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Renewable Energy 75 (2015) 722e732

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Renewable Energy

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Performance analysis of Azimuth Tracking Fixed Mirror SolarConcentrator

Longlong Li a, Huairui Li b, Qian Xu b, Weidong Huang b, *

a Department of Chemistry, University of Science and Technology of China, Hefei, Anhui 230026, PR Chinab Department of Earth Chemistry and Environmental Science, University of Science and Technology of China, Hefei, Anhui 230026, PR China

a r t i c l e i n f o

Article history:Received 5 April 2014Accepted 23 October 2014Available online

Keywords:Azimuth trackingFixed mirror solar collectorMean annual net heat efficiencyDesign parametersOptical performance

* Corresponding author. Tel.: þ86 13856945896.E-mail addresses: lill@mail.ustc.edu.cn (L.

(W. Huang).

http://dx.doi.org/10.1016/j.renene.2014.10.0620960-1481/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

The fixed mirror solar collector (FMSC) fixes reflector and mobiles receiver to collect solar energy.However, this type of concentrator has a low efficiency and short operating duration in practical ap-plications. In this paper, we propose to install the FMSC on an azimuth tracking device (ATFMSC) and thereflectors are arranged by intermission to avoid the shading of neighbor reflector for incidence angle ofless than 10� to improve its optical performance. Through the integration of the reflected solar radiationdistribution function over any reflection point, and then the whole collector aperture, we develop theanalytical expressions of various system efficiencies to numerically simulate the performance of ATFMSCwith evacuated tube receiver in different design parameters. It is validated by the ray tracing results. Theresult shows that the mean annual net heat efficiency of the whole system would be up to 61% with theoperating temperature of 400 �C, which is higher than parabolic trough collector and traditional FMSC.This is because the longitudinal incidence angle of ATFMSC always remains zero by tracking the sunazimuth, so the end loss of the concentrator can be avoided and enables it to operate with high efficiencycontinually.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The fixed mirror solar concentrator (FMSC) moves the receiverover the circle path to track the sun and provide moderate tem-perature heat without moving the reflectors. Comparing with theconventional concentrators based on tracking the sun by the re-flectors, it is an obvious advantage for FMSC system that thestructure of the collector is greatly simplified to makes this type ofconcentrator has a great potential in industrial fields.

The concept of FMSC was first proposed by Russell [1]. In theearly 1970s, Williams [2] tested a 2.1*2.4 m model at Georgia Techand reported its performance, the results suggested that it could beused for industrial medium process heat (80�250 �C). In 1974,Kumar [3] numerically calculated the daily solar energy collectedunder the sunny day model and the assumption that all reflectedrays were intercepted by the receiver. In 1982, Maolong Che andYunian Tang [4] analyzed the geometry of the FMSC and the in-fluences of design parameters on the optical performance.

Li), huangwd@ustc.edu.cn

In 1985, Nicola's and Dura'n [5,6] proposed a numerical methodto analyze the flux density distribution on the receiver, and ob-tained the distribution characteristics and intercept factors forsome moments and typical design parameters. The results showthat the instantaneous optical efficiency of the system varies atdifferent moments and different days, and the FMSC with concavemirrors has larger mean concentration factor than that with flatmirrors which has been certified subsequently through experiment[7]. Recently, Moll et al. proposed to fix a parabolic reflector toreplace a group of mirrors, but this kind of design abandoned thevirtue that a group of mirrors can focus the rays reflected from theircenters to the center of the receiver whether the incidence angle is0 or not for the FMSC [8,9]. Nadal and Moll [10] developed a ray-tracing procedure to calculate the incidence angle modifier (IAM)and then applied it to calculate the flux distribution of the receiverin some typical cases, the result shows that the performance of thesystem decreases with the solar incidence angle.

It is hard for FMSC to maintain a high efficiency with highoperating temperature, and the efficiency of the system decreasessignificantly when the concentrator operating at times away fromthe noon because of the large incidence angle [9]. In this paper, wepropose to install the FMSC (including the receiver) on an azimuthtracking device to improve the system efficiency, we call it Azimuth

L. Li et al. / Renewable Energy 75 (2015) 722e732 723

Tracking Fixed Mirror Solar Concentrator (ATFMSC). We also pro-pose to add an interval between adjacent reflectors to avoid theshading by the neighbor mirror for the incidence angle of less than10

�.In current research results, only the annual net solar energy of

the system is calculated to evaluate its performance, no method isdeveloped to evaluate its efficiency variations with the solar inci-dence angle for different days. We developed an optical model tooptimize the design parameters of ATFMSC through the net solarenergy collected over a year, and to evaluate its performance withdifferent operating conditions in typical equinoxes and solstices ofLatitude-30�N.

2. Method of study

The FMSC is composed of a set of long narrow reflectors fixed upon the concave cylindrical surface, and each of them is tilted towardthe cylinder axis with the inclination angle of a quarter of its

Fig. 1. Optical process demonstration of sun rays incident on: (a) reflectors' centers,and (b) any point P at the reflector.

corresponding central angle. The receiver is also mounted on thereference cylindrical surface and moves along the circle path totrack the sun. It can be easily demonstrated that [2] the centralsunlight would be focused sharply regardless of incidence angles.

The optical efficiency is an important parameter for analyzingoptical systems. The FMSC is a kind of linear focus concentratingcollector, and in general, there are three methods to calculate theflux distribution at the receiver for its optical efficiency: the coneoptics method [11], ray tracing method [12,13] and semifiniteintegration method [14], they are all computationally expensiveand time consuming for engineering design. In this paper, wedevelop a method proposed in linear Fresnel collector study andbased on the integral of any reflection point [15,16] to directlycalculate the optical efficiency of ATFMSC. Moreover, the expres-sions for cosine factor, intercept factor, receiver efficiency, shadingand blocking factor and net heat efficiency are also derived. The keyto calculating instantaneous optical efficiency of the system is asfollows.

2.1. Instantaneous optical efficiency of ATFMSC

We first calculate the instantaneous optical efficiency for a pointon the reflector based on the integration of the effective solar ra-diation distribution function, and then obtain the instantaneousoptical efficiency of the system through the integration over thewhole concentrator aperture. Our method takes into account theeffect of cosine factor which will be occurred when the incidentsunlight is not perpendicular to the concentrator aperture. Moredetails are demonstrated as follows.

Fig. 1(a) shows the cross section of the ATFMSC, the coordinatesx, y direct toward the directions tangent and perpendicular to thevertex O of central mirror, i.e., the coordinate origin. It is assumedthat all mirrors are symmetrically positioned about the concen-trator axis and of the same width of 1 m, and all dimensionsconsidered are referred to its width D ¼ 1 m. The mirrors arelabeled from i ¼ 1 to ±(nq þ 1)/2 in each direction. In addition, therim angle F is the half angle subtended by the outmost reflector.

As shown in Fig. 1(b), points M and N represent the two edges ofthe receiver. The M1N1 is the reference plane passes through thecentre of receiver S and perpendicular to the solar central ray PQreflected from a point P on the reflector. The reflected solar centralray strikes at points Q and Q1 of planes MN and M1N1 respectively,so the angles NPQ (41) and QPM (42) can be calculated by thefollowing geometric relationship for the case of di > fi:

tan 41z

w2 cos

�q� qi

2

�� DP cos miðdi�fiÞ

fi

2R cos�q� qi

2

�

¼ w4R

�DP cos

�qþ qi

4

�fi

þDP cos

�qþ qi

4

�

2R cos�q� qi

2

� ; (1)

tan 42z

w2 cos

�q� qi

2

�� DP cos miðdi�fiÞ

fi

2R cos�q� qi

2

�

¼ w4R

þDP cos

�qþ qi

4

�fi

�DP cos

�qþ qi

4

�

2R cos�q� qi

2

� ; (2)

Table 1Typical design parameters for ATFMSC.

Parameter Value

Reference radius R 33 mWidth of receiver w 1.5 mWidth of mirrors D 1 mConcentrator inclination j (rad) p/4Initial incidence angle without mutual shading ф (deg) 10Reflectivity of reflector r 0.92Product of absorptivity and transmittance a 0.888Atmospheric transparency p 0.8Operating temperature T (oC) 400Total optical error s (mrad) 4

L. Li et al. / Renewable Energy 75 (2015) 722e732724

where w is the width of the receiver; R is the radius of referencecircle; DP is the distance between any point P and the symmetryaxis of the ith reflector; q is the incidence angle at the vertex ofcentral mirror, q ¼ p/2 � h, where h is the solar altitude; qi is thecentral angle of the ith mirror as shown in Fig. 1 (a); mi ¼ q þ qi/4, isthe incidence angle at the vertex of the ith mirror, and is viewed asthe incidence angle of the ith reflector; di is the distance betweencentre of the ith mirror and centre of the receiver, di¼ 2Rcos (q� qi/2) and fi is the focal length of the ith reflector [17]

fizri2cos mi (3)

where ri represents the radius of the ith cylindrical mirror; cosmi isthe cosine factor of the ith reflector and can be expressed as

hcos i ¼ cos mi (4)

We evaluate the error caused by above approximation Equations(1)e(4), and find that the relative error is less than 0.1%, this meansthese approximation equations are accurate enough for the nextperformance evaluation and optimization of the system.

So, the instantaneous optical efficiency for point P is

hP ¼ IPDNI

¼

Z 42

�41

DNI$r$a$hcos i$Beff�q�dq

DNI

¼ rahcos i

Z42

�41

Beff�q�dq; (5)

where IP is the instantaneous solar energy captured by the receiver,DNI is the Direct Normal Irradiation, r is the reflectivity of the entirecollector, a is the product of absorptivity and transmittance of thereflected solar central ray striking on the receiver, both r and a areviewed as constant here, 41 and 42 are the maximum acceptanceangles for solar radiation reflected from point P, and can be ob-tained by (1) and (2). Beff(q) is the normalized distribution of re-flected solar brightness [18], which is calculated as following:

We take advantage of polynomial model to simulate the solarradial radiation distribution based on experimental data [19], anduse exponent decreasing model in Buie et al. [20] to simulate thecircumsolar region to obtain the radial distribution of the solarradiation B(d), d is the angle between any incidence ray and rayfrom the center of the sun. For linear focus solar collectors, we cansimply convert B(d) into linear distribution form according to thefollowing equation [21]:

Blinear�d⊥� ¼ Z dd‘B

�d�; d ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2⊥ þ d2‘r

(6)

where Blinear is the transverse distribution of incidence solarbrightness, in Wm-2rad�1，d with subscripts II and ⊥ are longitu-dinal and transverse angles respectively. Besides, the angleswithout subscripts represent the transverse ones generally.

In the real concentrating system, there are optical errors such asmirror contour error in the transverse direction scontour, the spec-ular error sspecular and the tracking error stracking, which are typi-cally 1.7 mrad [22], 0.5 mrad and 2 mrad [23] respectively.According to the equation [21] s2optic ¼ 4*s2contour þ s2specular

þs2tracking, the typical value of the total optical error is 4.0 mrad. By

convolving the distribution of the total optical error and thetransverse distribution of the solar brightness, the distribution ofeffective reflected solar brightness Beff(q) can be given by:

Beff�q� ¼ 1 ffiffiffiffiffiffiffip

Zdq0 exp

�q0

2!Blinear

�q� q0

�(7)

soptic 2p 2s2optic

Additionally, in order to improve the computing speed, Beff(q) isexpressed as polynomial in this paper [16]:

Beff�q� ¼Xi¼n

i¼0

ciqi (8)

where q represents the angle of any reflected ray which is associ-ated with the reflected direction of the incidence ray from thecenter of the sun for each point on the reflector, while ci is thesimulation coefficient, and n is the highest degree of thepolynomial.

Here, the optical errors are characterized by a Gaussian distri-bution model [24], and then the effective solar radiation distribu-tion function is obtained by convolution of the transversal errordistribution and the transversal solar radiation distribution.

The instantaneous intercept factor for point P can be calculatedas:

gP ¼Z42

�41

Beff�q�dq (9)

For the ith reflector, the instantaneous intercept factor can beobtained accordingly

gi ¼

Z D2

�D2

gPdx

D(10)

Moreover, in practical optical systems, the existing shading andblocking can lead to system performance degradation. Shading iscaused by the neighbor reflector or the receiver, and blocking oc-curs if a reflector blocks the sun rays reflected by a neighbor anddirected to the receiver. The shading and blocking factor of the ithreflector can be approximately viewed as the ratio of the areawhere the solar central ray is not shaded and blocked to theaperture area [25], it can be expressed as follows:

hsbi ¼ 1� DDi

D(11)

where DDi represents the part shaded or blocked by adjacent re-flectors or the receiver. Through the evaluation of mean annual netheat efficiency of the system with design parameters listed inTable 1, it is proved that the error introduced by this approximationis smaller than 0.1%.

Therefore, the instantaneous optical efficiency of the ithreflector can be expressed as

L. Li et al. / Renewable Energy 75 (2015) 722e732 725

Z D2

hPdx

hopti ¼ hsbi�D

2

D(12)

Based on the integration of the effective solar radiation distri-bution function over any reflection point, and then the whole col-lector aperture, we develop and obtain the analytical expressions ofthe cosine factor, shading and blocking factor, intercept factor,receiver efficiency and net heat efficiency of the ATFMSC tonumerically simulate its performance in different designparameters.

The instantaneous cosine factor of the system is

hcos ¼Pnq

i¼1 hcos i

nq: (13)

where nq is the total number of reflectors.Then, the instantaneous optical efficiency of thewhole system is

hopts¼Pnq

i¼1hopti$DNIPnq

i¼1DNI¼raPnq

i¼1hsbihcos i

Z D2

�D2

Z 42

�41

Beff ðqÞdqdx

nq$D;

(14)

The instantaneous shading and blocking factor of the system is

hsb ¼ 1�Pnq

i¼1 hcos i$DDiPnq

i¼1 hcos i$D(15)

In addition, the instantaneous intercept factor of the system canbe expressed as

gs ¼

Pnq

i¼1 hcos i

Z D2

�D2

Z 42

�41

Beff ðqÞdqdx

D$Pnq

i¼1 hcos i; (16)

2.2. Mean annual efficiencies of the whole system

The receiver we adopted consists of several vacuum tubes inZig-Zag form [26], and for a given working temperature, its heatloss qloss per area can be seen as constant, the total heat loss isproportional to the area of the receiver. The qloss can be calculatedaccording to Patnode's equation [27] based on SEGS data

qloss ¼ a0 þ a1T þ a2T2 þ a3T

3 þ DNI$�b0 þ b1T

2�; (17)

where T is the working temperature of the receiver (�C) andassumed to be of 400 �C herein, the parameters ai and bi are con-stants and quoted from reference literature [27], the DNI isexpressed by sunny day model

DNI�t� ¼ Is

�t�$p^m (18)

where Ist(t) ¼ 1367*[1 þ 0.034*cos(2pn/365)], and it is the solarirradiation outside the Earth's atmosphere (W/m2), n representsthe nth day in the year from Jan.1st, m is the air mass,m ¼ f1229þ ½614 sinðhÞ�2g1=2 � 614 sinðhÞ, h is the solar altitude, pis the atmospheric transparency with the value of 0.8 herein.

Above all, we can get the annual net solar energy collected byper unit area of collector

Z365� � � � � � � � � qlossðtÞ$gðtÞ$w�

Qnetyear ¼0

hopt t $DNI t $nx t $g t �nq$D

dt:

(19)

where nx is the ratio of the actual DNI to the DNI based on thesunny daymodel, so, nx is equal to 1 for the sunny day. Here we useg(t) to represent the running state of the system at moment t, and itis assumed that the system starts running when the net solar en-ergy Qnet is greater than a threshold value of 200 W/m2.

g�t� ¼ 1 for Qnet > ¼ 200 w=m2

g�t� ¼ 0 for Qnet <200 w=m2 (20)

Thus, the mean annual cosine factor of the system can bedefined and calculated as

hcosyear ¼

Z 365

0hcosðtÞ$DNIðtÞ$nxðtÞ$gðtÞdtZ 365

0DNIðtÞ$nxðtÞ$gðtÞdt

(21)

The mean annual shading and blocking factor of the system is

hsbyear ¼

Z 365

0hsbðtÞ$hcosðtÞ$DNIðtÞ$nxðtÞ$gðtÞdtZ 365

0hcosðtÞ$DNIðtÞ$nxðtÞ$gðtÞdt

(22)

The mean annual intercept factor of the system is

gyear ¼

Z 365

0gðtÞ$hsbðtÞ$hcosðtÞ$DNIðtÞ$nxðtÞ$gðtÞdtZ 365

0hsbðtÞ$hcosðtÞ$DNIðtÞ$nxðtÞ$gðtÞdt

(23)

The mean annual receiver efficiency of the system can beexpressed as follows

hryear ¼QnetyearZ 365

0ðr$a$hcosðtÞ$gðtÞ$hsbðtÞ$DNIðtÞ$nxðtÞ$gðtÞÞdt

;

(24)

In the morning or late afternoon, the collected solar energy isnot enough to get the system to start working at requested tem-perature, and this part of solar energy is not able to supply availableenergy and thereby discarded. The mean annual efficiency relatedto the discard loss hdyear is calculated as following:

hdyear ¼

Z 365

0DNIðtÞ$nxðtÞ$gðtÞdtZ 365

0DNIðtÞ$nxðtÞdt

(25)

So, the mean annual net heat efficiency of the system is

htyear ¼ r$a$hcosyear$gyear$hsbyear$hryear$hdyear (26)

In addition, the instantaneous efficiencies of the whole systemcan be defined and calculated in the similar way.

Fig. 2. Concentration efficiency of fixed mirror solar concentrators: experimental dataand model simulation.

L. Li et al. / Renewable Energy 75 (2015) 722e732726

3. Model validation

3.1. Validation with experimental data

The experimental data is from Richard Williams [2], main pa-rameters of the experiment: Receiver width ¼ 7 in, mirrorwidth ¼ 4 in, mirror number ¼ 23; radii of the referencecircle ¼ 63.95 feet; installation in north-south direction, the inci-dence angle in north-south direction ¼ 0; the simulated parame-ters: the optical error ¼ 0; the circumsolar ratio ¼ 0.02.

The concentration efficiency, as simulated with the model andmeasured experimentally, is shown in Fig. 2. The solar angle wasmeasured from the eastern horizon from the north-south orientedfixed mirror concentrator. The model simulated concentrated effi-ciency was calculated from 27.5 to 90�. Since the fixed mirrorconcentrator designed for the experiment was symmetrical aboutits center line, the theoretical concentrated efficiency for solar an-gles between 152.5 and 90� were simply mirror images of thosebetween 27.5 and 90�. The comparing between the model experi-mental data and model simulation shows litter difference.

Fig. 3. Comparison of intercept factors of the 13th reflector at normal incidence be-tween our model and Soltrace for different optical errors. The solar profile is ofGaussian distribution, and design parameters needed are listed in Table 1.

3.2. Validation with ray tracing

The ray tracing code Soltrace is developed by the NREL and hasthe ability to analyze the optical performance of various solar col-lectors [28]. In Fig. 3, the intercept factors of the 13th reflector atnormal incidence for different optical errors are compared betweenour method and Soltrace, both based on the Gaussian sun shapemodel. Essential design parameters are listed in Table 1. It can beseen that the intercept factor decreaseswith the optical error, and ishighly consistent between our method and Soltrace, with theaverage and maximum errors of 0.06% and 0.2% respectively.

We also compare with Nadal's result [9] based on the FMSCsystem. For the FMSC system with flat receiver, through the de-rivative of intercept factor of the system gs with respect to thedistance between any point on the receiver and the centre of thereceiver, we can yield the normalized flux density distributions onthe receiver at the normal incidence. The result is compared withthat from ray tracing method [9] in the case of N ¼ 25, s ¼ 8 mrad,R ¼ 12.5, CSR ¼ 0.05, and w ¼ 2.975. Here, the sun shape isdetermined by Buie [29], and the diffuse radiation is not taken intoaccount.

As shown in Fig. 4, the normalized flux density distributions onthe receiver obtained with our method coincide with those of raytracing model rather well; both reach the maximum values at thecentre of the receiver, and decrease with the distance to the centre.The results demonstrate that the method we proposed to simulatethe performance of ATFMSC is reliable.

4. Layout of the reflectors

It should also be noted that for traditional FMSC, the reflectorsare closely mounted on the concave cylindrical surface (no mutualshading or blocking among reflectors at normal incidence), and itadds the optical performance penalty of the system due to shadingand blocking. In this paper, we propose to add an interval betweenadjacent reflectors to avoid the shading by the neighbor mirror forthe incidence angle is less than a certain value of ф. If the centralangle qi of the ith reflector is known, then the central angle of thei þ 1st reflector can be calculated with the following equation [30].

2RD

¼cos�f� qiþ1þqi

8

�cos�qiþ1�qi

8

�

cos�fþ qiþ1þqi

2

�sin�qiþ1�qi

2

� ; (27)

Fig. 4. Normalized flux density on the receiver at normal incidence determined by ourmethod and Nadal model in the case of: number of mirrors N ¼ 25, optical errors ¼ 8 mrad, reference radius R ¼ 12.5, circumsolar ratio CSR ¼ 0.05, collector widthw ¼ 2.975, product of absorptivity and transmittance a ¼ 1, and total reflectivityr ¼ 0.92, all dimensions considered are referred to mirror width D ¼ 1, the sun shape isof Buie solar profile model.

L. Li et al. / Renewable Energy 75 (2015) 722e732 727

We simulate the performance of the ATFMSC system at differentinitial incidence angles ф. It can be found in Fig. 5 that the meanannual net heat efficiency of the system htyear increases obviouslywith initial incidence angle ф for ф < 10

�and becomes slower for

ф � 10�, the reason is: the reflection area that is not shaded or

blocked first becomes larger apparently with initial incidence anglefor ф < 10

�because of the increasing gap between reflectors, and

then it increases slightly due to the tiny increment of gap forф � 10

�, besides, both cosine factor and intercept factor always

decrease with the initial incidence angle. On the other hand, thecollector width W increases rapidly with ф, especially for ф � 10

�.

Taking into account the land occupation in practice, here we takeф ¼ 10

�as the advisable layout for further analysis.

Fig. 6. Effect of optical error on mean annual efficiencies of the system and otherparameters are given in Table 1.

5. Results and analysis

5.1. Optimization

It is assumed that the system starts working as the net solarenergy collected is greater than 200 W/m2. Fig. 6 shows the per-formance of our advisable layout with different optical errors. It canbe seen that less solar energy will be intercepted by the receiver forthe system with poor optical quality, which would cause theintercept factor to decrease from 97% to 94% as the optical errorvaries from 2 to 4 mrad, and reduce to about 71% when the opticalerror is 10 mrad. For the net heat efficiency, it drops slightly foroptical errors less than 4 mrad, and then falls to about 43% sharplyfrom 61%, as the optical error increases to 10 mrad. So, the opticalquality of the system has a major impact on its performance. In thepractical concentrating system, there are optical errors such asmirror contour error in the transversal direction, specular error andtracking error, the typical optical error of 4.0 mrad is used in ourstudy.

For the receiver, its width should be large enough to interceptmore sunlight reflected from reflectors but not too wide, this isbecause the existing heat loss is proportional to its width. Fig. 7shows the system performance reaches the optimum when itswidth is approximately equal to 1.5 m. At the beginning, the solarenergy intercepted by the receiver increases dramatically with itswidth, and its heat loss as well as the shading are both small, somore net solar energy is collected and the mean annual net heatefficiency increases until goes for the maximum; by contrast, thenet heat efficiency decreases as the receiver width is larger than1.5 m, this is because the intercept factor of the system increases

Fig. 5. Mean annual efficiencies and collector width W, as a function of the initialincidence angle ф from which there is no shading by the adjacent mirror, other designparameters are given in Table 1.

slightly and is close to the 100%, while the heat loss and the shadingresulted from the receiver increase proportionally.

As shown in Fig. 8, the reference radius also has an effect oncollector performance, the mean annual net heat efficiency of thesystem first increases and reaches the maximumvalue of about 61%with the reference radius of 33 m, then decreases with referenceradius. For the case of small reference radius, although less sun raysare intercepted by the receiver with the increase of reference radiusdue to the increasing distance between the receiver and reflectors,both the cosine factor and the shading and blocking factor increasewith reference radius, and are greater than the decrease of inter-cept factor. As the reference radius becomes larger than 33 m, theintercept factor is dominant to affect the collector performance anddecreases much rapidly with reference radius, resulting in thedegradation of collector performance.

The concentrator inclination angle also has a great impact on thesystem performance through the cosine factor and shading andblocking factor of the system and it can be expressed as pi/2-h-J.As shown in Fig. 9, the cosine factor, and shading and blockingfactor increase firstly with the inclination, and then decreases. Themaximum mean annual net heat efficiency is achieved when theinclination angle is of 45�, at which the incidence angle remainssmall for most of the times and the maximum mean annual cosinefactor and shading and blocking factor are achieved.

Moreover, as can be seen in Fig. 10, on summer solstice, thecosine factor and shading and blocking factor remain large around

Fig. 7. Effect of receiver width on mean annual efficiencies of the system and otherparameters are given in Table 1.

Fig. 8. Effect of reference radius on mean annual efficiencies of the system and otherparameters are given in Table 1.

Fig. 9. Effect of concentrator inclination on mean annual efficiencies of the system andother parameters are given in Table 1.

L. Li et al. / Renewable Energy 75 (2015) 722e732728

noon for small inclination angles, this enables the system to collectmore net solar energy during this time, but the starting time towork is delayed and the total working period in the whole day isshortened remarkably; on the contrary, the system can work

Fig. 10. Cosine factor and net energy curves depending on the inclination as a functionof time on summer solstice and other parameters are given in Table 1.

continuously for a longer time at large inclination angle of 45� andmore solar energy is collected in the whole day, even though lessenergy is harvested around noon. On the equinoxes, the cosinefactor is high from the beginning with the inclination angle of 45�,so the system is able to start to collect solar energy earlier, and thenet solar energy collected around noon is very close to themaximum value obtained at other inclination angles, as shown inFig. 11. So, taking into account these factors, the ATFMSC with theinclination angle of 45� has the better optical performance than atother inclination angles.

5.2. Concentrator performance

We analyze the performance of ATFMSC with different oper-ating conditions in typical equinoxes and solstices of Latitude-30�N. The typical design parameters for ATFMSC are listed inTable 1.

We simulate the instantaneous cosine factor, intercept factor,receiver efficiency, shading and blocking factor and net heat effi-ciency of the system at different incidence angles as shown inFig. 12(a), the shading and blocking factor first increases and rea-ches the maximum value, then decreases with the incidence angle.It should be noted that the shading and blocking factor is not of themaximum value at normal incidence because of the shadow by thereceiver, the maximumvalue occurs at the incidence angle of about13�, at which there is the minimum shading and blocking caused bythe receiver and neighbor reflectors, and the significant reductionlater is resulted from the greater shadow or blocking by mutualreflectors. The cosine factor decreases with the incidence angleobviously. Due to these effects, the net heat efficiency of the systemfirst increases slightly and then decreases with incidence angle.

Fig. 12(b) shows the variation of system performance on sum-mer solstice. Due to the large incidence angle during the earlymorning, noon and later afternoon, both the cosine factor and theshading and blocking factor are rather small, resulting in the lownet heat efficiency. It should be noted that a slight decrease of thenet heat efficiency happens at around 9 a.m. because of the shadowof the receiver, at which the incident sun rays are normal to theconcentrator. At noon, the net heat efficiency of the system is rathersmall because of the high incidence angle.

The similar results can be obtained at equinoxes as shown inFig.12(c), but the net heat efficiency only decreases slightly at noon,and maintains a high efficiency in most times. This attributes to thehigher shading and blocking factor and cosine factor at lowerincidence angles.

Atwinter solstice, the incidence angle decreases over time in themorning and reaches the minimum at noon, the similar situation

Fig. 11. Cosine factor and net energy curves depending on the inclination as a functionof time on the equinoxes and other parameters are given in Table 1.

Fig. 12. Instantaneous efficiencies of the system curves depending on: (a) Incidence angle, (b) summer solstice, (c) equinoxes, (d) winter solstice during running period with thecollected net solar energy of greater than 200 W/m2. The system is designed with parameters listed in Table 1.

L. Li et al. / Renewable Energy 75 (2015) 722e732 729

occurs in the afternoon. Thus, the net heat efficiency of the systemis high around the noon because of the high values of both cosinefactor and shading and blocking factor as shown in Fig. 12(d). Inaddition, it can also be found that the longest operation periodoccurs at summer solstice, followed by the equinoxes, and thewinter solstice is the shortest case.

The ATFMSC system is mainly used to provide medium tem-perature heat, Fig. 13 indicates that the mean annual net heat ef-ficiency of the system remains more than 60% when the operationtemperature is less than 400 �C. Moreover, a mean net heat effi-ciency of almost 50% at the design temperature of 600 �C can beachieved, which may be used for electric power generation.

Moreover, we also study the performance of the system at theinclination angle of 30�. Fig. 14 clearly shows that the system per-forms worse than that of inclination angle of 45� for differentoperating temperatures, for example, the net heat efficiency of thesystem is 4% lower when the operating temperature is 400 �C. Itshould be noted that the mean annual cosine factor increasesslightly with operating temperature, and this is because at higher

Fig. 13. Mean annual efficiencies of the system at different operating temperatures atinclination angle of 45� , the other design parameters are given in Table 1.

operating temperatures, the running period is reduced and thesystem will not work when the cosine factor is small, which con-tributes to the higher mean annual cosine factor. However, thedecreasing discard efficiency will lead to a lower mean annual netheat efficiency of the whole system.

The above analysis shows that the mean annual net heat effi-ciency can be approximately up to 61% for the optimized ATFMSCsystem when the operating temperature is 400 �C.

5.3. Performance comparison with the other solar system

5.3.1. Performance comparison with FMSC systemBased on the experimental results, William [2] predicted that

the net heat efficiency of the FMSC system can reach 50% if theoperating temperature is 260 �C. As the higher the operatingtemperature, the higher the heat loss of the receiver, if the oper-ating temperature rises to 400 �C, the efficiency of the FMSC systemwill be much less than 50%. It is much lower than the ATFMSC

Fig. 14. Mean annual efficiencies of the system at different operating temperatures atinclination angle of 30� , the other design parameters are given in Table 1.

L. Li et al. / Renewable Energy 75 (2015) 722e732730

system which has about 61% efficiency in the same operatingtemperature. The study by Nadal [9] demonstrated that theinstantaneous cosine factor, intercept factor of the FMSC system areall greatly influenced by the longitudinal incidence angle, resultingin the optical efficiency decreases rather greatly, while the effect oflateral incidence angle is very small. For East-West orientationsystem installed at latitude 40〫in Nadal's paper, the optical effi-ciency for the early morning or late evening will decrease to about10% of the value at noon as the longitudinal incidence angle in-creases to about 80〫. For North-south orientation system, theoptical efficiency at noon of winter solstice decreases to about halfof the maximumvalue due to the large longitudinal incidence angleof 63.50. By mounting the concentrator (including the receiver) onan azimuth tracking device to follow the sun throughout the day,the longitudinal incidence angle always remains zero, so enablesthe concentrator to operate with high efficiency continually.Furthermore, the mean annual shading and blocking factor of theATFMSC system is significantly higher than that of the FMSC, due tothe optimized arrangement of mirrors on cylindrical surface. So theperformance of the ATFMSCwill be much better than FMSC system.

5.3.2. Performance comparison with the parabolic trough collectorParabolic trough solar concentrator is a kind of mature solar

thermal concentrating system. The collector is provided with one-axis solar tracking, usually the horizontal north-south axis, toensure that the solar beam falls parallel to its axis and hencemaintain a high efficiency [31,32]. Similarly, the cosine factor ofparabolic trough solar concentrator with north-south axis trackingsystem also changes with time, and the mean annual cosine factoris smaller than 0.9, inferior to 0.95 of ATFMSC system and leading tothe reduction of both intercept factor and optical performance ofcollector, the reason is: the northesouth axis tracking system isonly able to track the sun in the transverse direction, but for theATFMSC system, the solar azimuth is tracked, and the systeminclination angle is adjusted to obtain better performance. Inaddition, the receiver of the trough collector is of vacuum tube [33],all reflectors of the ATFMSC system share one receiver, whichmeans its heat loss per unit area is smaller than that of the troughsystem. Based on these two facts, the mean annual net heat effi-ciency of the parabolic trough collector was predicted about 50% insimilar conditions. So the ATFMSC has much better efficiency thanordinary solar trough system.

5.3.3. Performance comparison with the solar islandsSolar Islands is a kind of collector with all mirrors and receiver

fixed on the azimuth tracking device [34,35], which turns thewhole concentrating system to track the sun azimuth. So, the solarrays are vertically incident upon reflectors and reflected to thereceiver, the reflected solar ray from a reflector center strikes intothe center of the receiver. Although the shadowing and blockingcan be avoided with the transversal solar incidence angle of 0, thetotal efficiency of Solar Islands has not been certified to be high.This is because the cosine factors are small for most of the time,especially in winter solstice. At 40� latitude, the maximum cosinefactor for winter solstice is only 0.595 which is less than the min-imum cosine factor of the ATFMSC analyzed in our study.

5.3.4. Performance comparison with the solar dish collectorsThe following two point focusing technologies are the most

probably solar concentrating systems which can compete withATFMSC: the paraboloidal dish collector [36] and the solar towercollector with heliostats [37]. However, they both need to move thereflectors by two-axis sun tracking device [37], and cannotconstruct large scale collectors because of their high elevations andwind pressures [38,39]. ATFMSC just moves the concentrator

horizontally by sun azimuth tracking device, and do not need tomove the concentrator in elevation, moreover, the receiver is muchsmaller than the concentrator and can be moved more easily to ahigh position both in technique and cost. So ATFMSC and SolarIslands may be the only two possible concentrating systems can beused for large scale applications and reduce the tracing costs andimprove the reliability because a single tracker can be applied for alarge scale reflector. But ATFMSC has a higher efficiency than solarislands.

6. Conclusion

In this paper, we propose to install the FMSC on an azimuthtracking device, it is called Azimuth Tracking Fixed Mirror SolarConcentrator (ATFMSC), and develop analytical expressions of thesystem efficiencies to numerically simulate the performance ofATFMSC with flat receiver in different design parameters. The ex-pressions of the instantaneous cosine factor, optical efficiency,intercept factor, receiver efficiency, shading and blocking factor andnet heat efficiency are established through the integration of theeffective solar radiation distribution function for a reflection pointand then the whole concentrator aperture, and then the designparameters are optimized to maximize the annual collected solarenergy. We compare the concentration efficiency with the modeland the experimental data; the intercept factors of the 13threflector at normal incidence for different optical errors betweenour method and ray tracing code Soltrace. Moreover, as the sunshape is simulated with Buie model, the normal flux density on thereceiver at normal incidence is compared with Nadal's result in thesame cases. All results demonstrate that themethodwe proposed isreliable.

It should be noted that the reflectors are mounted on theconcave cylindrical surface in terms of no mutual shading for theincidence angle of no larger than 10� to improve its optical per-formance. Moreover, the effects of concentrator inclination j,reference radius R and receiver width w are also analyzed andoptimized, and the results show that the ATFMSC achieves theoptimum optical performance with their values of p/4, 33 m, 1.5 mrespectively, and all dimensions considered are referred to its widthD ¼ 1 m. On this optimum design, we analyze the ATFMSC per-formance in different operating conditions, the results show that:

a) Mean annual net heat efficiency of the system decreases withthe optical error, due to the decreasing intercept factor.

b) Cosine factor decreases with incidence angle, while theshading and blocking factor first increases and then de-creases with incidence angle, so the net heat efficiency isincreased slightly and then decreased with incidence angle.

c) At inclination angle of 45�, the system canwork continuouslyfor longer and more solar energy is collected in the wholeday than at small inclination angles on summer solstice, eventhough less energy is harvested around noon; on the equi-noxes, the system starts to collect solar energy earlier, andthe net solar energy collected around noon is very close tothe maximumwith other inclination angles. So, the ATFMSCsystem at inclination angle of 45� has the best opticalperformance.

d) In the range of medium temperature heat utilization, themean annual net heat efficiency of the system is of at least61% at operating temperature of 400 �C. The result showsthat the ATFMSC can be used for industrial process heat.

The results demonstrate that the maximum annual net heatefficiency of ATFMSC system can be reached 61%, which is betterthan parabolic through concentrator as well as traditional fixed

L. Li et al. / Renewable Energy 75 (2015) 722e732 731

mirror solar concentrator, together with the advantages in con-struction complexity and costs, bothmake this type of concentratorhas a great potential for commercial applications.

Nomenclature

ai,bi parameters in calculation of heat lossC vertex of the ith reflectordi distance between the center of ith mirror and the receiverDP distance of point P to the axis of the ith reflectorDDi the parts shaded or blocked by adjacent reflectors and

receiverD the width of mirrorsDNI direct normal irradiance, W/m2

fi focus length of the ith reflectorh solar altitude, radIsc solar constant, W.m�2

M, N edges of the receiverM1N1 line passing through B and perpendicular to BRn the number of a day in the year from Jan.1stnq the number of the mirrorsnx ratio of the actual DNI to the DNI based on the sunny day

modelg running state of the system at moment tO origin of the coordinatep atmospheric transparencyP any point on the reflectorQ the incident ray from the center of the sun after reflection

striking at Q in plane MNQ1 the central incident ray after reflection striking at Q1 in

plane M1N1R reference circle radiusS centre of the receiverT working temperature, �Ct time, sw receiver widthW width of collectorX direction tangent to the vertex O of central mirrorY direction perpendicular to the vertex O of central mirror41 NPQ, rad42 QPM, radqi the central angle of ith mirror, radmi incidence angle of the ith mirror, radFi the rim angle of ith mirror, radhsbi the instantaneous shading and blocking factor of the ith

reflectorhsb the instantaneous shading and blocking factor of the

systemhcosi the instantaneous cosine factor of the ith reflectorhcos the instantaneous cosine factor of the systemhP optical efficiency of the solar ray reflected from the point

Phopti the instantaneous optical efficiency of the ith reflectorhopts the instantaneous optical efficiency of the whole systemhryear the mean annual receiver efficiency of the systemhsbyear the mean annual shading and blocking factor of the

systemhcosyear the mean annual cosine factor of the systemhdyear the mean annual discard efficiency of the systemhtyear the mean annual net heat efficiency of the systemBeff(q) normalized distribution of effective solar brightness, W/

(m2 rad)qloss heat loss per unit reflector area, W/m2

Qnetyear annual net energy collected by the system per unitreflector area, J/m2

Greek symbolsa product of transmittance and absorptivity of the reflected

ray striking on the receivergP the instantaneous intercept factor for point Pgi the instantaneous intercept efficiency of the ith reflectorgs the instantaneous intercept factor of the systemgyear the mean annual intercept factor of the systemq the incidence angle on the center of the central mirror,

radr reflectivity of point Ps total optical error, mradф original incidence angle at which there is no mutual

shading of mirrors, degreej concentrator inclination, rad

Subscripts and superscriptscos cosineeff effectivei the ithr receivers systemP a point at the reflectoropt opticalsc solar constantsb shading and blockingt total

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