect to the thermal deformation due to the uneven solar flux on the STC receivers, Cao et al. carried out thermal performance and stress analyses of the cavity receiver in the PTCs [34].Wang et al. developed a thermodynamic model to predict the thermal deformation of different cases for the preheating, boiling and superheating sections of a DSG-PTC loop [35].Zhang et al. carried out optical sensitivity analysis of geometrical deformation on the PTC with the MCRT method [36].
There are few studies both compared the optical and thermal performances of different STC types.To give directions for scholars and engineers on choosing the STC type in different application fields, the frequently used STC types, viz. the PTC, the CPC, the SUC and the TTC, are compared in the present study.Their optical performances are first simulated using the MCRT method to obtain solar radiation distributions on the receivers.Their thermal performances, viz. the temperature distribution, the thermal deformation and equivalent stress, are then compared by applying the STCs to an SWH system.

PTC
Figure 1 shows the schematic of a PTC model.According to Figure 1, P is the focal point of the parabola and φ is the circumference angle of the absorber tube.The reflector of the PTC is defined as [37]: The PTC structure is determined by the focal length OP (viz.f in Equation ( 1)), the opening width W and the absorber tube radius R.
The concentration ratio C of the PTC is defined as the ratio between the solar collecting area A a and the absorber tube area A r :

CPC
The cross-section of a CPC is shown in Figure 2. According to Figure 2, the reflector of the CPC is symmetrical with the focal line, which is composed of two groups of evolventcurves (BC and CD) and parabolic curves (AB and DE).The curve CD is defined as [37]: The curve DE is defined as [37]: where, where θ is the acceptance half-angle of the CPC.
The concentration ratio C of the CPC is defined as:

SUC
The cross-section of a SUC is shown in Figure 3.The SUC is composed of two symmetrical curves AC and BD and a line CD.

C D(a1,c1
) The curve BD is defined as [32]: where, (x b , y b ) is the coordinate of N.
The concentration ratio C of the SUC is defined as:

TTC
The cross-section of a TTC is shown in Figure 4.According to Figure 4, the reflector is an isosceles trapezoid.The main parameters of the TTC are the reflecting surface inclination angle δ, the centre height of the absorber tube H, the vertical height of the reflector h, the opening width of the concentrator W and the radius of the absorber tube R. The concentration ratio C of the TTC is defined as: To identify the surface radiation distribution uniformity on the absorber tube, the uniform index U is defined, which is [32]: where B max and B min are the maximum and minimum solar radiation on the absorber.

Geometry of the STCs
The geometry parameters of the STCs are summarized in Table 1.According to Table 1, the PTC concentration ratio is much larger than the other three STCs'.The PTCs are recognized as high-concentration reflectors, mainly used for high-temperature solar thermal utilization, such as solar thermal power generation, auxiliary heat source, etc.The CPC, SUC and TTC are recognized as low-concentration reflectors, mainly used for domestic SWH, solar photocatalysis or PV power generation.

Simulation Methods
The simulation methods in this research are shown in Figure 5.The solar ray tracing results of four STCs are first simulated using the MCRT method in TracePro software.The simulation process and input parameters are shown in Figure 5.The obtained solar radiation distribution on the absorber tube outer surface is then introduced into ANSYS Fluent to simulate the thermal performance of the STCs, viz. the absorber wall temperature distribution and the fluid temperature field.The thermal simulation results are then introduced as the thermal condition into ANSYS Workbench to simulate the static structure deformation and stress of the STCs' absorber tubes.

Model Validation
The MCRT method is used to simulate the optical performances of the STCs.The correctness of the developed models and the MCRT method in this paper is verified by comparing the simulated distribution of heat flux in the circumferential direction of the absorber outer surface with reported data in the literature.The PTC absorber heat flux distribution from Ref. [38] and the simulated results in the present study are compared in Figure 6.It is found in Figure 6 that good agreement is found between the simulated results and the results in Ref. [38].So, the mathematical model built in this study and the MRCT method is ready to simulate the optical performances of the four STCs.
During the MCRT simulation, several assumptions are applied: (1) Parallel light is emitted perpendicular toward the STC opening surface with a uniform wavelength; (2) The incident solar radiation intensity is 1000 W/m 2 ; (3) The tracking error of the STC is ignored, and the solar incidence angle is 0 • ; (4) The number of incident light is 1,000,000; (5) Only direct radiation is considered in the simulation, and (6) The reflectivity of STC reflector is assumed to be 1.

PTC Results
The solar ray tracing results, the average solar radiation at the cross-section, and the radiation distribution of the absorber upper and lower surfaces of the PTC are shown in Figure 7a-d respectively.According to Figure 7a,b, the average solar radiation at the cross-section of the PTC absorber is symmetrically distributed at 180 • , and there are two peaks located at 55 • and 310 • respectively.The heat flux of the PTC absorber is uneven along the circumferential direction, with a maximum value of 66,535 W/m 2 at 55 • and 310 • , and the minimum value of is 0 W/m 2 in 70 • -90 • and 270 • -290 • .Statistical results show that the average heat flux on the PTC absorber tube is 22,591.33W/m 2 .The circumferential radiation distribution of the absorber tube can be divided into four areas, viz. the shielding area, the increasing area, the attenuation area and the direct incidence area.According to an enlarged view of the absorber tube in Figure 7a, the incident light is blocked by the absorber tube at 0 • , leading to a sudden drop of heat flux, which is called the shielding area.Due to the increase of reflected light, the heat flux density increases continuously at 0 • -55 • , which is called the increasing area.The heat flux decreases rapidly due to the continuous reduction of reflected light near 70 • , which is called the attenuation area.Only direct radiation can be absorbed by the absorber at 90 • -180 • , whose heat flux is the initial solar radiation of 1000 W/m 2 .This range is called the direct incidence area.According to Figure 7c,d, the radiation distribution along the radial direction of the absorber is uniform.

CPC Results
The solar ray tracing results, the average circumferential solar radiation at the crosssection, and the radiation distribution of the CPC absorber upper and lower surfaces along the radial direction are shown in Figure 8a-d respectively.According to Figure 8a,b, the average solar radiation at the cross-section of the CPC absorber is symmetrically distributed at 180 • , and there are two peaks located at 65 • , 300 • and one secondary peak at 0 • respectively.At 65 • and 300 • , most of the incident light is focused after one reflection, whereas at 0 • (also 360 • ), the incident light reaches the absorber after multiple reflections resulting in greater reflection loss at 0

SUC Results
The solar ray tracing results, the average solar radiation at the cross-section, and the radiation distribution of the SUC absorber upper and lower surfaces along the radial direction are shown in Figure 9a-d respectively.It can be seen from the figures that more uniform solar radiation distribution is found on the SUC receivers than those of the PTC and CPC receivers.The circumferential radiation distribution is also symmetrical at 180 • , and it is fluctuant in a small range between 3577 W/m 2 and 4610 W/m 2 , with the average being 3941.54W/m 2 .Moreover, it is found that the heat flux on the upper surface of the absorption tube is greater than the initial incident radiation of 1000 W/m 2 , which indicates that the SUC's reflector can reflect incoming light to the range of 0 • to 360 • .According to previous analyses, the PTC and CPC mainly concentrate the incident light on both sides of the lower surface of the absorber tube and they mainly absorb direct solar radiation on the upper surface.So, their heat flux on the lower surface is greater than that on the upper surface.However, the heat flux on the upper surface of the SUC heat absorption tube is greater than that on the lower surface within the range of 90 • to 270 • .This is because the lower surface of the absorber only absorbs reflected light, whereas both direct and reflected light are received on the upper surface of the absorber.The superposition of the two kinds of radiation makes the upper surface heat flux larger than the lower surface heat flux.The difference between the maximum and the minimum heat flux of the SUC is 1033 W/m 2 , which denotes the high uniformity of the SUC.

TTC Results
The solar ray tracing results, the average circumferential solar radiation at the crosssection, and the radiation distribution of the TTC absorber upper and lower surfaces along the radial direction are shown in Figure 10a-d respectively.Compared with the PTC and CPC, the radial and the circumferential radiation distributions of the TTC are more uniform, because of their symmetrical structure.In addition, there is a dark area at the bottom of the absorber in the range of −30 • -+30 • , where incident and reflected light are shielded by the absorber.There are two peaks of 1171.9W/m 2 at 125 • and 235 • respectively.There are also two secondary peaks of 991.31 W/m 2 at 75 • and 285 • .The reason for their difference is that the absorber only absorbs the reflected radiation at the secondary peak, while there is a superposition of direct and reflected radiation at the peak.Only direct incident light reaches the absorber in the range of 160 • -180 • .The average heat flux on the TTC absorber is only 845.91 W/m 2 .

Absorber Tube Radiation Distribution Comparison
To compare the optical performance of the four STCs, four evaluation parameters, namely, the central angle of reflected light, the non-light central angle, the light reception ratio and the uniformity, are proposed.The central angle of reflected light refers to the maximum range of reflected light on the surface of the absorber.The non-light central angle refers to the region where the absorber does not receive any light.The light reception ratio refers to the proportion of the amount of light received on the surface of the absorber to the amount of incident light.The uniformity is calculated according to Equation (10).Comparison results of the optical performance of four STCs are shown in Table 2.   2, it is found that the PTC has the worst uniformity among the four STCs, which is only 0.32%.Its reflected light distributes in the region of 140 • , where the heat flux on the absorber tube is greater than the initial solar radiation of 1000 W/m 2 .In addition, there is a non-light incident region of 40 • on the PTC absorber.The reflected light distribution range of the CPC is 160 • .But the actual angle range where the heat flux is greater than 1000 W/m 2 is 100 • , and there is a non-light incident region of 20 • on the CPC absorber.Compared with the CPC and the PTC, the light distribution on the TTC absorber tube is more uniform.but the TTC's light reception ratio is poorer, whose no-light area reaches 1/6 of the whole surface.Compared with the other three STCs, the SUC holds the highest uniformity of 87.38%, and its reflected light can reach every location of the absorber.

Performance Evaluation in an SWH System
To compare the thermal performance of the PTC, CPC, SUC and TTC, the STCs are introduced into an SWH system.The cold water comes into the SWH system from one side of the absorber, and goes out of the SWH system on the other side, as shown in Figure 11.Five typical cross-sections are used to compare their thermal performances.The parameters used in the simulation are summarized in Table 3.The top and bottom wall temperatures of four STCs in the SWH systems are shown in Figure 12a-d.It can be seen from the figures that the PTC wall temperatures are much higher than the other three STCs' even though the PTC inlet water velocity is twice the other three STCs', due to the much higher concentration ratio of the PTC.Because of the solar radiation distribution as shown in Figures 7 and 8, the top wall temperatures of the PTC and CPC are much smaller than those at the bottom.But, according to Figure 12d, the top wall temperature of the TTC is larger than that at the bottom, which is also a result of the solar radiation distribution as shown in Figure 10.The temperature uniformity at the cross-section of the SUC is the best among the four STCs.And the highest wall temperatures match well with the STCs' solar concentration ratio in Table 1.Note: 1 As the PTC concentration ratio is much higher than the other three STCs, its inlet velocity is larger than the others, otherwise the water temperature may be over 100 • C.  The cross-section temperatures of the four STCs in the SWH systems are shown in Figure 13.It can be seen from Figure 13Aa-f that, due to the high solar flux at the bottom of the PTC, the wall and water temperatures at the bottom are much higher than those in the other locations.And according to Figure 13Ag, the hot water temperature area linearly increases and finally takes nearly 1/5 of the cross-section at the outlet.Figure 13Ah shows the water temperature rises gradually along the horizontal central cross-section.The PTC holds the highest water temperature rise among the four STCs.
Three peaks can be found at the cross-sections of the CPC according to Figure 13Ba-f, which are at the bottom and 60 • off the bottom.That fits well with the solar flux distribution in Figure 8b.The water temperatures near these areas are slightly higher than those in the other areas.The vertical cross-section temperature in Figure 13Bg shows the water temperature near the bottom is slightly higher than that in the other area, and the hot area takes nearly 1/10 of the cross-section at the water outlet.According to Figure 13Ca-f, the water temperature near the absorber is higher than that in the centre due to the uniform solar irradiation.As shown in Figure 13Cg,h, the water temperature along the horizontal cross-section is similar to that along the vertical cross-section, which validates the best uniformity of the SUC.
Unlike the other three STCs, according to Figure 13D, the water temperature near the top wall of the TTC is higher than that near the bottom, which is a result of the solar flux distribution as shown in Figure 10.The water temperature rise of the TTC is the smallest among the four STCs.

Thermal Deformation Comparison
The steel absorber thermal deformation and equivalent stress of the four STCs in the SWH systems are shown in Figure 14A-D respectively.The STCs' thermal deformations are positively correlated to their wall temperatures in Figure 12, with the highest thermal deformation of 9.9322 mm in PTC and the smallest thermal deformation of 0.2007 mm in TTC.And the receivers deform towards the direction of low-temperature areas.Due to the uniform temperature distribution, the radial deformation of the SUC is much larger than those of the other three STCs.It can also be found from Figure 14A-Db that, the smallest equivalent stress of 7.6525 MPa is found in the SUC, which is beneficial to the long-term operation of the SWH system.The highest equivalent stress of 165.52 MPa is found in PTC, due to the highest solar flux and wall temperature.

STC Application
The STCs can be combined with different receivers, including circular tubes 25,26], flat plates [22][23][24], cavity receivers [27][28][29][30][31], etc.The STCs are used to increase the solar radiation intensity on absorbers for SWH [3,7,26], DSG [11,35], desalination [6,13,14] and solar thermal/PV power generation [5,22,24,25] purposes.After optical, thermal and deformation comparisons of four STCs, it can be found that the PTC has the highest concentration ratio and heating performance.The PTC can be combined with all absorber types for heating purposes as the incoming heat flux is high and can reach the full area of the absorber.The PTC is thus suggested to be used in high-temperature thermal, DSG, and concentrated PV (CPV) purposes.The CPC, especially the truncated CPC has a low concentration ratio but benefits from its simple structure and low cost.The CPC can be combined with all the absorber types as sunlight can reach all areas of its receiver.It can be used for low-temperature solar thermal applications, such as the SWH and solar desalination.The CPC can also be used for solar photocatalysis purposes, as the truncated reflector can also reflect the diffused solar radiation, in which the ultraviolet radiation is high.Cao et al. have designed and constructed a pilot direct solar photocatalyst water-splitting system for hydrogen generation using the CPC modules [39,40].The SUC can hardly be combined with the flat plate receivers due to its special-designed geometry.The SUC have the highest uniformity, thus it can be used for solar PV purposes.The SUC is also suggested to be used in solar photocatalysis purposes, where photocatalysts are evenly dispersed in the solution.Too high local solar radiation or uneven solar radiation distribution on the absorber would lead to agglomeration and disability of the photocatalysts [32,33].The TTC holds high uniformity but low light reception and concentration ratio.Considering its simple structure and low cost, it is usually combined with flat plate receivers and used for solar PV purposes.The detailed STC characteristics and their suggested application fields are summarized in Table 4.

Conclusions
In the present study, optical and thermal performances of four STC types, viz. the PTC, the CPC, the SUC and the TTC, are compared using numerical simulation methods.The main conclusions of this study are summarized as follows: (1) As a concentrator with a high concentration ratio, PTC has the highest solar flux with two peaks at 55 • and 305 • on the receiver cross-section.The CPC can generate the highest heat flux among the three low concentrators (viz.the CPC, the SUC and the TTC).There are two peaks at 65 • and 300 • and one secondary peak at 0 • on the cross-section of the CPC.There is a dark area at the bottom of the TTC at the angle of −30 • -+30 • due to the absorber shielding.More uniform solar radiation distribution is found on the SUC receivers than that on the other three STCs.(2) The PTC has the worst uniformity of 0.32% among the four STCs.Compared with the CPC and the PTC, the light distribution on the TTC absorber tube is more uniform.But the TTC's light reception ratio is poorer, whose no-light area reaches 1/6 of the whole surface.The SUC holds the highest uniformity of 87.38%, and its reflected light can reach every location of the absorber.(3) The PTC wall temperatures are much higher than the other three STCs' even though the PTC inlet water velocity is twice the other three STCs'.The top wall temperatures of the PTC and CPC are much smaller than those at the bottom.The top wall temperature of the TTC is larger than that at the bottom.The temperature uniformity at the cross-section of the SUC is the best among the four STCs.The water temperatures inside the STCs are directly in response to their wall temperature, with the highest temperature rise in the PTC and the smallest in the TTC.(4) The STCs' thermal deformations are positively correlated to their wall temperatures.
And the receivers deform towards the direction of low-temperature areas.The radial deformation of the SUC is much larger than those of the other three STCs.The highest equivalent stress is found in PTC.The smallest equivalent stress is found in the SUC, which is beneficial to the long-term operation of the SWH system.

Figure 2 .
Figure 2. Schematic of a CPC model.

Figure 3 .
Figure 3. Schematic of a SUC model.

Figure 5 .
Figure 5. Simulation methods in this study.
Circumference angle of absorber tube / ϕ (°) Heat flux (W/m 2 ) Results from Liu et al., 2017 Simulated results in this sutdy

Figure 6 .
Figure 6.Comparison of PTC radiation distribution on the absorber outer surface between the simulated results in this study and Liu, et al., 2017 (Ref.[38]).

2 )Figure 7 .
Figure 7. (a) PTC solar ray tracing results (including an enlarged view of the absorber tube), (b) average circumferential solar radiation at the cross-section, (c) Upper surface solar radiation distribution along the radial direction, (d) lower surface solar radiation distribution along the radial direction.

Figure 8 .
Figure 8.(a) CPC solar ray tracing results, (b) average circumferential solar radiation at the crosssection, (c) upper surface solar radiation distribution along the radial direction, (d) lower surface solar radiation distribution along the radial direction.

Figure 9 .
Figure 9. (a) SUC solar ray tracing results (including enlarged view near the absorber tube), (b) average circumferential solar radiation at the cross-section, (c) upper surface solar radiation distribution along the radial direction, (d) lower surface solar radiation distribution along the radial direction.

-Figure 10 .
Figure 10.(a) TTC solar ray tracing results, (b) average circumferential solar radiation at the crosssection, (c) Upper surface solar radiation distribution along the radial direction, (d) lower surface solar radiation distribution along the radial direction.

Figure 11 .
Figure 11.Schematic of the STC receiver in the SWH system.

3. 4 . 1 .
Comparison of Top and Bottom Wall Temperature Distributions tracing results as shown in Figure10

Figure 12 .
Figure 12.The top wall and bottom wall temperatures of (a) the PTC, (b) the CPC, (c) the SUC and (d) the TTC in the SWH systems, where the above figure and below figure refer to the top wall and bottom wall respectively, and the left side and right side are the water inlet and outlet respectively.

Figure 14 .
Figure 14.The steel absorber thermal deformation and equivalent stress of (A) the PTC, (B) the CPC, (C) the SUC and (D) the TTC in the SWH systems where (a) is the thermal deformation (Unit: mm), (b) is the equivalent stress (Unit: MPa).Note that the shadow in the figures are the location of the original tube receivers.

Table 1 .
Geometrical parameters of the concentrators.

Table 2 .
Comparison of optical results of four STCs.

Table 3 .
Parameters in thermal performance system of the four STCs.