Effect of the Orientation Schemes of the Energy Collection Element on the Optical Performance of a Parabolic Trough Concentrating Collector

: While the circular shape is currently the proven optimum design of the energy collection element (ECE) of a parabolic trough collector, that is yet to be conﬁrmed for parabolic trough concentrating collectors (PTCCs) like trough concentrating photovoltaic collectors and hybrid photovoltaic/thermal collectors. Orientation scheme of the ECE is expected to have signiﬁcant effect on the optical performance including the irradiance distribution around the ECE and the optical efﬁciency, and therefore, on the overall energy performance of the PTCC. However, little progress addressing this issue has been reported in the literature. In this study, a thorough investigation has been conducted to determine the effect of the orientation schemes of ECE on the optical performance of a PTCC applying a state-of-the-art Monte Carlo ray tracing (MCRT) technique. The orientation schemes considered are a ﬂat rectangular target and a hollow circular, semi-circular, triangular, inverted triangular, rectangular and rectangle on semi-circle (RSc). The effect of ECE defocus, Sun tracking error and trough rim angle on the optical performance is also investigated. The MCRT study reveals that the ECE orientation schemes with a curved surface at the trough end showed much higher optical efﬁciency than those with a linear surface under ideal conditions. ECEs among the linear surface group, the inverted triangular orientation exhibited the highest optical efﬁciency, whereas the ﬂat and triangular ones exhibited the lowest optical efﬁciency, and the rectangular one was in between them. In the event of defocus and tracking errors, a signiﬁcant portion of the concentrated light was observed to be intercepted by the surfaces of the rectangular and RSc ECEs that are perpendicular to the trough aperture. This is an extended version of a published work by the current authors, which will help to design an optically efﬁcient ECE for a parabolic trough concentrating collector.


Introduction
Rapid depletion of fossil fuels, global warming and ecological imbalance are driving society to use green and renewable alternatives like solar energy. Efficient harnessing of this vast energy sources has the potential to fulfil the world's energy demand [1][2][3]. A significant amount of research has already been undertaken to develop technologies for harnessing this abundant energy source [4][5][6][7][8][9][10][11][12][13][14][15][16]. Concentrating solar power technologies including the PTCC, parabolic dish, linear Fresnel reflector and heliostat field are competitive candidates to the conventional sources in terms of carbon emissions though the investment cost is still higher [17].

Physical Model
To facilitate direct validation of the MCRT model with measured optical efficiency, a proven and widely used parabolic trough collector, Luz Solar II (LS2), was modelled in this study. The LS2 collector was used in the Solar Energy Generating System III-VII 150 MW plants, Kramer Junction, (CA, USA). It was tested on the AZTRAK rotating platform at Sandia National Laboratory [18]. Figure 1a,b show the cross-sectional views of the collector and the ECE of the LS2 collector. The geometry and the optical properties of the collector components are given in Table 1. A closed-end plug was inserted in the tube of the collector to increase the flow velocity of the Syltherm 800 silicone oil heat transfer fluid (HTF). If the difference between the HTF mean working temperature and the ambient temperature is very low, the heat loss from the receiver is negligible, and the collected thermal energy is almost equal to that of the absorbed optical energy by the receiver. To facilitate direct validation of the MCRT model with measured optical efficiency, a proven and widely used parabolic trough collector, Luz Solar II (LS2), was modelled in this study. The LS2 collector was used in the Solar Energy Generating System III-VII 150 MW plants, Kramer Junction, (CA, USA). It was tested on the AZTRAK rotating platform at Sandia National Laboratory [18]. Figures 1a,b show the cross-sectional views of the collector and the ECE of the LS2 collector. The geometry and the optical properties of the collector components are given in Table 1. A closed-end plug was inserted in the tube of the collector to increase the flow velocity of the Syltherm 800 silicone oil heat transfer fluid (HTF). If the difference between the HTF mean working temperature and the ambient temperature is very low, the heat loss from the receiver is negligible, and the collected thermal energy is almost equal to that of the absorbed optical energy by the receiver.  Table 1). Table 1. Geometric configuration, optical properties and loss factors of the Luz Solar 2 parabolic trough collector [19]. In the table, suffix PT, RT, P, GT, max, min, σ, geom and gen refer the parabolic trough, receiver tube, close ends plug as flow restriction device, glass tube, maximum, minimum, sigma, geometry and general respectively.  Table 1). Table 1. Geometric configuration, optical properties and loss factors of the Luz Solar 2 parabolic trough collector [19].

Parameters Value
Concentrator width, W = 5 m Concentrator length, L PT  In the table, suffix PT, RT, P, GT, max, min, σ, geom and gen refer the parabolic trough, receiver tube, close ends plug as flow restriction device, glass tube, maximum, minimum, sigma, geometry and general respectively. Relying on this assumption, near optical efficiency of the LS2 collector was measured for different conditions [18]. Several test conditions were chosen for the current MCRT model as presented in Table 2. The support structure was ignored in the MCRT model.

Ray Tracing Technique
The cylindrical coordinate system, rβZ, was used for circular surfaces and the Cartesian coordinate system, XYZ, was used for the rest of the collector components including the trough. The LS2 collector was modelled as per the design presented in Table 1. In order to minimise the computational expense, following assumptions were made: (a) The irradiance distribution varies across the axis and uniform along the axis of the ECE. Therefore, the length of the modelled collector was 50 cm. (b) The trough was locally specularly reflective, and the light incident was perpendicular on the trough aperture. A biconic surface for the parabolic trough (PT) and two concentric annular volume objects for the receiver tube (RT) and the glass tube (GT) from the object library were adapted to model the collector system. Enabling 'surface coating' option, a specular reflectance of the mirror, glass transmittance with anti-reflection coating and light absorptance with black chrome spectral selective coatings on the ECE material were developed (refer to Table 1). In addition to these, incidence angle (θ • ) depended light absorptance (α) in the ECE material was also considered in the model, such that the actual light absorptance at the point of incidence was αcosθ. The ray-tracing algorithm and the light interaction among the components are presented in Figure 2a,b, respectively.
The steps in Figure 2a are self-explanatory; the rhombuses and rectangles in the flow chart represent the arguments and Monte Carlo decision of the arguments respectively. The Monte Carlo decision was based on the sunshape, optical properties and geometry of the collector components, and the laws of reflection and refraction. The azimuth angle and the deflection angle of the modelled sunshape, I (ϕ), (see Equation (1) [66][67][68]) were 2π and 0.266 • respectively. The incident location and the direction vectors of the rays are denoted by P (P x , P y , P z ) and D (D x , D y , D z ) respectively. Considering a mean absolute error of 0.08-0.1% in the concentrator reflectance and the glass transmittance, 5 × 10 7 to 10 8 rays/m 2 of aperture area of the collector were traced for the model. A grid sensitivity test was carried out, and it was observed that the final MCRT result significantly varied with the variation of the number of rays instead of the grid of the collector components. As the irradiance distribution varies across the axis and uniform along the axis as assumed, the width × length of each grid was 1.32 mm × 12.5 mm. Light reflection on the mirror and transmission through the glass were followed by Fresnel's law, Equation (2), and Snell's law, Equation (3).
where, ϕ is the deflection angle.
where, D, N, n GT and n air are the direction vector, normal vector, and refractive indices of the glass and air respectively. The normal vectors, N PT and N GT were calculated from Equations (4) and (5) respectively: where, P is the incident points of the rays in Cartesian coordinate system and f is focal length of the mirror.
where, D, N, nGT and nair are the direction vector, normal vector, and refractive indices of the glass and air respectively. The normal vectors, NPT and NGT were calculated from Equations (4) and (5) respectively: where, P is the incident points of the rays in Cartesian coordinate system and f is focal length of the mirror.

Validation of the Model
The selected test conditions (see Table 2) were modelled, and optical efficiencies were calculated using the current MCRT model and compared against the measured ones as shown in Figure 3a. On the other hand, the irradiation distribution around the receiver surface was verified with Jeter's published analytical model for an ideal collector of 20× geometric concentration (GC) and 90 • rim angle [40][41][42]. Jeter calculated local concentration ratio (LCR) (see Equation (6)): where, I(l) is local irradiance, l is any point on the concerned surface of the ECE and C opt is product of collector optical properties and other factors. For the Jeter's collector, C opt was considered unity, the daily normal irradiation (DNI) was 1 kW/m 2 and solar disk was 7.5 mrad. LCR profile calculated using Jeter's analytical model and current MCRT model with and without glass envelop around the receiver tube are shown in Figure 3b. Our model slightly overestimates the optical efficiency (see Figure 3a) which is reasonable given the simplifying assumptions and experimental error because of incident angle modifier in the measured near optical efficiency calculated by Dudley et al. [18] (see the last column in Table 2). Figure 3b shows an excellent match between the LCR profiles calculated using the current MCRT model and Jeter's analytical model. So, it can be claimed that the current MCRT model is validated.

Validation of the Model
The selected test conditions (see Error! Reference source not found.) were modelled, and optical efficiencies were calculated using the current MCRT model and compared against the measured ones as shown in Figure 3a. On the other hand, the irradiation distribution around the receiver surface was verified with Jeter's published analytical model for an ideal collector of 20× geometric concentration (GC) and 90° rim angle [40][41][42]. Jeter calculated local concentration ratio (LCR) (see Equation (6)): where, I(l) is local irradiance, l is any point on the concerned surface of the ECE and Copt is product of collector optical properties and other factors. For the Jeter's collector, Copt was considered unity, the daily normal irradiation (DNI) was 1 kW/m 2 and solar disk was 7.5 mrad. LCR profile calculated using Jeter's analytical model and current MCRT model with and without glass envelop around the receiver tube are shown in Figure 3b. Our model slightly overestimates the optical efficiency (see Figure 3a) which is reasonable given the simplifying assumptions and experimental error because of incident angle modifier in the measured near optical efficiency calculated by Dudley et al. [18] (see the last column in Table 2). Figure 3b shows an excellent match between the LCR profiles calculated using the current MCRT model and Jeter's analytical model. So, it can be claimed that the current MCRT model is validated.

Geometry of the Studied ECE Orientation Schemes
The orientation scheme of the ECE of the validated MCRT model was adapted for different geometric configurations including a circular, semicircular, flat, triangular, inverted triangular, rectangular and Rectangle-on-Semicircular (RSc) as shown in Figure 4. The cross section of each scheme is shown in the figure. The rim angle, optical properties of the collector and the projected area of each ECE were equal to those of the LS2 collector (see Table 1

Geometry of the Studied ECE Orientation Schemes
The orientation scheme of the ECE of the validated MCRT model was adapted for different geometric configurations including a circular, semicircular, flat, triangular, inverted triangular, rectangular and Rectangle-on-Semicircular (RSc) as shown in Figure 4. The cross section of each scheme is shown in the figure. The rim angle, optical properties of the collector and the projected area of each ECE were equal to those of the LS2 collector (see Table 1). By this way, their geometric concentrations (=trough aperture width/ECE outer perimeter) are calculated as 24.1, 29.46, 75.76, 32.05, 32.05, 18.94 and 21.22 respectively with a 5 m wide trough. The glass envelope was not considered in this study. Usually a part of the outer periphery of these receivers are irradiated from concentrated light from the trough, a part from the direct Sun, and the rest remains shaded. All these ECEs, except the circular one, consist of one or more plain surfaces. Depending on the relative positions between the ECE and the trough, these surfaces were named: Trough See Surface (TSS), Trough No-see Surface (TNS) and Trough Partially-see Surface or perpendicular to the trough aperture (TPS) as shown in Figure 4. Respective LCR profile and optical efficiency were calculated applying the MCRT model detailed in the following section. concentrated light from the trough, a part from the direct Sun, and the rest remains shaded. All these ECEs, except the circular one, consist of one or more plain surfaces. Depending on the relative positions between the ECE and the trough, these surfaces were named: Trough See Surface (TSS), Trough No-see Surface (TNS) and Trough Partially-see Surface or perpendicular to the trough aperture (TPS) as shown in Figure 4. Respective LCR profile and optical efficiency were calculated applying the MCRT model detailed in the following section.

Calculated LCR Profile around the ECEs and Respective Optical Efficiency
Local irradiance around the ECE was estimated using the current MCRT model, and LCR profile across the respective surfaces was calculated using (6). From the LCR profile, peak light concentration (Cpeak), base concentration (Cbase), mean concentration (Cmean) and mean absolute deviation (MAD) of each surface, and optical efficiency (ηopt) of each ECE were calculated using the Equations (8)- (10).
where, Δw was the grid width and n was the number of grids or data points across each surface of the ECE, and W was the aperture width of the trough. As the glass envelop was not modelled, the transmittance was assumed unity. The aperture width was 5 m for a 68° rim angle collector and 7.36 m for a 90° rim angle collector. The lengths of the trough and the ECE were equal. The MCRT results of each ECE orientation scheme are discussed in the following subsections.

Circular
This is the most common and proven ECE geometry of a PTC. However, a circular ECE for a TCPVT collector can be made using flexible solar cells or arranging PV facets in circular orientation [26]. The entire circumference of the receiver tube was considered as the TSS, and the trough focus is

Calculated LCR Profile around the ECEs and Respective Optical Efficiency
Local irradiance around the ECE was estimated using the current MCRT model, and LCR profile across the respective surfaces was calculated using (6). From the LCR profile, peak light concentration (C peak ), base concentration (C base) , mean concentration (C mean ) and mean absolute deviation (MAD) of each surface, and optical efficiency (η opt ) of each ECE were calculated using the Equations (8)- (10).
where, ∆w was the grid width and n was the number of grids or data points across each surface of the ECE, and W was the aperture width of the trough. As the glass envelop was not modelled, the transmittance was assumed unity. The aperture width was 5 m for a 68 • rim angle collector and 7.36 m for a 90 • rim angle collector. The lengths of the trough and the ECE were equal. The MCRT results of each ECE orientation scheme are discussed in the following subsections.

Circular
This is the most common and proven ECE geometry of a PTC. However, a circular ECE for a TCPVT collector can be made using flexible solar cells or arranging PV facets in circular orientation [26]. The entire circumference of the receiver tube was considered as the TSS, and the trough focus is considered along the ECE axis. The LCR profile is generally somewhat "M" shaped, bi-symmetric with double peaks (C peak ≈ 63.5×) with C mean ≈ 22× as shown in Figure 5. Almost half of the outer surface of the circular ECE is away from the trough that makes complete shading (C base = 0) with high MAD (≈23.7) as presented in Table 3. The optical efficiency of a circular ECE was estimated to be about 80%.

Semi-Circular
As almost half of the outer surface of the circular ECE is away from the trough and receives direct Sun, this part can be replaced by a flat surface to make a semi-circular ECE. By this way, unlike the circular ECE, a semicircular ECE has a distinct TSS and TNS. The trough focus is still on the same location, that is toward the center of the TSS that coincide with the centerline of the TNS. However, by this way, no significant improvement in the optical efficiency could be achieved (still about 80%). Although the actual LCR profile on the TSS of the semicircular ECE is the same (see Figure 5) of that of the circular one, since the surface (the TSS) is separated from the TNS, the MAD is dropped to ≈16 (see Table 3). As the TNS gets direct sun, the optical efficiency of this surface directly depends on the optical properties of the associated components. For the investigated collector it is about 1.02%.  In the table, Cbase, Cpeak and Cmean stand for the base, peak and mean light concentration respectively, and the MAD stand for mean absolute deviation.

Flat
A flat rectangular target at the focus of the trough with active [23] or passive cooling [21,22] system forms a flat ECE. The LCR profile is measured for the TSS assuming the trough focus onto the axis or centerline of that surface. The TNS is assumed either insulated and/or carry some form of active or passive cooling system (see Figure 4). The LCR profile was calculated to be Gaussian (see

Semi-Circular
As almost half of the outer surface of the circular ECE is away from the trough and receives direct Sun, this part can be replaced by a flat surface to make a semi-circular ECE. By this way, unlike the circular ECE, a semicircular ECE has a distinct TSS and TNS. The trough focus is still on the same location, that is toward the center of the TSS that coincide with the centerline of the TNS. However, by this way, no significant improvement in the optical efficiency could be achieved (still about 80%). Although the actual LCR profile on the TSS of the semicircular ECE is the same (see Figure 5) of that of the circular one, since the surface (the TSS) is separated from the TNS, the MAD is dropped to ≈16 (see Table 3). As the TNS gets direct sun, the optical efficiency of this surface directly depends on the optical properties of the associated components. For the investigated collector it is about 1.02%.  In the table, Cbase, Cpeak and Cmean stand for the base, peak and mean light concentration respectively, and the MAD stand for mean absolute deviation.

Flat
A flat rectangular target at the focus of the trough with active [23] or passive cooling [21,22] system forms a flat ECE. The LCR profile is measured for the TSS assuming the trough focus onto the axis or centerline of that surface. The TNS is assumed either insulated and/or carry some form of active or passive cooling system (see Figure 4). The LCR profile was calculated to be Gaussian (see

Semi-Circular
As almost half of the outer surface of the circular ECE is away from the trough and receives direct Sun, this part can be replaced by a flat surface to make a semi-circular ECE. By this way, unlike the circular ECE, a semicircular ECE has a distinct TSS and TNS. The trough focus is still on the same location, that is toward the center of the TSS that coincide with the centerline of the TNS. However, by this way, no significant improvement in the optical efficiency could be achieved (still about 80%). Although the actual LCR profile on the TSS of the semicircular ECE is the same (see Figure 5) of that of the circular one, since the surface (the TSS) is separated from the TNS, the MAD is dropped to ≈16 (see Table 3). As the TNS gets direct sun, the optical efficiency of this surface directly depends on the optical properties of the associated components. For the investigated collector it is about 1.02%.  In the table, Cbase, Cpeak and Cmean stand for the base, peak and mean light concentration respectively, and the MAD stand for mean absolute deviation.

Flat
A flat rectangular target at the focus of the trough with active [23] or passive cooling [21,22] system forms a flat ECE. The LCR profile is measured for the TSS assuming the trough focus onto the axis or centerline of that surface. The TNS is assumed either insulated and/or carry some form of active or passive cooling system (see Figure 4). The LCR profile was calculated to be Gaussian (see

Semi-Circular
As almost half of the outer surface of the circular ECE is away from the trough and receives direct Sun, this part can be replaced by a flat surface to make a semi-circular ECE. By this way, unlike the circular ECE, a semicircular ECE has a distinct TSS and TNS. The trough focus is still on the same location, that is toward the center of the TSS that coincide with the centerline of the TNS. However, by this way, no significant improvement in the optical efficiency could be achieved (still about 80%). Although the actual LCR profile on the TSS of the semicircular ECE is the same (see Figure 5) of that of the circular one, since the surface (the TSS) is separated from the TNS, the MAD is dropped to ≈16 (see Table 3). As the TNS gets direct sun, the optical efficiency of this surface directly depends on the optical properties of the associated components. For the investigated collector it is about 1.02%.  In the table, Cbase, Cpeak and Cmean stand for the base, peak and mean light concentration respectively, and the MAD stand for mean absolute deviation.

Flat
A flat rectangular target at the focus of the trough with active [23] or passive cooling [21,22] system forms a flat ECE. The LCR profile is measured for the TSS assuming the trough focus onto the axis or centerline of that surface. The TNS is assumed either insulated and/or carry some form of active or passive cooling system (see Figure 4). The LCR profile was calculated to be Gaussian (see

Semi-Circular
As almost half of the outer surface of the circular ECE is away from the trough and receives direct Sun, this part can be replaced by a flat surface to make a semi-circular ECE. By this way, unlike the circular ECE, a semicircular ECE has a distinct TSS and TNS. The trough focus is still on the same location, that is toward the center of the TSS that coincide with the centerline of the TNS. However, by this way, no significant improvement in the optical efficiency could be achieved (still about 80%). Although the actual LCR profile on the TSS of the semicircular ECE is the same (see Figure 5) of that of the circular one, since the surface (the TSS) is separated from the TNS, the MAD is dropped to ≈16 (see Table 3). As the TNS gets direct sun, the optical efficiency of this surface directly depends on the optical properties of the associated components. For the investigated collector it is about 1.02%.  In the table, Cbase, Cpeak and Cmean stand for the base, peak and mean light concentration respectively, and the MAD stand for mean absolute deviation.

Flat
A flat rectangular target at the focus of the trough with active [23] or passive cooling [21,22] system forms a flat ECE. The LCR profile is measured for the TSS assuming the trough focus onto the axis or centerline of that surface. The TNS is assumed either insulated and/or carry some form of active or passive cooling system (see Figure 4). The LCR profile was calculated to be Gaussian (see In the table, C base , C peak and C mean stand for the base, peak and mean light concentration respectively, and the MAD stand for mean absolute deviation.

Semi-Circular
As almost half of the outer surface of the circular ECE is away from the trough and receives direct Sun, this part can be replaced by a flat surface to make a semi-circular ECE. By this way, unlike the circular ECE, a semicircular ECE has a distinct TSS and TNS. The trough focus is still on the same location, that is toward the center of the TSS that coincide with the centerline of the TNS. However, by this way, no significant improvement in the optical efficiency could be achieved (still about 80%). Although the actual LCR profile on the TSS of the semicircular ECE is the same (see Figure 5) of that of the circular one, since the surface (the TSS) is separated from the TNS, the MAD is dropped to ≈16 (see Table 3). As the TNS gets direct sun, the optical efficiency of this surface directly depends on the optical properties of the associated components. For the investigated collector it is about 1.02%.

Flat
A flat rectangular target at the focus of the trough with active [23] or passive cooling [21,22] system forms a flat ECE. The LCR profile is measured for the TSS assuming the trough focus onto the axis or centerline of that surface. The TNS is assumed either insulated and/or carry some form of active or passive cooling system (see Figure 4). The LCR profile was calculated to be Gaussian (see Figure 5) at ideal conditions with MAD of 27.36, while the C peak and C mean were estimated to be ≈88× and 22.61× respectively as presented in Table 3. However, the optical efficiency of a flat ECE was predicted to be only 26.19%.

Triangular
The LCR profile was developed for two locations of the trough focus: (i) toward the centerline of the TSS and (ii) toward the tip of the triangle as shown in Figure 6. At the first focus condition, the LCR profile of the TSS and the optical efficiency of the ECE were found almost the same to that of the flat one (see Figure 6 and Table 4). The estimated η opt was marginally higher (0.31%) only because of the direct sun on the inclined TNSs. At the second focus condition, because of defocus effect, though the η opt falls by ≈0.5 point, the irradiance distribution improves significantly as the estimated MAD drops by almost 50% (≈14.6) with C peak of ≈53×.  Table 3. However, the optical efficiency of a flat ECE was predicted to be only 26.19%.

Triangular
The LCR profile was developed for two locations of the trough focus: (i) toward the centerline of the TSS and (ii) toward the tip of the triangle as shown in Figure 6. At the first focus condition, the LCR profile of the TSS and the optical efficiency of the ECE were found almost the same to that of the flat one (see Figure 6 and Table 4). The estimated ηopt was marginally higher (0.31%) only because of the direct sun on the inclined TNSs. At the second focus condition, because of defocus effect, though the ηopt falls by ≈0.5 point, the irradiance distribution improves significantly as the estimated MAD drops by almost 50% (≈14.6) with Cpeak of ≈53×.

Inverted Triangular
An inverted triangular [27,29] ECE has two inclined TSSs. At ideal condition with no tracking error, the LCR profiles of these two surfaces were assumed to be the mirror image of each other. The profiles were developed for three locations of the trough focus as shown in Figure 7: (a) focus toward the centroid of the triangle, (b) focus toward the centerline of the TNS (the base of the triangle) and (c) focus beyond the TNS of the triangle.
As the ECE moves toward the trough from the first location to the next and so forth, the defocus effect takes place. As a result, the Cpeak decreases gradually and moves away from the trough send of    Table 3. However, the optical efficiency of a flat ECE was predicted to be only 26.19%.

Triangular
The LCR profile was developed for two locations of the trough focus: (i) toward the centerline of the TSS and (ii) toward the tip of the triangle as shown in Figure 6. At the first focus condition, the LCR profile of the TSS and the optical efficiency of the ECE were found almost the same to that of the flat one (see Figure 6 and Table 4). The estimated ηopt was marginally higher (0.31%) only because of the direct sun on the inclined TNSs. At the second focus condition, because of defocus effect, though the ηopt falls by ≈0.5 point, the irradiance distribution improves significantly as the estimated MAD drops by almost 50% (≈14.6) with Cpeak of ≈53×.

Inverted Triangular
An inverted triangular [27,29] ECE has two inclined TSSs. At ideal condition with no tracking error, the LCR profiles of these two surfaces were assumed to be the mirror image of each other. The profiles were developed for three locations of the trough focus as shown in Figure 7: (a) focus toward the centroid of the triangle, (b) focus toward the centerline of the TNS (the base of the triangle) and (c) focus beyond the TNS of the triangle.
As the ECE moves toward the trough from the first location to the next and so forth, the defocus effect takes place. As a result, the Cpeak decreases gradually and moves away from the trough send of  Table 3. However, the optical efficiency of a flat ECE was predicted to be only 26.19%.

Triangular
The LCR profile was developed for two locations of the trough focus: (i) toward the centerline of the TSS and (ii) toward the tip of the triangle as shown in Figure 6. At the first focus condition, the LCR profile of the TSS and the optical efficiency of the ECE were found almost the same to that of the flat one (see Figure 6 and Table 4). The estimated ηopt was marginally higher (0.31%) only because of the direct sun on the inclined TNSs. At the second focus condition, because of defocus effect, though the ηopt falls by ≈0.5 point, the irradiance distribution improves significantly as the estimated MAD drops by almost 50% (≈14.6) with Cpeak of ≈53×.

Inverted Triangular
An inverted triangular [27,29] ECE has two inclined TSSs. At ideal condition with no tracking error, the LCR profiles of these two surfaces were assumed to be the mirror image of each other. The profiles were developed for three locations of the trough focus as shown in Figure 7: (a) focus toward the centroid of the triangle, (b) focus toward the centerline of the TNS (the base of the triangle) and (c) focus beyond the TNS of the triangle.
As the ECE moves toward the trough from the first location to the next and so forth, the defocus effect takes place. As a result, the Cpeak decreases gradually and moves away from the trough send of  Table 3. However, the optical efficiency of a flat ECE was predicted to be only 26.19%.

Triangular
The LCR profile was developed for two locations of the trough focus: (i) toward the centerline of the TSS and (ii) toward the tip of the triangle as shown in Figure 6. At the first focus condition, the LCR profile of the TSS and the optical efficiency of the ECE were found almost the same to that of the flat one (see Figure 6 and Table 4). The estimated ηopt was marginally higher (0.31%) only because of the direct sun on the inclined TNSs. At the second focus condition, because of defocus effect, though the ηopt falls by ≈0.5 point, the irradiance distribution improves significantly as the estimated MAD drops by almost 50% (≈14.6) with Cpeak of ≈53×.

Inverted Triangular
An inverted triangular [27,29] ECE has two inclined TSSs. At ideal condition with no tracking error, the LCR profiles of these two surfaces were assumed to be the mirror image of each other. The profiles were developed for three locations of the trough focus as shown in Figure 7: (a) focus toward the centroid of the triangle, (b) focus toward the centerline of the TNS (the base of the triangle) and (c) focus beyond the TNS of the triangle.
As the ECE moves toward the trough from the first location to the next and so forth, the defocus effect takes place. As a result, the Cpeak decreases gradually and moves away from the trough send of  Table 3. However, the optical efficiency of a flat ECE was predicted to be only 26.19%.

Triangular
The LCR profile was developed for two locations of the trough focus: (i) toward the centerline of the TSS and (ii) toward the tip of the triangle as shown in Figure 6. At the first focus condition, the LCR profile of the TSS and the optical efficiency of the ECE were found almost the same to that of the flat one (see Figure 6 and Table 4). The estimated ηopt was marginally higher (0.31%) only because of the direct sun on the inclined TNSs. At the second focus condition, because of defocus effect, though the ηopt falls by ≈0.5 point, the irradiance distribution improves significantly as the estimated MAD drops by almost 50% (≈14.6) with Cpeak of ≈53×.

Inverted Triangular
An inverted triangular [27,29] ECE has two inclined TSSs. At ideal condition with no tracking error, the LCR profiles of these two surfaces were assumed to be the mirror image of each other. The profiles were developed for three locations of the trough focus as shown in Figure 7: (a) focus toward the centroid of the triangle, (b) focus toward the centerline of the TNS (the base of the triangle) and (c) focus beyond the TNS of the triangle.
As the ECE moves toward the trough from the first location to the next and so forth, the defocus effect takes place. As a result, the Cpeak decreases gradually and moves away from the trough send of  Table 3. However, the optical efficiency of a flat ECE was predicted to be only 26.19%.

Triangular
The LCR profile was developed for two locations of the trough focus: (i) toward the centerline of the TSS and (ii) toward the tip of the triangle as shown in Figure 6. At the first focus condition, the LCR profile of the TSS and the optical efficiency of the ECE were found almost the same to that of the flat one (see Figure 6 and Table 4). The estimated ηopt was marginally higher (0.31%) only because of the direct sun on the inclined TNSs. At the second focus condition, because of defocus effect, though the ηopt falls by ≈0.5 point, the irradiance distribution improves significantly as the estimated MAD drops by almost 50% (≈14.6) with Cpeak of ≈53×.

Inverted Triangular
An inverted triangular [27,29] ECE has two inclined TSSs. At ideal condition with no tracking error, the LCR profiles of these two surfaces were assumed to be the mirror image of each other. The profiles were developed for three locations of the trough focus as shown in Figure 7: (a) focus toward the centroid of the triangle, (b) focus toward the centerline of the TNS (the base of the triangle) and (c) focus beyond the TNS of the triangle.
As the ECE moves toward the trough from the first location to the next and so forth, the defocus effect takes place. As a result, the Cpeak decreases gradually and moves away from the trough send of  Table 3. However, the optical efficiency of a flat ECE was predicted to be only 26.19%.

Triangular
The LCR profile was developed for two locations of the trough focus: (i) toward the centerline of the TSS and (ii) toward the tip of the triangle as shown in Figure 6. At the first focus condition, the LCR profile of the TSS and the optical efficiency of the ECE were found almost the same to that of the flat one (see Figure 6 and Table 4). The estimated ηopt was marginally higher (0.31%) only because of the direct sun on the inclined TNSs. At the second focus condition, because of defocus effect, though the ηopt falls by ≈0.5 point, the irradiance distribution improves significantly as the estimated MAD drops by almost 50% (≈14.6) with Cpeak of ≈53×.

Inverted Triangular
An inverted triangular [27,29] ECE has two inclined TSSs. At ideal condition with no tracking error, the LCR profiles of these two surfaces were assumed to be the mirror image of each other. The profiles were developed for three locations of the trough focus as shown in Figure 7: (a) focus toward the centroid of the triangle, (b) focus toward the centerline of the TNS (the base of the triangle) and (c) focus beyond the TNS of the triangle.
As the ECE moves toward the trough from the first location to the next and so forth, the defocus effect takes place. As a result, the Cpeak decreases gradually and moves away from the trough send of

Inverted Triangular
An inverted triangular [27,29] ECE has two inclined TSSs. At ideal condition with no tracking error, the LCR profiles of these two surfaces were assumed to be the mirror image of each other. The profiles were developed for three locations of the trough focus as shown in Figure 7: (a) focus toward the centroid of the triangle, (b) focus toward the centerline of the TNS (the base of the triangle) and (c) focus beyond the TNS of the triangle.
As the ECE moves toward the trough from the first location to the next and so forth, the defocus effect takes place. As a result, the C peak decreases gradually and moves away from the trough send of the TSS to the TNS end as shown in Figure 7. Because of the defocus effect, the MAD also decreases significantly from 22 to 7.3 (see Table 4). However, interestingly, the C mean was estimated to be the highest at the second focus condition that was ≈23.08× in compared to 22.42× and 20.41× for the rest of focus the conditions. The η opt was also found higher at this focus condition (see Table 4). the TSS to the TNS end as shown in Figure 7. Because of the defocus effect, the MAD also decreases significantly from 22 to 7.3 (see Table 4). However, interestingly, the Cmean was estimated to be the highest at the second focus condition that was ≈23.08× in compared to 22.42× and 20.41× for the rest of focus the conditions. The ηopt was also found higher at this focus condition (see Table 4).

Rectangular (Square)
The LCR profile of TSS and TPSs were investigated for three locations of the trough focus on the ECE as shown in Figure 8: (i) focus toward the centroid of the rectangle, (ii) focus in middle of the centroid and the TNS and (iii) focus on the TNS.
As the ECE moves toward the trough from the first location to the next, the concentrated irradiation gradually shifts from TSS to the TPSs of the ECE. As a result, the downward open parabolic LCR profile of the TSS gradually flattens, and the Cpeak and Cbase of the TPSs gradually move from the TSS end to the TNS end diminishing the shading (see Figure 8). Significant improvement was observed in MAD of the TSS, which dropped rapidly from 5.44 to 0.94 (see Table 5). As the table shows for the TSS, though the Cbase was almost fixed at 9×, the Cpeak and Cmean were dropped from ≈26 and ≈18× to ≈13.5 and 11.7× respectively that caused almost 36% drop of partial optical efficiency of the surface. With the movement of the ECE towards the trough, the estimated partial optical efficiency ratio between the TSS and TPSs was found varied between 1.82 and 0.85. The total ηopt dropped from 33.6% to 30.5%, which is still higher than the flat and triangular ECEs. The rectangular ECE is even better than the inverted triangular ECE in case of irradiance distribution on the TSS though the ηopt is slightly lower (refer the MAD and ηopt in Tables 4 and 5). Because of the least MAD of the LCR profile of the TSS at the third focus location, effect of tracking error and rim angle on the optical performance of the ECE was further investigated.
As an example, σ = 0.3° and ψ = 90° in place of 68° on the LCR profile and ηopt was investigated as shown in Figure 9 and Table 6. Figure 9a shows, tracking error has insignificant effect on the LCR profile of the TSS, and the table shows that the Cmean and MAD are almost equal to that without tracking error (see Table 6). However, the σ adversely affects the LCR profiles of the TPSs (see Figure  9b), and the ηopt of the ECE dropped almost by 2 points (see Table 6). Despite this, the ηopt of the rectangular ECE is still higher than that of the flat and triangular ones (refer Tables 4 and 6).
On the other hand, light from the increased surface area of the trough of 90° rim angle (width of the trough increased from 5 m to 7.36 m) was concentrated on the TPSs instead of the TSS. Therefore, though the Cmean of the TSS was almost fixed (≈11.7×) that was increased almost by 1.8 times for the

Rectangular (Square)
The LCR profile of TSS and TPSs were investigated for three locations of the trough focus on the ECE as shown in Figure 8: (i) focus toward the centroid of the rectangle, (ii) focus in middle of the centroid and the TNS and (iii) focus on the TNS. TPSs (see Table 6). However, the overall ηopt is decreased from 28.63% to 26.98% that is because the collector geometry for this study was adapted from the LS2 collector for which the 68° rim angle is optimum.
(a) (b)  As the ECE moves toward the trough from the first location to the next, the concentrated irradiation gradually shifts from TSS to the TPSs of the ECE. As a result, the downward open parabolic LCR profile of the TSS gradually flattens, and the C peak and C base of the TPSs gradually move from the TSS end to the TNS end diminishing the shading (see Figure 8). Significant improvement was observed in MAD of the TSS, which dropped rapidly from 5.44 to 0.94 (see Table 5). As the table shows for the TSS, though the C base was almost fixed at 9×, the C peak and C mean were dropped from ≈26 and ≈18× to ≈13.5 and 11.7× respectively that caused almost 36% drop of partial optical efficiency of the surface. With the movement of the ECE towards the trough, the estimated partial optical efficiency ratio between the TSS and TPSs was found varied between 1.82 and 0.85. The total η opt dropped from 33.6% to 30.5%, which is still higher than the flat and triangular ECEs. The rectangular ECE is even better than the inverted triangular ECE in case of irradiance distribution on the TSS though the η opt is slightly lower (refer the MAD and η opt in Tables 4 and 5). Because of the least MAD of the LCR profile of the TSS at the third focus location, effect of tracking error and rim angle on the optical performance of the ECE was further investigated. Table 5. Irradiance distribution and optical efficiencies of each surfaces of a rectangular receiver at different defocus conditions.  ≈0.88 -≈1.02 - Table 6. Effect of tracking error (σ°) and rim angle (ψ°) on the optical performance of a rectangular receiver while the trough focus toward the centerline of the TNS, .  ≈0.88 -≈1.02 - Table 6. Effect of tracking error (σ°) and rim angle (ψ°) on the optical performance of a rectangular receiver while the trough focus toward the centerline of the TNS, .    ≈0.88 -≈1.02 - Table 6. Effect of tracking error (σ°) and rim angle (ψ°) on the optical performance of a rectangular receiver while the trough focus toward the centerline of the TNS, .    ≈0.88 -≈1.02 - Table 6. Effect of tracking error (σ°) and rim angle (ψ°) on the optical performance of a rectangular receiver while the trough focus toward the centerline of the TNS, .  As an example, σ = 0.3 • and ψ = 90 • in place of 68 • on the LCR profile and η opt was investigated as shown in Figure 9 and Table 6. Figure 9a shows, tracking error has insignificant effect on the LCR profile of the TSS, and the table shows that the C mean and MAD are almost equal to that without tracking error (see Table 6). However, the σ adversely affects the LCR profiles of the TPSs (see Figure 9b), and the η opt of the ECE dropped almost by 2 points (see Table 6). Despite this, the η opt of the rectangular ECE is still higher than that of the flat and triangular ones (refer Tables 4 and 6).
On the other hand, light from the increased surface area of the trough of 90 • rim angle (width of the trough increased from 5 m to 7.36 m) was concentrated on the TPSs instead of the TSS. Therefore, though the C mean of the TSS was almost fixed (≈11.7×) that was increased almost by 1.8 times for the TPSs (see Table 6). However, the overall η opt is decreased from 28.63% to 26.98% that is because the collector geometry for this study was adapted from the LS2 collector for which the 68 • rim angle is optimum. TPSs (see Table 6). However, the overall ηopt is decreased from 28.63% to 26.98% that is because the collector geometry for this study was adapted from the LS2 collector for which the 68° rim angle is optimum.

Rectangle on Semi-Circle (RSc) ECE
A Rectangle on Semi-circle (RSc) ECE was formed replacing the flat TSS of the rectangular ECE with a semicircular one because of high partial optical efficiency of circular TSSs were estimated for the circular and semicircular ECES. The LCR profiles were developed for two locations of the trough focus on the ECE: (a) focus on the center of the semi-circular TSS, and (b) on the TNS of the ECE as shown in Table 7.

Rectangle on Semi-Circle (RSc) ECE
A Rectangle on Semi-circle (RSc) ECE was formed replacing the flat TSS of the rectangular ECE with a semicircular one because of high partial optical efficiency of circular TSSs were estimated for the circular and semicircular ECES. The LCR profiles were developed for two locations of the trough focus on the ECE: (a) focus on the center of the semi-circular TSS, and (b) on the TNS of the ECE as shown in Table 7. At the first focus location, the optical performance of the RSc ECE with a 68° rim angle trough was found similar to that of a semicircular one (see Table 7 and Figure 10a). Increase of trough rim angle from 68° to 90° rather decreases the overall optical efficiency of the ECE by ≈1% from 79.3%. However, the LCR profile of the TSS improves significantly as the MAD drops from 16.7 to 8.83, Cmean increases drastically from 43× to 60× and the TPSs contribute 3.46% of optical efficiency with 5.18 MAD (see Table 7 and Figure 10c).
An induced tacking error of 0.3° caused a reduction ηopt by 1-2% for both troughs, and adversely affected the light distribution. The adversity on the LCR profile was observed to be much higher for the ECE with a trough of 90° ψ than that of 68° one (see Table 7).

Rectangle on Semi-Circle (RSc) ECE
A Rectangle on Semi-circle (RSc) ECE was formed replacing the flat TSS of the rectangular ECE with a semicircular one because of high partial optical efficiency of circular TSSs were estimated for the circular and semicircular ECES. The LCR profiles were developed for two locations of the trough focus on the ECE: (a) focus on the center of the semi-circular TSS, and (b) on the TNS of the ECE as shown in Table 7. At the first focus location, the optical performance of the RSc ECE with a 68° rim angle trough was found similar to that of a semicircular one (see Table 7 and Figure 10a). Increase of trough rim angle from 68° to 90° rather decreases the overall optical efficiency of the ECE by ≈1% from 79.3%. However, the LCR profile of the TSS improves significantly as the MAD drops from 16

Rectangle on Semi-Circle (RSc) ECE
A Rectangle on Semi-circle (RSc) ECE was formed replacing the flat TSS of the rectangular ECE with a semicircular one because of high partial optical efficiency of circular TSSs were estimated for the circular and semicircular ECES. The LCR profiles were developed for two locations of the trough focus on the ECE: (a) focus on the center of the semi-circular TSS, and (b) on the TNS of the ECE as shown in Table 7. At the first focus location, the optical performance of the RSc ECE with a 68° rim angle trough was found similar to that of a semicircular one (see Table 7 and Figure 10a). Increase of trough rim angle from 68° to 90° rather decreases the overall optical efficiency of the ECE by ≈1% from 79.3%. However, the LCR profile of the TSS improves significantly as the MAD drops from 16 At the first focus location, the optical performance of the RSc ECE with a 68 • rim angle trough was found similar to that of a semicircular one (see Table 7 and Figure 10a). Increase of trough rim angle from 68 • to 90 • rather decreases the overall optical efficiency of the ECE by ≈1% from 79.3%. However, the LCR profile of the TSS improves significantly as the MAD drops from 16.7 to 8.83, C mean increases drastically from 43× to 60× and the TPSs contribute 3.46% of optical efficiency with 5.18 MAD (see Table 7 and Figure 10c). At the second focus location, the defocus effect causes a drastic reduction in the ηopt, which was estimated to be 12%+ for the 68° trough rim angle and 22%+ for the 90° trough rim angle with or without 0.3° tracking error (see Table 7). In general, the defocus effect improved the LCR distribution of the TSS significantly; the LCR profile fairly flatten (see Figure 10b) and the MAD dropped to 2 for both trough rim angles, and increased the TPSs' contribution to the ηopt considerably (see Table 7).
Interestingly, though there was an overall drop in the ηopt by around 3%, and a 0.3° tracking error was observed to increase the Cmean and partial optical efficiency of the TSS. The partial optical efficiency of the TSS was increased by 0.85% and 2.23% respectively for the troughs of rim angle 68° and 90° (see Table 7). The improvement in the uniformity of the LCR profile due of the induced defocus was gone because of the tracking error (see Figure 10d and Table 7).

Discussion
In this section, a comparison among the studied ECEs is made based on their relative optical performance. This study found that the optical efficiency of the ECEs with curved TSS (i.e., circular, semicircular and RSc) is much higher than that of linear TSS (i.e., flat, triangular, inverted triangular and rectangular). The reason is the incidence angle dependent light absorptance of the ECE material. Average incidence angle of the concentrated light on the linear TSS is much higher than that of the   An induced tacking error of 0.3 • caused a reduction η opt by 1-2% for both troughs, and adversely affected the light distribution. The adversity on the LCR profile was observed to be much higher for the ECE with a trough of 90 • ψ than that of 68 • one (see Table 7).
At the second focus location, the defocus effect causes a drastic reduction in the η opt , which was estimated to be 12%+ for the 68 • trough rim angle and 22%+ for the 90 • trough rim angle with or without 0.3 • tracking error (see Table 7). In general, the defocus effect improved the LCR distribution of the TSS significantly; the LCR profile fairly flatten (see Figure 10b) and the MAD dropped to 2 for both trough rim angles, and increased the TPSs' contribution to the η opt considerably (see Table 7).
Interestingly, though there was an overall drop in the η opt by around 3%, and a 0.3 • tracking error was observed to increase the C mean and partial optical efficiency of the TSS. The partial optical efficiency of the TSS was increased by 0.85% and 2.23% respectively for the troughs of rim angle 68 • and 90 • (see Table 7). The improvement in the uniformity of the LCR profile due of the induced defocus was gone because of the tracking error (see Figure 10d and Table 7).

Discussion
In this section, a comparison among the studied ECEs is made based on their relative optical performance. This study found that the optical efficiency of the ECEs with curved TSS (i.e., circular, semicircular and RSc) is much higher than that of linear TSS (i.e., flat, triangular, inverted triangular and rectangular). The reason is the incidence angle dependent light absorptance of the ECE material. Average incidence angle of the concentrated light on the linear TSS is much higher than that of the curved one, and causes cosine loss in light absorptance. ECE materials with incidence angle independent light absorptance would increase the optical efficiency of the collector.
In the event of defocus, as well as tracking error, a part of the concentrated light usually escape the TSS of the ECE. Resultant optical loss and the overall optical efficiency relies on the interceptance of this light by any active surface like TPS of the ECE. The rectangular and RSc type ECEs with TPSs are more likely to exhibit better interceptance to this escaped light than the flat, semicircular and triangular ECEs. However, defocus event improves the uniformity of the LCR profile (decrease the MAD) of the TSS, though the C mean of the surface decreases.
Being flat and parallel to the trough aperture, the TNS of the semicircular, inverted triangular, rectangular and RSC ECEs allows uniform one sun photovoltaic application. Other than these, the direct sun incident on the curved surface of the circular ECE and inclined surfaces of the triangular ECE are likely to be highly non-uniform and causes shading effect to photovoltaic application.
Although the flat ECE has the highest C peak and C mean at the ideal optical condition, being a single surface (TSS only), it is the most vulnerable to any optical loss event like defocus and tracking errors. In addition to this, the HTF has the least contact area with the TSS, and therefore, the cooling effect should be the least compared to the rest of the studied ECEs in general.

Conclusions
Characteristics of the irradiance distribution around the outer surface of different ECE orientation schemes and optical efficiency of a PTCC were investigated computationally applying MCRT technique. The ECE orientation schemes include a circular, semicircular, flat, triangular, inverted triangular, rectangular and RSc. The surfaces of the ECEs are called TSS (surface at the trough end), TNS (opposite to the TSS) and TPS (perpendicular to the trough aperture). The irradiance profile around the ECEs was calculated as LCR profile, from which C base , C peak , C mean , uniformity of the LCR profile as MAD, and η opt were calculated. The effect of ECE defocus, sun tracking error and trough rim angle were also investigated for selective cases. From this study, following conclusions can be made: (i) The ECE orientation schemes (circular, semicircular and RSc) with curved TSS showed much higher η opt (>79%) than those (flat, triangular, inverted triangular and rectangular) with linear TSS (η opt ranging between 25% and 38%) in ideal conditions (i.e., no defocus and tracking error) with a trough of 68 • rim angle (ii) Among the linear TSS group, the optical efficiency was calculated to be highest for the inverted triangular ECE (around 37%), lowest for the flat and triangular ECEs (≈26%), and somewhere in middle for the rectangular ECE (in between 30.5% and 33.5%). (iii) The MAD of the TSS of the rectangular ECE was estimated to be the lowest, ranging from 0.94 at defocus condition to 5.44 at ideal condition. In the event of tracking error and with a trough of higher rim angle, the MAD was found increased insignificantly, only increased to 1.05 from 0.94. That is the uniformity of the irradiance distribution (i.e., the LCR profile) on the TSS of a rectangular ECE was found to be the best among the studied ECE orientation schemes. (iv) In the event of defocus and tracking error, a significant portion of the concentrated light was observed to be intercepted by the TPSs. Therefore, based on this study, ECEs among the curved TSS group, it is clear that the RSc would be more efficient than the circular and semicircular ones. Similarly, among the linear TSS group the rectangular one would be more efficient than the remaining ECEs of the group. (v) Sun tracking error and the ECE defocus have negative effect on the optical efficiency of the collector in general. However, ECE defocus helps to reduce the MAD that is the non-uniformity of the LCR profile of the TSS. (vi) Study of the effect of 90 • trough rim angle over a 68 • one for the rectangular and RSc ECEs implied that there is an optimum rim angle for each collector design; trough rim angle larger than that would increase the total concentrated light on the ECE and, for few instances, the uniformity of the LCR profile of the TSS, but will decrease the optical efficiency of the collector.
The current study provides useful information for designing optically efficient ECE for a parabolic trough concentrating solar collector. The current study will be continued to investigate the effect of structural deformity and light incident angle on the optical performance, and relative thermal performance of the ECEs under certain environmental conditions. The daily and year round energy potential of the optimum ECE is worthy to be exploring and will be a step forward towards the energy sustainability.
Author Contributions: M.I. conceived, designed and accomplished the research, and drafted the article under the principle supervision of A.K., who was the leader of the research team. P.Y. was the associate supervisor for the research program.
Funding: This article is supported by QUT research grant.

Acknowledgments:
The authors like to acknowledge the support provided by Suvash Chandra Saha, Senior Lecturer at University of Technology Sydney, during conducting the research.