Effects of deformation of cylindrical compound parabolic concentrator(CPC) on concentration characteristicsRongji Xu⁎, Yusen Ma, Meiyu Yan, Chao Zhang, Shuhui Xu, Ruixiang WangBeijing Engineering Research Center of Sustainable Energy and Buildings & Research Centre of Urban Underground Space and Utility Tunnel, Beijing University of CivilEngineering and Architecture, Beijing 100044, China
A R T I C L E I N F O
Keywords:Solar energyMonte Carlo ray tracing (MCRT) methodHeat fluxTruncationIncidence angle
A B S T R A C T
Compound parabolic concentrators (CPC) have immense potential for middle-and-low temperature heat col-lection of solar energy. In this study, the propagation process of solar radiation rays in a CPC–tubular solarcollector was simulated using an in-house MATLAB code by the Monte Carlo ray tracing (MCRT) method. Theeffects of ray incidence angle, CPC concentration ratio, and truncation residual ratio on the heat flux distributionon the tubular heat receiver were analyzed. Moreover, the effects of CPC deformations on concentration char-acteristics were analyzed, involving the rotation and translation of the reflective surface, the truncation of theinvolute starting point, and the position offset of the heat receiver. The heat flux distribution exhibited twoevident peaks on the heat receiver external surface due to the concentration effect of the CPC. A small internalrotation, a small internal or external translation of the reflective surface, and the truncation of the involutestarting point had mere effect on the heat flux distribution on the heat receiver. Therefore, the reflective surfacecan be allowed to have certain translation and internal rotation when machining a CPC. The downward,rightward, and leftward offsets of heat receivers should be avoided in the assembly of heat receivers and CPCs.
1. Introduction
Solar energy resource is an important component of future energybecause of its widespread distribution, abundance, cleanliness, and otherproperties for environmental protection (Kalogirou, 2004). However, thedensity of solar energy is easily affected by weather, season, and otherfactors due to its intermittence. Hence, solar energy systems are usuallyequipped with concentrators and uniaxial or biaxial tracking devices toimprove the utilization of solar energy. Tracking systems require highcontrol accuracy and considerable investment in necessary equipment.Non-imaging concentrators can be used to eliminate the demand fortracking devices in solar energy systems and thus reduce investment cost.The compound parabolic concentrator (CPC) is a non-imaging con-centrator that resembles the ideal concentrator closely. A CPC comprisestwo identical reflectors placed on each other’s focal point according to theedge-ray principle to exert an optical trap effect on the incidence rays ofincidence angles no more than the half acceptance angle (O'Gallagher,2008; Welford,1989; Winston and Welford, 1980). Moreover, the CPCperforms high concentration efficiency to collect light with various wa-velengths and partial scattering and therefore eliminates the need for atracking device. Thus, CPCs can be applied to the middle- and low-tem-perature heat collection of solar energy.
The concentration characteristics and optimization of CPCs andtheir various applications are current research hotspots. Winston andHinterberger (1975) proposed the principle of the radiation con-centration on a tubular receiver to achieve the maximum efficiency andthe design rules of CPCs suitable for tubular receivers. Tabor (1984)analyzed the effects of the radiate incidence angle, the uniform irra-diation, and the length of a reflector on actual concentrations in theoptical design principle of CPCs. Rabl (1976) proposed the ray reflec-tion frequency model of CPCs and studied the relationship between thereflection frequency and the concentration ratio. Abid et al. (2016)calculated the uniformity of solar radiation in a CPC receiver. Theyevaluated the uniformity of solar radiation in the receiver using heatflux and temperature distributions, optimized the uniformity of solarirradiation by intercepting the CPC, and determined the best ratio ofinterception according to the optical and thermal efficiencies. The mainobjective was to make the heat flux distribution on the heat receiversurface more uniform, to determine the optimal size of the CPC withoutsignificantly reduction in the heat collection efficiency, and to reducethe production cost. They provided a method to evaluate the uniformityof the heat flux density on the heat receiver surface. However, CPCdeformations and their effects on concentration characteristics were notconsidered. Baig et al. (2014) modeled and analyzed the performance of
https://doi.org/10.1016/j.solener.2018.10.001Received 27 April 2018; Received in revised form 21 September 2018; Accepted 1 October 2018
a linear asymmetrical CPC photovoltaic (PV) system based on di-electrics. Through the addition of a reflection film on the concentratoredge to change the solar incidence distribution, the average outputpower of the improved system increased by 16%. Zhang et al. (2017)proposed and optimized a biaxial tracking CPC with a special trunca-tion called the well-distributed CPC, which had a relatively high halfacceptance angle and a small height–width ratio. This CPC showed animproved light distribution uniformity on the acceptance surface. Xuanet al. (2017) proposed a new static asymmetric lens wall CPC, whichcomprised a mirror condenser and lens wall. The novel CPC could fullyutilize total internal reflection and specular reflection. The authors alsosimulated the flux distribution at different incidence angles. The resultsshowed that the flux was distributed uniformly with a minimal changeat smaller incidence angles. Conversely, the flux showed a non-uniformdistribution when the incidence angles were relatively large.
In terms of application, Tian et al. (2018) introduced the concept,design principle, and considerations of CPCs and summarized the re-lated research in four main applications since 2000. The researcherssuggested that the key considerations in designs should include thegeometry and edge effects of CPCs, the impact of manufacturing, andthe tracking system. Santos-González et al. (2017) built a numericalmodel of CPCs for heat collection and conducted the experimentalverification. The numerical model was based on the applications ofphysical laws and general empirical equations, which can be used as atool to design and optimize systems working with liquid water. Aguilar-Jiménez et al. (2018) conducted a comparative study on the heat col-lection performance of two CPCs positioned longitudinally in thenorth–south and east–west directions, and built a mathematical modelsubsequently. On average, the east-west CPC performed an optical
efficiency of 57.5%, whereas the north–south CPC’s optical efficiencyreached 51.3%. Acuña et al. (2016) established a mathematical modelfor a CPC absorption refrigeration system and studied the influence ofCPC collectors on system performance. The results suggested that theCPC collector can fully meet the requirements to provide the heatsource for the absorption refrigeration. Xu et al. (2017) proposed a CPCpulsating heat pipe solar collector, which used a CPC to increase theheat flux density at the evaporation section of the pulsating heat pipewith a diameter of 4 mm, realizing 3.4 focusing multiples. This raised anew idea to utilize CPCs for the middle-and-low temperature heatcollection of solar energy. Zhu et al. (2016) presented a new type ofCPC solar-air collector with flat micro-heat pipe arrays and tested thethermal performance of the new collector. The average efficiency wasapproximately 61% at the volume flow rate of 320 m3/h, the radiationof 799 W/m2, and the ambient temperature of 28.8 °C. Xie et al. (2016)designed a new type of CPC with four-fold geometric concentrationratios to develop a flat-plate concentrating PV thermal (CPV/T) systemwith low cost, high output, and no multiple reflections. Monte Carlosimulation was adopted to calculate the uniformity and optical effi-ciency of the new CPC. The overall efficiency of the CPV/T system withan optimized CPC can exceed 71%. Saini et al. (2018) built an N-par-tially covered PV thermal collector using a CPC as the concentrator.They also compared the annual generation capacity, heat collectioncapacity, and energy efficiency of five types of PV cell systems made ofdifferent materials.
Errors may occur when a concentrator is fabricated and assembledwith a heat receiver, which can affect concentration characteristics.Yong et al. (2011) integrated the manufacturing and assembly errors ofa parabolic butterfly concentrator with the surface slope errors to
Nomenclature
C concentration ratioCt CPC truncation residual ratioD diameter of the heat receiver, mmem energy when the mth photon in the ith partition of the heat
receiver external surface was absorbed, WH distance from the aperture of the full CPC to the center of
the receiver, mmHt distance from the aperture of the truncated CPC to the
center of the receiver, mmE initial energy of a ray, WL length of heat receiver tubes, mmM mth photon in the ith partition of the heat receiver ex-
ternal surfaceMi number of photons absorbed in the ith partition of the heat
receiver external surfaceN normal vector at the intersection pointqi average heat flux density in the ith partition of the heat
receiver external surface, W/m2
q average heat flux density on the heat receiver externalsurface, W/m2
qj heat flux density on the heat receiver external surface, W/m2
Si area of the ith partition of the heat receiver external sur-face, m2
U photon incidence direction vectorUx direction vector of a random photon along X-axis directionUy direction vector of a random photon along the Y-axis di-
rectionU’ direction vector of photons after reflectionLm translation distance of the reflective surface, mmLu moving distance of the heat receiver, mmxmax maximum value within the range of values in X-axis
direction of the incidence plane, mmxmin minimum value within the range of values in X-axis di-
rection of the incidence plane, mmxi incidence location of a random photon in X-axis direction,
mmyi incidence location of a random photon in Y-axis direction,
mmy0 vertical distance between the center of the heat receiver
and the aperture of the CPC, mmZ number of partitions on the heat receiver external surfacen number of photons
Greek symbols
absorption ratio of the heat receiverdimensionless number for reflective surface translationdimensionless number for heat receiver position offsetcentral angle corresponding to the heat receiver externalsurfaceglass transmissivitydistance from a point on the reflection surface curve to thecorresponding pointcut on the heat absorbing circle
σ truncation of the starting point of the CPC point involuteφ involute spreading angleµ reflectivity of the reflective surfacer number of ray reflectionsν absorption rate of the heat receiverω rotation of the concentration surface
a half acceptance angle of the CPCangle of light incidence, °
u dimensionless number for heat receiver position offseta random number
′ a random number
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analyze the radiant flux characteristics of the focus area. The surfaceslope errors include mirror tilt, tracking, and mirror reflection on thereflection surface. The results showed that the surface slope errors wi-dened the radiation flux distribution, reduced the peak value, whilemaintained the energy balance of the system. Sarmah et al. (2014)studied the optical losses under different incidence angles and differentbeam positions at the concentrator aperture. Compared with the theo-retical value of the CPC optical efficiency loss, the results suggested thatthe incomplete paraboloid and surface roughness caused by machiningerrors can lead to light loss and scattering. Moreover, the micro dis-location produced in the assembly of concentrators and solar cells mayincrease the optical loss near the half acceptance angle and reduce thewhole system performance.
The study of the concentration characteristics of CPCs, which canaffect the efficiency of heat receivers or PV systems directly, is the basisfor their application. The machining mode and precision of CPCs andthe assembly precision of heat receivers have influences on the con-centration characteristics of CPCs. The objective of this paper is toprovide fundamental data for the practical application and processingof CPCs. In Section 1, the propagation and absorption processes of solarradiation light in a CPC tube solar collector were simulated by theMCRT method. The calculation condition and verification of the modelwhich included the detail of the CPC deformations were introduced inSection2. The influences of the CPC concentration ratio and truncationresidual ratio, as well as the ray incidence angle, on the heat flux dis-tribution on the external surface of the heat receiver were studied andanalyzed in Section 3. Based on the results, Section 4 focuses on that theeffects of CPC deformations on CPC concentration characteristics wereanalyzed in terms of the rotation and translation of the reflective sur-face, the truncation of the involute starting point, and the positionoffset of the heat receiver. Finally, the conclusions were presented inSection 5.
2. CPC concentration model and calculation method
2.1. Introduction to the CPC model
The CPC using a circular tube as the heat receiver is axisymmetric.The reflective surface of the CPC consists of two involute segments: theAB segment and the BC segment (as shown in Fig. 1). The parameterequation of the right half curve is as follows (Khonkar and Sayigh,1995).
The equation of the involute AB is
=
= +
X
Y
(sin cos )
( sin cos )
D
D2
2 (1)
where +( )0 a 2 .The equation of the involute BC is
=
= +
X A
Y A
(sin cos )
( sin cos )
D
D2
2 (2)
where +( )a a232 ,
=+ +
+A
cos( )1 sin( )
a a
a
2
(3)
As shown in Fig. 1, B is the connection point of the two involutesegments on the right half curve. The origin of the coordinate system isat the center of the heat absorbing tube. D represents the diameter ofthe heat absorbing tube. φ designates the angle of the line between apoint on the reflection surface curve and the center of the circular tube,and the starting direction takes the negative Y-axis direction. meansthe distance from a point on the reflection surface curve to the corre-sponding cutting point on the heat receiver circle. The angle a between
the straight line BE and the symmetry axis of the CPC is called the halfacceptance angle. This angle represents the maximum incidence anglewhich allows incident rays at the aperture of the CPC to be reflected tothe heat receiver (Chen et al., 2015). If a incidence angle is less than a,the corresponding ray can be reflected to the heat receiver by the CPC.Rays whose incidence angles are beyond the half acceptance anglecannot be absorbed. The half acceptance angle a determines the con-centration ratio directly and thus is an important parameter of the CPC.The concentration ratio of the CPC is defined as
=C 1sin( )a (4)
The CPC has a larger reflective surface than other concentrators.However, its upper reflective surface is almost parallel to the symme-trical axis Thereby the increase in the reflective surface height leads to amere increase in the concentration ratio. Therefore, in practice, theinefficient upper part is often cut off. The ratio of the distance betweenthe truncated CPC aperture and the heat receiver center to the originaldistance is defined as the truncation residual ratio Ct (Chen et al., 2015)(Fig. 1). H means the distance from the full CPC aperture to the receivercenter. And Ht represents the distance from the truncated CPC apertureto the receiver center.
=C HHt
t(5)
When the CPC was truncated, the aperture surface of the CPC be-comes smaller. The amount of sunlight coming into the CPC is reduced,and the actual concentration ratio of the CPC becomes smaller.However, a small amount of light over half acceptance angle can stillreach the receiver and be absorbed as shown in Fig. 2. There are twocases of optical path, one of which reaches the CPC reflective surfaceand is reflected to the receiver, the other part is absorbed directly by thereceiver.
2.2. Calculation method
The CPC concentration characteristics are analyzed by the MCRTmethod using an in-house MATLAB code. A flow diagram of the pro-gram is shown in Fig. 3. The overall programming concept is as follows.We regard the CPC aperture surface as the initial surface for the solarradiation entering the heat collector. We then simulate the reflectionand absorption processes of the solar radiation inside the heat collectorusing many random path photons. Each photon is assumed to carry aspecific energy, whose position and direction are generated by a spe-cific probability function. The number of photons on the heat receivingsurface is calculated by tracking the path of each photon. In addition,
Fig. 1. Schematic of CPC.
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the heat flux distribution on the heat receiver external surface is cal-culated according to the energy property of the photons.
The simulation of the concentration and heat collection processes ofthe CPC by the MCRT method mainly comprises three processes:random photon initialization, random photon reflection process, andrandom photon absorption.
(1). Random photon initialization
Random photon initialization refers to the determination of the initialposition and incidence direction of the random photons by the probabilityfunction model. To determine the position of the random photons, theradiation on the initial surface of the solar heat receiver is assumed to bedistributed uniformly and the surface is regarded as the incidence surfacefor the random photons in the simulation. The incidence positions xi and yiof the random photons are both uniformly distributed on the incidentsurface of the concentrator. The probability model (Cui et al., 2011) isestablished as follows, where xmin and xmax are the value range in the Xdirection of the incident surface, and are random numbers which areuniformly distributed in the range [0, 1]. y0 is the vertical distance be-tween the heat receiver center and the CPC aperture.
= +x x x x( )i min max min (6)
=y yi 0 (7)
The schematic of the CPC incident surface is shown in Fig. 4:To determine the incidence direction, if the probability function of
the initial photon direction vector on the incident surface follows auniform distribution, the initial direction vectors of the random photonson the incident surface (Sarmah et al., 2014) are expressed as follows,where Ux and Uy represent the initial direction vectors of the incidentrandom photons in the direction of X and Y respectively.
=U cos( )x (8)
=U sin( )y (9)
(2). Reflection process of the random photons
Two possible scenarios describe the random photon motion afterentering the CPC. In the first scenario, the photons are directly absorbedby the heat receiver. In the second scenario, the photons are absorbedby the heat receiver or eliminated after single or multiple reflections bythe CPC (complete energy loss). The simulation of the photon reflectionpath in the CPC requires the intersection coordinates of the photon, theCPC reflector, and the reflection vector. The intersection coordinatesare obtained by combining the equations for the CPC and the linearequations for the random photons. After the completion of the solution,the initial value [x0, y0] of the photon position is updated to the in-tersection coordinates for the next calculation. The solution of the re-flection vector is represented as follows. It is assumed to be a mirrorreflection on the CPC, and the reflections of all photons on the reflectivesurface meet the reflection law. After the random photon is reflected bythe CPC, its direction vector is updated to the reflection directionvector. The reflection direction vector is obtained by the followingformula (Tan, 2007):
=U U N N U2 ( · ) (10)
where U designates the incident direction vector of the photon, Nmeans the normal vector at the intersection, and U’ represents the di-rection vector of the reflected photon. (N · U) < 0; if (N · U) > 0, thenthe normal vector takes the negative value in the calculation.
Fig. 2. Light transmission process of the truncated CPC.
Fig. 3. Flow diagram of MCRT program.
Fig. 4. Schematic of the incidence surface.
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(3). Energy statistics for the heat receiver
Energy loss occurs during the photon propagation. For example, inthe reflection process, the reflector of the CPC has a certain reflectivity.When a photon hits the CPC, the portion of photon energy loss dependson the proportion (reflectivity). Moreover, when the photon hits theheat receiver, its energy is not totally absorbed, which means that thephoton energy is absorbed by a certain proportion (absorption rate). Inthe program calculation, the parameters of the reflectivity and the ab-sorption rate are set to simulate the energy loss in the photon propa-gation process. The amount of energy that rays can carry after fourreflections is set as the threshold. Rays with lower energy are elimi-nated. The threshold value is Eμ4, where E is the initial photon energyand μ is the reflectivity of the reflective surface. When the photons arereflected or absorbed, the MCRT method is used for ray tracing. Theexternal surface of the heat receiver is divided into Z partitions. Then,the photon number in each partition and the energy intensity duringabsorption are calculated to obtain the heat flux distribution on theexternal surface of the heat receiver.
= =qe
Sim
Mm
i
1
i
(11)
= × ×e E µmr (12)
= ×S D L72i (13)
where (1 ≤ i≤ 72).The heat receiver circumference is divided into 72 parts equally. Mi
means the number of the photons absorbed in the ith partition of theheat receiver external surface. M is the mth photon absorbed in the ithpartition of the heat receiver external surface. em represents the energywhen the mth photon in the ith partition of the heat receiver externalsurface is absorbed, W. Si means the area of the ith partition of the heatreceiver outer surface, m2. E designates the initial energy of a ray, W. μis the reflectivity of the reflective surface. r means the number of rayreflections and ν is the absorption rate of the heat receiver. L designatesthe length of heat receiver tubes, mm.
The average heat flux density on the heat receiver can be calculatedby the following equation:
= = =
=
qe
Sj
i
Z
m
Mm
i
Zi
1 1
1
i
(14)
To simplify the simulation, we established the following assump-tions.
(1). The non-parallel rays were not considered. When considering thelight non-parallelism (32′) and surface roughness, compared withthe parallel light, the heat flux density distribution was not muchdifferent and the peak decreased slightly.
(2). The intensity of the solar radiation took 1000 W/m2. Each random
photon carried the equal amount of energy. The number of photonswas n. Thus the energy amount of a single photon was 1000/n W/m2.
(3). The CPC had an ideal curved surface with mirror reflection, whosereflectivity was 0.9.
(4). The axial heat flux on the heat receiver surface was uniformlydistributed. The axial length of the heat receiver was taken as theunit length when calculating the heat flux density on its externalsurface.
(5). A random photon was reflected at most 4 times. When the reflec-tion frequency exceeded 4 times, the photon is energy free. Thedifference of the average heat flux of the receiver including pho-tons reflected more than four times or not is no more than 30 W/m2. On the one hand, the ratio of the photon reflected over fourtimes was low, on the other hand the direction of the photon wasdifficult to make sure after four times reflections in reality.
(6). The photons reaching the heat receiver were absorbed in propor-tion, and the absorption rate was 0.85.
3. Calculation condition and verification
3.1. Calculation condition
The concentration characteristics of the cylindrical CPC were si-mulated by the MCRT method. This simulation comprised two parts:the concentration characteristics of the CPC without deformations andthe influence of deformations on CPC concentration characteristics. Theeffects of the different incidence angles, concentration ratios, andtruncation residual ratios of the CPC without deformations on the heatflux distribution on the heat receiver were studied. The specific con-ditions are shown in Table 1.
Under different truncation residual ratio, the maximum acceptanceangle of CPC changes, for example, when concentrating ratio is 4 andthe truncation residual ratio is 0.3, its corresponding maximum ac-ceptance angle of incidence is 36°. As shown in Fig. 2, the incidenceangle is 25°, which is larger than half acceptance angle (14.5°). Some ofthe light hits the receiver directly; some reaches the CPC involute part,and is reflected to the receiver. In order to understand this situationmore clearly, in Table 1, the corresponding angle between the halfacceptance angle and maximum acceptance angle is considered.
For the CPC with deformations, the influences of CPC machiningmode and accuracy and the assembly precision of the heat receiver onCPC concentration characteristics were mainly considered. CPC ma-chining is difficult due to its complexity, especially when a high accu-racy module line is required. The main machining forms includesrolling, cutting, and 3D printing. Rolling is a relatively common form inindustrial processing. After rolling, the CPC module line may be de-formed because of the elasticity of the mirror stainless steel or thealuminum plate. The possible deformations are the rotation andtranslation of the reflective surface. The relative position precision ofthe assembly of the receiver and CPC also affects CPC concentrationcharacteristics. In this study, the effects of CPC deformations on con-centration characteristics were analyzed in terms of the rotation and
Table 1Simulation conditions for the concentration characteristics of the CPC without deformations.
Concentration ratio, C Half acceptance angle, ° Incidence angle, ° Truncation residual ratio
translation of the reflective surface, the truncation of the involutestarting point, and the position offset of the heat receiver. As shown inFig. 5, the detail information of the four kinds of deformations are in-troduced as below:
Rotation of the reflective surface: Taking the involute startingpoint as the circle center, the two CPC reflective surfaces rotate in thedirection of aperture increase and decrease respectively. Rotation de-formation is common in actual CPC machining. The main CPC manu-facturing technology is rolling. The mirror stainless steel plate is themost popular material, which extends outward naturally under stress.Thus the external rotation is important, whose offset values are usuallypositive. Meanwhile, there is also transition rolling to reserve the pos-sibility of elastic deformation. If the backward elastic deformation isinsufficient, the internal rotation should be considered, correspondingto negative offset values. The probability of internal deformation isrelatively small. So in the current work, fewer negative offset values areselected than positive offset values.
Translation of the reflective surface: The CPC reflective surface ismoved along the X direction horizontally. The translation of the re-flective surface is difficult to avoid in actual machining. Considering themirror plate thickness, when a CPC is processed, the position of theinvolute starting point moves. which is equivalent to the offset distanceof the plate. So outward translation is important, with positive offset
values. In few cases, the mirror plate is grooved along a bendline andthen bent to keep the right position of the reflective surface. If thegroove is very deep, the reflective surface will be translated inwardafter bent. The corresponding offset value is negative.
Truncation of the involute starting point: If the involute startingpoint is retained, the heat receiver will contact the CPC when as-sembled, which will increase the heat dissipation. High assembly pre-cision will increase CPC machining difficulty exponentially. To as-semble the heat receiver, the involute starting point is usually truncatedand the position of the heat receiver remains the same, thereby af-fecting the light reflection path. To analyze the effect of cut off of theinvolute starting point on heat flux density, this simulation starts from atiny cut off (machining accuracy), which is then increased to 28.5%D.This means that the involute starting point is totally truncated and theCPC bottom is flat.
Position offset of the heat receiver: The receiver movement isgenerated during processing and practical application. The receiver canbe curved due to the gravity and thermal stress. The heat receiverchanges its position from the cross section. Movement is selected as atypical deformation of the heat receiver. If the involute starting point isretained, the heat receiver will contact the CPC when assembled, whichwill increase the heat dissipation. High assembly precision will increasethe CPC machining difficulty exponentially. In practice, the involute
Fig. 5. CPC deformations (a) Rotation of the CPC reflective surface (b) Translation of the CPC reflective surface (c) Truncation of the starting point (d) Position offsetof the heat receiver.
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starting point is always removed. So, in the process of assembly, thegravity and thermal stress may cause the downward offset of the heatreceiver and the downward movement is always combined with thetruncation of the involute starting point.
Table 2 lists the values of the four deformations and the incidenceangles of rays. The detail information and equations for the deforma-tions are listed in Appendix. The basic parameters of the CPC are takenas follows: a concentration ratio of 4, a truncation residual ratio of 0.5,and a half acceptance angle of 14.5°. To make the result referential, weconducted a non-dimensional treatment on the CPC deformation vari-ables. The linear deformation variables were divided by the heat re-ceiver external diameter, and the angle deformation variables weredivided by the half acceptance angle (14.5°).
3.2. Program verification
To verify the correctness and reliability of the in-house MATLABcode, we compared its simulation results with the results from the lit-erature (Abid et al., 2016). The contrast conditions were as follows: aCPC concentration ratio of 2.5, a truncation residual ratio of 1, a re-flectivity of 0.918, a heat receiver radius of 10 mm, an absorption rateof 0.9, a ray incidence angle of 0°, and an incident photon number of5000. Fig. 6 shows the heat flux distribution on the external surface ofthe heat receiver. Where, q means the heat flux density on the externalsurface of the heat receiver, represents the central angle of the heatreceiver external surface, and r designates the radius of the heat re-ceiver. According to the figure, the simulation results of the MCRTmethod agreed with the results from the literature (Abid et al., 2016).This outcome indicated that the results of the MCRT method were ac-curate and the required precision was fulfilled.
4. Concentration characteristics of the CPC without deformations
4.1. Influence of the concentration ratio on the heat flux density of the heatreceiver
As the incidence angle changes, Fig. 7 shows the variation law of theaverage heat flux density on the heat receiver external surface underdifferent concentration ratios and a truncation residual ratio of 0.5.According to the figure, when the ray incidence angle was smaller thanthe half acceptance angle, the average heat flux density on the heatreceiver performed a nearly linear increase with the increase of theconcentration ratio. When the incidence angle was larger than the halfacceptance angle, the average heat flux density decreased from the
order of thousands watts per square meters to that of hundreds wattsper square meters. Because for a full CPC, once the ray incident angleexceeded the half acceptance angle, the ray hitting the CPC was nolonger collected, which led to a sharp decrease in heat flux. However,for a truncated CPC, once the ray incident angle exceeded the half ac-ceptance angle, part of the rays could still reach the receiver. So finally,
Table 2Simulation conditions for the concentration characteristics of the deformed CPC.
No. Deformation form Deformation mode Offset value Non-dimensionalization Ray incidenceangle, °
1 Rotation of the reflectivesurface
Opposite rotation taking theinvolute starting point ascenter
Upward movement 0/0.5/1 (mm) 0/0.125/0.25 0/14Downward movement* 0/0.5/1 (mm) 0/0.125/0.25 0/14Leftward movement 0/0.5/1 (mm) 0/0.125/0.25 0/14Rightward movement 0/0.5/1 (mm) 0/0.125/0.25 0/14
* Downward movement of the heat receiver is combined with the truncation of the involute starting point and the offset value is same.
Fig. 6. Comparison between the simulated photon distribution by the MCRTmethod and the results from the literature (Abid et al., 2016).
Fig. 7. Effect of the concentration ratio on the average heat flux density on theheat receiver external surface (Ct = 0.5).
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the average heat flux density reduced to 0 W/m2 gradually. For ex-ample, when the concentration ratio took 4 and the incidence angletook 12° (the half acceptance angle took 14.5°), the heat flux densitywas 2600 W/m2. When the concentration ratio took 6 and the incidenceangle took 12° (the half acceptance angle took 9.6°), the heat fluxdensity was 700 W/m2.
Fig. 8 shows the heat flux distribution on the heat receiver externalsurface under different concentration ratios when the truncation re-sidual ratio took 0.5 and the incidence rays were vertical. As shown inthe figure, the heat flux was symmetrically distributed in the cir-cumferential direction and the distribution was extremely uneven. Twoevident peaks of heat flux distribution appeared in the circumference.The highest peaks were concentrated within the ranges of 60–120° and240–300° on the circle side. The secondary peaks appeared within theranges of 0–60° and 300–360° at the circle bottom. As the concentrationratio rise, the peaks and the secondary peaks of the heat flux distribu-tion on the heat receiver external surface closed to the upper middlepart of the circular tube. This phenomenon results from the increases inthe width of the CPC aperture and the amount of the incidence light onthe reflector upper part and the upward shift of the reflected light,which are caused by the growth of the concentration ratio. When theconcentration ratios were 3, 4, and 5, the maximum heat flux densitywere close and increased with a further increase in the concentrationratio. When the concentration ratio was 8, the peak heat flux densityreached 27,000 W/m2.
4.2. Effect of the truncation residual ratio on the heat flux density on theheat receiver
With the changes in the truncation residual ratio, Fig. 9 shows thevariation law of the average heat flux density on the heat receiver ex-ternal surface under different incidence angles. As shown in the figure,when the incidence angle was smaller than the half acceptance angle, theaverage heat flux density on the heat receiver increased gradually as thetruncation residual ratio rise. The overall variation range was less than300 W/m2. When the incidence angle is 12° and 14°, with the increase ofthe truncation residual ratio the heat flux density first increases and thendecreases around 2.4 × 103 W/m2. The reason is that as the angle of theincident ray increased, both the amount of light coming into the CPC andthe number of light reflections changed. Therefore, the greater thetruncation residual ratio was, the lower the curve slope would be, andthe smaller the concentration effect on the incident rays would be. Whenthe truncation residual ratios took 0.3 and 0.4, if the incidence angleswere smaller than the half acceptance angle, the change range of the heatflux density on the heat receiver external surface under different incidentangles was lower than 250 W/m2. With the increase of the incidence
angle, the average heat flux density reduced. When the incidence angletook 15° and exceeded the half acceptance angle, a small amount of lightcould be absorbed by the receiver due to truncation, causing the decreaseof the heat flux density. When the truncation residual ratio was over 0.7,the average heat flux density decreased to 300 W/m2 with the increase ofthe interception ratio.
Fig. 10 shows the heat flux distribution on the heat receiver externalsurface under different truncation residual ratios when the concentra-tion ratio took 4 and the incidence rays were vertical. According to thefigure, the heat flux distribution kept even with the increase in thetruncation residual ratio, while the heat flux density at the heat receiverbottom increased slightly. This phenomenon resulted from the increasein the CPC effective height. As the truncation residual ratio increased,the CPC can reflect the incidence rays from the higher parts to the heatreceiver bottom after multiple reflections, increasing the amount of theabsorbed light. Thus the heat flux density is improved.
Fig. 11 presents the effect of different concentrations and truncationresidual ratios on the average heat flux density on the heat receiverexternal surface when the incident rays were vertical. According to thefigure, the average heat flux density has a consistent change trend withthe truncation residual ratio under different concentration ratios. Withthe increase in the truncation residual ratio, the average heat fluxdensity increased first and then kept even. With the increase in theconcentration ratio, the effect range of the truncation residual rationarrowed. For example, when the concentration ratio was 8, the heatflux density on the receiver increased from 4.6 ×103 W/m2 to
Fig. 8. Heat flux distribution on the tubular heat receiver under differentconcentration ratios (Ct = 0.5, θ= 0°).
Fig. 9. Effect of the truncation residual ratio on the average heat flux density onthe heat receiver external surface under different incidence angles (C= 4).
Fig. 10. Heat flux distribution on the heat receiver external surface underdifferent truncation residual ratios (C= 4, = 0°).
R. Xu et al. Solar Energy 176 (2018) 73–86
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4.9 ×103 W/m2 with the increase of the truncation residual ratio from0.3 to 0.8, and the change was 6.5%. However, for the concentrationratio was 1.5, the heat flux density on the receiver increased from1.1 ×103 W/m2 to 1.2 ×103 W/m2 and the change was 9%. This var-iation range was larger than that under a concentration ratio of 8.Therefore, a small truncation residual ratio could be selected when theconcentration ratio was high. Correspondingly, a high truncation re-sidual ratio could be used when the concentration ratio was low.
4.3. Effect of the incidence angle on the heat flux density on heat receiver
Fig. 12 shows the variation law of the average heat flux density onthe heat receiver external surface with various incidence angles underdifferent truncation residual ratios (the concentration ratio took 4).According to the figure, when the incidence angle was smaller than thehalf acceptance angle (14.5°) and the truncation residual ratio waslarger than 0.5, as the incidence angle increased, the heat flux densityof the heat receiver decreased slowly with a change less than 400 W/m2. When the truncation residual ratio was smaller than 0.5, the heatflux density fluctuated as the incidence angle increased. That’s becauseas the incidence angle increased, both the amount of light ming into theCPC and the number of light reflections changed, and the heat fluxdensity change was affected by both. When the incidence angle waslarger than the half acceptance angle, with the further increase in theincidence angle, the heat flux density decreased from the order ofthousands watts per square meter to that of hundreds watts per squaremeter first, and then decreased to 0 W/m2 gradually.
Fig. 13 shows the heat flux distribution on the tubular heat receiverwith a concentration ratio of 4 and a truncation residual ratio of 0.5under different incidence angles. According to the figure, with the in-crease in the incidence angle, the heat flux density was no longer dis-tributed symmetrically as the peak heat flux density shifted and in-creased. When the incidence angle was 12°, the peak heat flux densityreached 42,000 W/m2. When the incident angle was larger than the halfacceptance angle (14.5°), the average heat flux density reduced to about500 W/m2.
5. Effects of the CPC deformations on concentrationcharacteristics
5.1. Effect of the rotation of the CPC reflective surface
Fig. 14 shows the effect of the rotation of the CPC reflective surfaceon the average heat flux density on the heat receiver external surface.According to the figure, the rotation of the reflective surface had a
beneficial effect on the average heat flux density on the heat receiverwhen the incidence angles θ were 0° and 8°. The average heat fluxdensity increased with the increase in the rotation angle due to theexternal rotation increase the CPC aperture width and the quantity ofincidence rays. For example, the heat flux increased about 20% when
Fig. 11. Effect of the truncation residual ratios on the average heat flux densityon the heat receiver external surface under different concentration ratios( = 0°).
Fig. 12. Effect of the incidence angle on the average heat flux density on theheat receiver external surface (C= 4).
Fig. 13. Heat flux density distribution on the heat receiver external surfaceunder different incidence angles (C= 4, Ct = 0.5).
Fig. 14. Effect of the rotation of the CPC reflective surface on the average heatflux density on the heat receiver external surface (C= 4, Ct = 0.5).
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the external angle is 0.345. As internal rotation brought fewer incidentrays, the heat flux density on the heat receiver surface decreased. Whenthe incidence angle (14°) was close to the half acceptance angle (14.5°),the heat flux density on the heat receiver decreased from 2.4 × 103 W/m2 to 8.0 × 102 W/m2 with the increase of the rotation angle from zeroto 0.069. Because most of the rays were reflected out of the CPC fromthe parts above the heat receiver and cannot be absorbed. However,when the reflective surface rotated internally, the reflected raysreached the heat receiver and were absorbed. Therefore, when the in-cidence angle was close to the half acceptance angle, internal rotationjust had a slight effect on the average heat flux density. Such as, whenthe internal rotation angle is 0.069 the heat flux density decreased from2.4 × 103 W/m2 to 2.3 × 103 W/m2. When the incidence angle in-creased to 16°, the rays reflected out of the CPC during external rotationincreased, and the average heat flux density on the heat receiver de-creased accordingly. With further increases in the incidence angle andthe external rotation angle, the heat flux density on the heat receiverincreased linearly with a change range less than 300 W/m2, which wastoo small to utilize. The average heat flux density in internal rotationshowed the same change rule as that of the heat flux density in externalrotation. As the internal rotation angle increased, the heat flux densitydecreased linearly. In conclusion, concentration characteristics wereonly slightly influenced by the internal rotation of the reflective surfaceand substantially affected by the external rotation. Therefore, whenmanufacturing CPCs by rolling, the external rotation of the formed CPCshould be avoided. In other words, light is more likely to hit the re-ceiver at large angles. According to the edge ray principle, the externalrotation of the reflecting surface reduces the half acceptance angle ofCPC. In fact, the incidence angle of the ray changes at all times, and theexternal rotation increases the probability that the incidence angle ofthe ray exceeds the half acceptance angle. The total energy obtained bythe heat receiver decreases taking long working time.
5.2. Effect of the translation of the CPC reflective surface
Fig. 15 shows the effect of the translation of the CPC reflectivesurface on the average heat flux density on the heat receiver externalsurface. According to the figure, a small external translation of the CPCreflective surface only had a small influence on the average heat fluxdensity on the heat receiver external surface. Even when the incidenceangle (14°) was close to the half acceptance angle (14.5°) and the re-lative displacement of the reflective surface was 0.125, the average heatflux density on the heat receiver decreased by approximately 300 W/m2. When the incidence angle (15°) was larger than the half acceptanceangle (14.5°), the change was the same as that of the heat flux densityfor the CPC without deformations, and the heat flux density on the heatreceiver was approximately 520 W/m2. However, when the translationof the CPC reflector was 0.12, a part of the incident rays can reach theheat receiver external surface by reflection and be absorbed. Thus, theaverage heat flux density on the heat receiver can reach 1800 W/m2.That is, the external translation of the CPC reflector was advantageouswhen the incidence angle was large.
When the internal translation of the reflective surface was 0.0125and the incident rays were vertical, the effect was negligible. However,when the incidence angle (14°) was close to the half acceptance angle(14.5°), the heat flux density decreased from 2400 W/m2 to 1700 W/m2
because a part of the second reflected rays was reflected out of the CPCfrom the parts above the heat receiver. When the incidence angle ex-ceeded the half acceptance angle, the outward relative shift of the re-flective surface in the range of 0–0.05 had little influence on the heatflux density. The heat flux density fluctuated around 500 W/m2.
5.3. Effect of the trunction of the involute starting point
Fig. 16 shows the effect of the truncation of the involute startingpoint on the average heat flux density on the heat receiver external
surface. According to the figure, the truncation of the involute startingpoint only had a slight effect on the heat flux density. With the increasein the truncation length, the heat flux density decreased by less than350 W/m2. When the incident angle is 0°, the average heat flux densitydecreases as the truncation length increases. When the truncationlength is 28.5%D, the heat flux density is 2127.3 W/m2, decreasing by18.6% compared with the full CPC. When the incidence angle was 14°and the relative truncation length was 28.5%D, the average heat fluxdensity decreased from 2441.2 W/m2 to 2281.0 W/m2, decreasing by6.5% compared with the full CPC. The effect of starting point truncationof involute on average heat flux with incidence angles between 0° and14° is nearly same. No data was added for image clarity. Because theCPC involute part mainly reflected the rays after multiple reflections tothe heat receiver bottom. In actual propagation, photon energy atte-nuated completely after multiple reflection, and the propagation di-rection also deflected. According to the calculation, if a photon hadbeen reflected more than 4 times, its energy was 0. Therefore, the in-volute starting point only had a mere effect on reflection and con-centration. It can be truncated by 28.5%D, as the truncated startingpoint had little influence on the heat flux density. Even the maximumheat flux decreased by 18.6%, cut off of the involute starting point isrecommended since the processing technic and processing cost anddifficulty should also be considered.
5.4. Effect of the position offset of the heat receiver
The effect of the position offset of the heat receiver on the averageheat flux density on its external surface is shown in Fig. 17. Accordingto the figure, when the incident rays were vertical ( = 0°), the upwardmovement of the heat receiver affected the heat flux density con-siderably. With the increase in the moving distance, the heat fluxdensity decreased. The changes of the heat flux density, caused by therightward, leftward and downward movements, were less than 400 W/m2. When the heat receiver was moved upward, the incident rays werereflected out of the CPC from the heat receiver bottom because ofmultiple reflections by the CPC. A gap between the heat receiver andthe CPC formed due to the movement. The greater the movement dis-tance was, the more the reflected light would be. Consequently, theheat flux density decreased. When the heat receiver was moved right-ward, the part of the rays reflected by the left part of the CPC wereabsorbed by the heat receiver. However, a small part of the rays wasreflected out through the gap between the heat receiver and the CPC.The same thing happened in the cases of the leftward movement andthe rightward movement. When the heat receiver was moved down-ward, the rays absorbed near the lowest point of the heat receiver
Fig. 15. Effect of the translation of the CPC reflective surface on average heatflux density on the heat receiver external surface (C= 4, Ct = 0.5).
R. Xu et al. Solar Energy 176 (2018) 73–86
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decreased. However, the reflection times of this part of the rays alsodecreased, which reduced the energy loss in the reflection. Therefore,when the heat receiver was moved downward, and the incident rayswere vertical, the heat flux density was changed by less than 50 W/m2.
When the incidence angle was close to the half acceptance angle(14.5°), the upward and leftward movements of the heat receiver hadlittle effect on the average heat flux density. When the relative movingdistance reached 0.25, the average heat flux density decreased by only50 W/m2. When the heat receiver is moved downward and rightward,the heat flux density on the surface decreased sharply. According toFig. 13, with the increase in the incidence angle, the area with high heatflux density on the heat receiver moved over the heat receiver circle.When the incidence angle was larger than the half acceptance angle, theincident rays were reflected out of the CPC from the upper part of theheat receiver circle. Thus, the heat receiver can absorb the rays with bigincidence angles if it is moved up. In addition, the upward movement ofthe heat receiver can make the rays, which were reflected to the CPCbottom normally, reflected out through the gap between the heat re-ceiving circle and the CPC. This part of the rays carried only a smallamount of energy after multiple reflection. Thus, the average heat fluxdensity kept even. When the incidence angle was 14°, a large amount oflight is emitted to the right surface. The light reflected from the right
surface can still be absorbed by the heat receiver with a moderateleftward movement. Therefore, under a leftward movement of 0.25, theheat flux density decreased by 500 W/m2 only, which was small com-pared with that under the right movement of the heat receiver.
The heat receiver can absorb the energy of the rays with big in-cidence angles when it is moved up. The energy can even exceed that ofthe rays with incidence angles smaller than the half acceptance angle.Fig. 18 shows the variation law of the average heat flux density on theheat receiver external surface under different incidence angles whenthe relative displacement of the heat receiver was 0.25. According tothe figure, when the incidence angle was smaller than the half accep-tance angle, the light leakage phenomenon between the heat receiverbottom and the CPC was severe. Thus, the heat flux density decreased.When the light incidence angle was between 14.5° and 15.5° (the halfacceptance angle is 14.5°) the heat flux density decreased by 13% withan upward movement of 0.125, and by 6% with an upward movementof 0.25. which meant increased the half acceptance angle. The heat fluxdensity decreased from 1500 W/m2 to about 300 W/m2 gradually as theincidence angle increased from 16° to 22°. In addition, with the increasein the moving distance, the reduction rate decreased. When the in-cidence angle reached 22°, the average heat flux density was the sameas that on the heat receiver without movement. In general, when the
Fig. 16. Effect of the truncation of the involute starting point on the averageheat flux density on the heat receiver external surface (C= 4, Ct = 0.5).
Fig. 17. Effect of the heat receiver position offset on the average heat flux density on the heat receiver external surface (a) Ct = 0.5, = 0° (b) Ct = 0.5, = 14.
Fig. 18. Effect of the upward movement of the heat receiver on the averageheat flux density on the heat receiver external surface (Ct = 0.5).
R. Xu et al. Solar Energy 176 (2018) 73–86
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heat receiver is moved upward and the incidence angles were smallerthan the half acceptance angle, the heat flux density decreased by about100–200 W/m2 with an offset value of 0.125 and by about 100–600 W/m2 with an offset value of 0.25. The heat flux density correlates to thehalf acceptance angle of the CPC negatively. Therefore, considering thevariations of the sunlight incidence angle and the CPC placement di-rection, the downward, rightward, and leftward offsets of the heat re-ceiver during the assembly of the heat receiver and the CPC must beavoided. Because of the different placement directions of the CPC, ifthere may be incident rays at the half acceptance angle, the heat re-ceiver should be moved up accordingly.
6. Conclusion
In this study, the concentration characteristics of a cylindrical CPCsolar heat collector were simulated by in-house MCRT method. In ad-dition, the effects of CPC deformations on concentration characteristicswere analyzed, in terms of the rotation and translation of the reflectivesurface, the truncation of the involute starting point, and the positionoffset of the heat receiver. Finally, the following conclusions weredrawn.
(1). Two evident peaks of heat flux distribution were observed in thecircumferential direction of the tubular heat receiver due to CPCconcentration. With the increase in incidence angle, the peak heatflux density increased. Moreover, the peak location moved to theupper part of the tubular heat receiver as incidence rays deflect.When the incidence angle was 12°, the peak heat flux densityreached 42,000 W/m2 with a concentration ratio of 4 and a trun-cation residual ratio of 0.5.
(2). When the incidence angle was smaller than the half acceptanceangle, as the concentration ratio increased, the average heat fluxdensity of the tubular heat receiver increased linearly. The peakand secondary peak heat flux densities were gathered at the uppermiddle part of the tubular heat receiver. This phenomenon is dueto the increase in the width of the aperture of the CPC, the increase
in the amount of light incident on the upper part of the reflectorand the upward shift of the reflected light as the concentrationratio increases. When the incidence angle was larger than the halfacceptance angle, the average heat flux density decreased sharply.
(3). The average heat flux density of the tubular heat receiver graduallyincreased and had a consistent change trend with the increase intruncation residual ratio under different concentration ratios,while the overall change was not significant. With the increase inconcentration ratio, the influence range of the truncation residualratio narrowed. Therefore, a large truncation residual ratio couldbe selected when the concentration ratio is small. Conversely, asmall truncation residual ratio could be selected when the con-centration ratio is high.
(4). The small internal rotation and small internal or external transla-tion of the reflective surface and the truncation of the involutestarting point affected the heat flux distribution on the heat re-ceiver merely. However, external rotation can bring a significantinfluence. Therefore, the reflective surface can have certaintranslations and internal rotations, but external rotations should beavoided when machining CPC.
(5). The downward, rightward, and leftward offsets of the heat receivershould be avoided in the assembly of heat receivers and CPCs. Ifthe incidence rays have a high incidence probability at a largeincidence angle, the heat receiver should be moved upward ac-cordingly in assembly.
Acknowledgements
The present work was supported by the National Natural ScienceFoundation of China (51506004), Beijing Municipal Natural ScienceFoundation (3162009), Scientific Research Project of BeijingEducational Committee (KM201410016001), Beijing Youth Top-notchTalent Support Program, Science and Technology Project of Beijing(Z171100000517007) and Fundamental Research Fund of BeijingUniversity of Civil Engineering and Architecture (X18101).
R. Xu et al. Solar Energy 176 (2018) 73–86
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App
endi
x:Th
eCP
Cde
form
atio
ns
Def
orm
atio
nO
ffset
valu
eSc
hem
atic
Equa
tion
ofth
ein
volu
teA
BEq
uatio
nof
the
invo
lute
BCBo
und
Rota
tion
ofth
ere
flect
ive
surf
ace
ω= =
++
XX
YY
YX
cos(
)sin
()
2sin
()
cos(
)sin
()
2cos
()
2
= =+
X Y
(sin
cos
)
(sin
cos
)
D
D2
2
= =+
+X
XY
YY
Xco
s()
sin(
)2s
in(
)co
s()
sin(
)2c
os(
)2
= =+
XA
YA
(sin
cos
)
(sin
cos
)
D
D2
2
=+
++
A/2
cos(
)1
sin(
)a
aa
AB:
+(
)0
a2
BC: +
()
aa
23 2
Tran
slat
ion
ofth
ere
flect
ive
surf
ace
L m=
+
=+
XL
Y
(sin
cos
)
(sin
cos
)
Dm
D2
2
=+
=+
XA
L
YA
(sin
cos
)
(sin
cos
)
Dm
D2
2
=+
++
A/2
cos(
)1
sin(
)a
aa
AB:
+(
)0
a2
BC: +
()
aa
23 2
Trun
catio
nof
the
star
ting
poin
toft
hein
volu
teσ
= =+
X Y
(sin
cos
)
(sin
cos
)
D
D2
2
=+
2(
sinco
s)
D 2
= =+
XA
YA
(sin
cos
)
(sin
cos
)
D
D2
2
=+
++
A/2
cos(
)1
sin(
)a
aa
AB:
+(
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2
BC: +
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aa
23 2
Posi
tion
offse
tof
the
heat
rece
iver
L u= =
+
X Y
(sin
cos
)
(sin
cos
)
D
D2
2
= =+
XA
YA
(sin
cos
)
(sin
cos
)
D
D2
2
=+
++
A/2
cos(
)1
sin(
)a
aa
AB:
+(
)0
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BC: +
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aa
23 2
Equa
tion
for
the
rece
iver
= =+
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cos
sin
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(02
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Not
ice:
All
the
equa
tions
are
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esse
din
rect
angu
lar
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te.X
′is
anin
depe
nden
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le,Y
′is
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,and
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ein
term
edia
teva
riab
les.
R. Xu et al. Solar Energy 176 (2018) 73–86
85
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