Applied Energy 115 (2014) 128–139

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Applied Energy

journal homepage: www.elsevier .com/ locate/apenergy

Research on the compensation of the end loss effect for parabolic troughsolar collectors

0306-2619/$ - see front matter � 2013 Published by Elsevier Ltd.http://dx.doi.org/10.1016/j.apenergy.2013.11.003

⇑ Corresponding author. Tel./fax: +86 871 65517266.E-mail address: liming@ynnu.edu.cn (M. Li).

Chengmu Xu a, Zhiping Chen a, Ming Li a,⇑, Peng Zhang b, Xu Ji a, Xi Luo a, Jiangtao Liu a

a Solar Energy Research Institute, Yunnan Normal University, Kunming 650092, Chinab Institute of Refrigeration and Cryogenics, MOE Key Laboratory for Power Machinery and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

h i g h l i g h t s

� The optical analysis on the end loss effect of PTC-HNSA is performed in detail.� A method to compensate the end loss effect of PTC-HNSA is proposed and the optical analysis is made.� We use typical weather data of some Chinese cities to calculate goel,d, goel,y, gioe,d and gioe,y.� The applicability conditions of the compensation method is analyzed and discussed.� Verified the feasibility of the compensation method by experiments with a 5-m long PTC.

a r t i c l e i n f o

Article history:Received 18 September 2012Received in revised form 4 October 2013Accepted 3 November 2013Available online 26 November 2013

Keywords:Solar energyPTCEnd loss effectEnd plane mirrorCompensation method

a b s t r a c t

In this paper, an optical analysis on the end loss effect of parabolic trough solar collector (PTC) with hor-izontal north–south axis (PTC-HNSA) is performed, and a method to compensate its end loss effect is pre-sented. The calculation formulae for the optical end loss ratio and the increased optical efficiency (theoptical collection efficiency increment of PTC system after this compensation method is used) arederived; the daily optical end loss ratio, yearly optical end loss ratio, daily increased optical efficiencyand yearly increased optical efficiency in different latitudes are calculated; the variation of optical endloss ratio and increased optical efficiency with trough’s length and latitude angles are analyzed and dis-cussed. It is indicated through the analyses that this compensation method is very applicable for regionswith the latitude over 25� (especially over 30�) and short trough collectors. In order to verify the feasibil-ity of the compensation method, a five-meter PTC-HNSA experimental system was built. The increasedthermal efficiency of the experimental system is measured, and the result that the experimental value(increased thermal efficiency) substantially agreed with the theoretical value (increased optical effi-ciency) is gained. All these works can offer some valuable references to the further study on high-effi-ciency trough solar concentrating systems.

� 2013 Published by Elsevier Ltd.

1. Introduction

Solar energy is a sort of energy with low density, intermittentand changing spatial distribution, and is quite different from con-ventional energy, so it requires more for its collection and applica-tion. Among the concentrating solar collectors, parabolic troughsolar collector (PTC) is one of the most matured technologies [1].As one of the middle and high temperature collectors, PTC can beapplied in the production and living fields such as thermal gener-ate electricity, industrial process heat, domestic hot water andspace heating, air-conditioning and refrigeration, desalinationand drying [2–8].

At present, PTC usually adopts single axis tracking. As the sun–earth line is not vertical to earth’s axis, the incident rays are gener-ally oblique, therefore, it is inevitable for the PTC to have cosineloss (cosine effect) [2,9–12], and besides, partial end light reflectedby one end of trough cannot be gathered to the absorber and alsocause end loss effect [6,8,12–14]. The end loss effect of PTC will in-crease with the increasing of cosine effect, especially for regionwith high latitude and short PTC, the end loss ratio is relativelyhigh. In application and research, this loss shall be consideredand shall be compensated by applicable measures. In some works,although the end loss was considered [6,8,12,13], no further dis-cussion and research on relevant compensation methods werementioned.

Aimed at the end loss effect of parabolic trough solar collectorwith horizontal north–south axis (PTC-HNSA), a method that caneffectively compensate its end loss effect is presented in this paper.

Nomenclature

cp specific heat (J/kg/�C)f focal distance (m)far focal distance at arbitrary point (m)hav average height of end plane mirror (m)I0 solar constantIb; I

0b solar beam radiation (W/m2)

K absorption constantL length of parabolic trough (m)Loel,ar optical end loss at arbitrary point (m)Loel,av average optical end loss (m)Lpta length of focal line formed by sunlight orderly reflected

by end plane mirror and parabolic mirror (m)Ltpa length of focal line formed by sunlight orderly reflected

by parabolic mirror and end plane mirror (m)N the day number of the year (day)Pt solar energy reflected by the parabolic trough (J)Q, Q0 heat transfer rate (J/s)Ti; T

0i inlet temperature (�C)

To; T0o outlet temperature (�C)

tS solar time (h)vw wind velocity (m/s)w aperture width of parabolic trough (m)x, y, z cartesian coordinates

Greek symbolsb tilt angle of PTC (�)d declination angle (�)

DT, DT0 temperature difference (temperature rise) between out-let and inlet (�C)

gioe increased optical efficiencygite increased thermal efficiencygioe,d daily increased optical efficiencygioe,y yearly increased optical efficiencygoel optical end loss ratiogoel,d daily optical end loss ratiogoel,y yearly optical end loss ratiogoesl optical end shade loss ratiogt;ins;g0t;ins instantaneous thermal efficiencyh, h0 incident angle (�)hz zenith angle (�)H rim angle (�)lm mass flow rate (kg/s)qp reflectivity of end plane mirrorqt reflectivity of parabolic trough reflectoru latitude angle (�)x hour angle (�)

AbbreviationsAM air massPTC parabolic trough solar collector(s)PTC-HNSA PTC with horizontal north–south axis

C. Xu et al. / Applied Energy 115 (2014) 128–139 129

Mainly analyzed the variation of the end loss and compensationeffect with the length of trough and latitude, and discussed theapplicable latitude scope for the compensation method. A 5-mPTC-HNSA experimental system was built, and the test experi-ments for the compensation effect were conducted. All these workscan offer some valuable references to the further study on high-efficiency trough solar concentrating systems.

2. Analysis for the end loss effect

2.1. Optical analysis

For the PTC-HNSA with the aperture width w, length L and spec-ular reflectivity qt, the solar energy reflected by the parabolictrough will be

Pt ¼ wLIbqt cos h ð1Þ

where Ib and h are respectively the solar beam radiation and theincident angle of sunlight. However, not all solar energy reflectedby parabolic trough is gathered on heat absorber tube, and partialenergy will be emitted from one end of trough and cause opticalend loss (see Fig. 1).

In Fig. 1, Let the equation of edge AOB of parabolic trough be

z ¼ x2

4fð2Þ

where f is the focal distance of parabolic trough, point C is a arbi-trary point at parabola AOB, DC is the normal passing point C; lightEC is the incident ray parallel to plane yOz, and light CG is its reflec-tive ray. Let the light HC as the incident ray being vertical to troughsurface (parallel to axis z) and light CF as its reflective ray, it is obvi-ously that the angle between HC and EC is the incident angle h. InFig. 1, FH is vertical to DC, EH is vertical to HC, and GF is verticalto FC, according to geometric relationship, it can be evidenced that

\FCG = \ECH = h. Let coordinates of point C and focus F respectivelyas (x, x2/4f) and (0, f), the distance between points C and F (focal dis-tance of point C) will be

far ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ ðf � x2=4f Þ2

q¼ ðx2 þ f 2Þ=4f ð3Þ

It can be gained that the optical end loss caused by sunlight re-flected by point C is

Loel;ar ¼ far tan h ð4Þ

The above equation shows that the optical end loss of certainpoint is related to its focal distance, and it is obviously that theaverage optical end loss of the whole trough’s end is absolutely re-lated with the average focal distance of parabolic trough. In Fig. 1,the coordinates of point A and B are respectively (�w/2, w2/16f)and (w/2, w2/16f), and then the average optical end loss of trough’send is

Loel;av ¼1w

Z w=2

�w=2far tan hdx ¼ w2 þ 48f 2

48ftan h ð5Þ

where (w2 + 48f2)/48f is the average focal distance of parabola AOB.It can be gained that the optical end loss ratio of PTC is

goel ¼Loel;av

L¼ w2 þ 48f 2

48fLtan h ð6Þ

The incident angle h can be given through the following several for-mulae [15].

cos h ¼ ðcos2 hz þ cos2 d sin2 xÞ1=2

ð7Þ

cos hz ¼ cos u cos d cos xþ sin u sin d ð8Þ

d ¼ 23:45 sin½360ðN þ 284Þ=365� ð9Þ

x ¼ 15ðtS � 12Þ ð10Þ

Fig. 1. The optical end loss effect for parabolic trough concentrator.

6 8 10 12 14 16 180.0

0.2

0.4

0.6

0.8

1.0

1.2

Sola

r bea

m ra

diat

ion

I b (k

W/m

2 )

Solar time tS (hours)

N=51,K=0.21 (TR) N=57,K=0.17 (TR) N=51 (MR) N=57 (MR)

Fig. 2. The diurnal variation of solar beam radiation (where TR and MR stands fortheoretical results and measurement results, respectively).

130 C. Xu et al. / Applied Energy 115 (2014) 128–139

where hz—zenith angle, d—declination angle, N—the day number ofthe year, x—hour angle, tS—solar time in hours, and u—latitudeangle.

The average optical end loss ratio from sunrise time tsr to sunsettime tss (daily optical end loss ratio) can be expressed as

goel;d ¼w2 þ 48f 2

48fL

R tss

tsrIb sin hdtSR tss

tsrIb cos hdtS

ð11Þ

It is defined as the ratio of the loss of total solar energy due toend loss effect and total solar radiation energy reflected by para-bolic trough from sunrise time to sunset time. While the yearlyoptical end loss ratio can be expressed as

goel;y ¼w2 þ 48f 2

48fL

P365N¼1

R tss

tsrIb sin hdtSP365

N¼1

R tss

tsrIb cos hdtS

ð12Þ

It is defined as the ratio of the loss of total solar radiation energydue to end loss effect and total solar energy reflected by parabolictrough within a year.

The sunrise time and sunset time are given as

tsr ¼ 12� cos�1ð� tan u tan dÞ=15 ð13aÞ

tss ¼ 12þ cos�1ð� tan u tan dÞ=15 ð13bÞ

For PTC with north–south oriented and the tilt angle of rotatingaxis be b, when calculating the optical end loss ratio, simply re-place latitude angle u with u � b [15] (it equals to changing lati-tude to u � b) in Eq. (10), i.e. rewrite Eq. (10) as

cos hz ¼ cosðu� bÞ cos d cos xþ sinðu� bÞ sin d ð14Þ

while it is not required to substitute u in Eq. (13), because the sun-rise and sunset time are not related to the inclination of collector.

2.2. Calculation results and analysis

Obviously, the optical end loss ratio goel is the function of L, w, f,N, u and tS. In order to better understand the change rule of endloss ratio of PTC system under single-axis tracking with relevantparameters, this section will carry out the related calculationsand analyses based on the theoretical model given above. Accord-ing to Eqs. (11) and (12), the parameter of solar beam radiation, Ib,is required for calculation of daily and yearly optical end loss ratio.While the value of Ib is changing all over the time. In sunny days,the solar beam radiation of the sunlight throughout a day can beexpressed as [15,16]

Ib ¼ I0 exp½�KðAMÞ� ð15Þ

where I0 (�1367 W/m2), K and AM (=1/coshz) are the solar constant(extraterrestrial radiation), the absorption constant of air and theair mass, respectively. Fig. 2 gives the comparison of measurementvalue and theoretical value of solar beam radiation, and it showsthat the value of K can be selected in the range of about 0.17–0.21.

The relevant calculation results of optical end loss ratio areshown in Figs. 3–7, and some calculation parameters are shownin Table 1 (except otherwise specified, such as Figs. 4 and 7).

Fig. 3 shows the variation of optical end loss ratio in the regionof latitude 35� with day number N and solar time tS, which indi-cates that optical end loss ratio is large in winter and small in sum-mer throughout the year and in the shape of ‘‘saddle’’. Fig. 4 showsthe variation of daily optical end loss ratio goel,d with the length ofparabolic trough L. Obviously, goel,d decreases with the increase ofL, and it varies significantly especially in the range of L < 15 m,while varies little in the range of L > 15 m. Fig. 5 gives the variationof daily optical end loss ratio goel,d with day number N. It showsthat in the range of about u < 15�, daily optical end loss ratio islarge in winter and summer and small in spring and fall while inthe range of about u > 15�, it is large in winter and small in sum-mer. Figs. 4 and 5 show that the end loss effect increases as latitudeincreases. In order to understand the deviation of theoretical re-sults and measurement results, we use typical hourly data of solarbeam radiation at different latitudes in 17 Chinese cities to calcu-late daily and yearly optical end loss ratio, and compare them withthe theoretical results, as shown in Figs. 6 and 7. Where the geo-graphic locations (latitudes and longitudes) of these 17 Chinese cit-ies are shown in Table 2. The results show that the measurementresults are substantially agreed with the theoretical results. Obvi-ously, in cloudy and rainy days, solar radiation intensity usually

50 100 150 200 250 300350

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0

22.5

810

1214

16

ηoe

l(%)

t S(ho

urs)

N (day)

Fig. 3. The variation of goel with N and tS (u = 35�).

5 10 15 20 25 30 35 4005

1015202530354045505560

η oel,d

(%)

L (m)

ϕ = 0o

ϕ = 10o

ϕ = 20o

ϕ = 30o

ϕ = 40o

ϕ = 50o

N = 354

Fig. 4. The variation of goel,d with L.

0 50 100 150 200 250 300 3500

5

10

15

20

25

30

η oel

,d (%

)

N (day)

ϕ = 0o ϕ = 5o ϕ = 10o

ϕ = 15o ϕ = 20o ϕ = 25o

ϕ = 30o ϕ = 35o ϕ = 40o

ϕ = 45o ϕ = 50o

Fig. 5. The variation of goel,d with N.

0 50 100 150 200 250 300 3500

5

10

15

20

25

30

35

η oel

,d (%

)

N (day)

MR TR (Guangzhou) MR TR (Beijing) MR TR (Manzhouli)

Fig. 6. The variation of goel,d with N. Where, MR and TR stands for measurementresults and theoretical results, respectively.

0 5 10 15 20 25 30 35 40 45 50 5502468

10121416182022

η oel,y

(%)

ϕ ( o )

TR MR (L = 5 m) TR MR (L = 10 m)

Fig. 7. The variation of goel,y with u. Where, TR and MR stands for theoretical resultsand measurement results, respectively.

Table 1Some calculation parameters.

Parameters Value

Length of parabolic trough L (m) 10Aperture width of parabolic trough w (m) 3Focal distance of parabolic trough f (m) 1.2Absorption constant K 0.18

C. Xu et al. / Applied Energy 115 (2014) 128–139 131

changes significantly in a day; however, when solar radiation valueis small, PTC system will also receive low solar energy, resulting ina small amount of end loss; moreover, the time when beam radia-tion suddenly appears is random. Therefore, the measurement re-sults are relatively well agreed with the theoretical results.Otherwise, the yearly optical end loss ratio changes little in therange of about u < 20� and significantly in the range of aboutu > 20�, and almost changes in a linear.

3. Compensation for the end loss effect

3.1. Compensation method and optical analysis

Due to cosine effect, all the single axis tracking PTC have endloss, except when the double axis tracking is adopted [8,17]. ForPTC with relatively short length, it is feasible to adopt two dimen-sional tracking; however, for PTC with relatively long length, it isdifficult to adopt the two dimensional tracking. Due to the end lossratio and trough length is in an inverse correlation, to reduce endloss ratio, the current PTC are usually made long. Making PTC prop-erly oblique [11,12], it can also decrease the end loss effect (seeFig. 8), but the trough with long length will increase the difficultyof construction and investment. Prolonging the heat absorber tubeproperly along the trough’s end can also compensate the opticalend loss (see Fig. 9), but the increasing of length will increase the

Table 2Longitudes and latitudes of 17 Chinese cities.

City name East longitude (�) North latitude (�)

Guangzhou 113.33 23.17Xiamen 118.07 24.48Ganzhou 115.00 25.87Hangzhou 120.17 30.23Shanghai 121.45 31.40Hefei 117.23 31.87Nanyang 112.58 33.03Zhengzhou 113.65 34.72Lanzhou 103.88 36.05Yinchuan 106.22 38.48Dalian 121.63 38.90Tianjin 117.07 39.08Beijing 116.47 39.80Datong 113.33 40.10Chengde 117.95 40.98Changchun 125.22 43.90Manzhouli 117.43 49.57

Fig. 9. Decrease the end loss effect of PTC by increasing the absorber tube.

132 C. Xu et al. / Applied Energy 115 (2014) 128–139

thermal loss, especially when it adopts cavity absorber tube, thethermal loss will be worse.

Here, we will introduce a method that can effectively compen-sate the end loss effect of PTC-HNSA in regions with relatively highlatitude (the following discussion is based on the example of PTC-HNSA on Northern Hemisphere).

As shown in Fig. 10, set a plane mirror with the width of para-bolic trough as its width and the focal distance of parabolic troughas the height at the north end of PTC-HNSA (rim angle [18]H 6 90�); thus, the light reflected from the north end of trough willbe reflected to the heat absorber tube by the plane mirror, such aslights ‘1’ and ‘2’ in Fig. 10, so then the optical end loss is compen-sated (ideal situation). Besides, the direct sunlight to the plane mir-ror will be reflected to the parabolic trough by the plane mirror,and then be reflected to the heat absorber tube, such as lights ‘3’and ‘4’ in Fig. 10, and then an additional compensation is gained,which will further improve the heat collection efficiency of system.

In Fig. 10, Let the equation of parabola AOB be Eq. (2), and theequation of beeline GH is

z ¼ f ð16Þ

For the convenience of analysis and discussion, let plane AOBHGbe equivalent to rectangle CDGH with a height of hav, and we canget

hav ¼1w

Z w=2

�w=2f � x2

4f

� �dx ¼ f � w2

48fð17Þ

Thus, the direct sunlight to end plane mirror, after twice reflec-tion, will form the focal line on heat absorber tube with a length of

Lpta ¼ hav tan h ð18Þ

The sunlight reflected to end plane mirror by parabolic troughwill finally form a focal line on the heat absorber tube with a lengthof

Fig. 8. Decrease the end loss effect of PTC by making it properly oblique.

Ltpa ¼ Loel;av ð19Þ

Let the reflectivity of end plane mirror be qp, in can be seenfrom Fig. 10 that setting an end plane mirror equals to increasingthe length of parabolic trough by (Lpta + Ltpa)qp. It can be gainedthat after setting the end plane mirror, the increased optical effi-ciency of PTC system can be expressed as

gioe ¼ðLpta þ LtpaÞqp

L� Loel;av¼

2fqp tan h

L� ðw2 þ 48f 2Þ tan h=48fð20Þ

It is similar to the previous analyses of daily and yearly opticalend loss ratio, the average increased optical efficiency from tsr to tss

(daily increased optical efficiency) can be expressed as

gioe;d ¼2fqp

R tss

tsrIb sin hdtSR tss

tsrIb L� w2þ48f 2

48f tan h� �

cos hdtS

ð21Þ

and the yearly increased optical efficiency can be expressed as

gioe;y ¼2fqp

P365N¼1

R tss

tsrIb sin hdtSP365

N¼1

R tss

tsrIb L� w2þ48f 2

48f tan h� �

cos hdtS

ð22Þ

Only the situation that the rim angle H of parabolic trough690� is analyzed above. For the situation of H > 90�, the maxheight of end plane mirror shall equal to the depth of trough (orbe larger than the depth of trough). When the height of plane mir-ror equals to the depth of trough, Eq. (16) will be expressed as

z ¼ w2

16fð23Þ

It can be gained that for PTC-HNSA with H > 90�, the increasedoptical efficiency to its optical end loss is

gioe ¼ðw2=12f þ f Þqp tan h

L� ðw2=48f þ f Þ tan hð24Þ

3.2. Calculation results and analysis

Obviously, the increased optical efficiency is the function of L,w, f, N, u, qp and tS. To better understand the change rule of in-creased optical efficiency of PTC system with the relevant parame-ters after setting an end plane mirror, this section will carry out therelated calculations and analyses based on the theoretical modelabove. The related theoretical results are shown in Figs. 11–15.For the convenience of comparison with the calculation results inSection 2.2, the parameter values in Table 1 are also used for thefollowing calculation (except otherwise specified, such as Figs. 12and 15), and takes qp = 0.85.

Fig. 11 shows the variation of increased optical efficiency in theregion of latitude 35� with N and tS, which indicates that the in-creased optical efficiency is also in the shape of ‘‘saddle’’ and largein winter and small in summer, the maximum value generally ap-pears at midday. In a certain period of time, usually during summerand fall, the increased optical efficiency appears negative value inthe early morning and late afternoon. Fig. 12 shows the variation

Fig. 10. Schematic diagram for compensation of optical end loss by setting an end plane mirror.

5 10 15 20 25 30 35 400

20

40

60

80

100

120

140

160

180

200

η ioe

,d (%

)

L (m)

ϕ = 0o

ϕ = 10o

ϕ = 20o

ϕ = 30o

ϕ = 40o

ϕ = 50o

N = 354

Fig. 12. The variation of gioe,d with L.

30

40

50

60

ioe,

d (%

)

ϕ = 0o ϕ = 5o ϕ = 10o

ϕ = 15o ϕ = 20o ϕ = 25o

ϕ = 30o ϕ = 35o ϕ = 40o

ϕ = 45o ϕ = 50o

C. Xu et al. / Applied Energy 115 (2014) 128–139 133

of increased optical efficiency with the length of trough. Obviously,the increased optical efficiency decreases with the increase of L,and it varies significantly especially in the range of L < 15 m, andvaries little in the range of L > 15. Fig. 13 shows the variation ofgioe,d with N, which is a negative for several days or even severalmonths (generally in summer and fall) throughout the year inthe region of the latitude less than about 33�. Figs. 12 and 13 showthat the increased optical efficiency increases with the increase oflatitude, and is large in winter and small in summer. Especially for5 m long PTC system, its compensation effect is quite obvious inthe regions of above latitude 30�. And just the same as Section2.2, here, we use the typical hourly data of solar beam radiationat different latitudes in 17 Chinese cities (see Table 2) to calculatedaily and yearly increased optical efficiency, and compare themwith theoretical results, as shown in Figs. 14 and 15. The resultsshow that the measurement results are substantially agreed withthe theoretical results. In addition, the yearly increased optical effi-ciency almost changes in a linear with latitude, and the fit result oftheoretical results in Fig. 15 are

gioe;y ¼ 0:498uþ 0:00277u2; for L ¼ 5 m; 0� < u < 55�

ð25aÞ

gioe;y ¼ 0:721u� 4:306; for L ¼ 5 m; u > 25� ð25bÞ

50 100 150 200250

300350

-505

10

15

20

25

30

35

40

45

810

1214

16

η ioe

(%)

t S (ho

urs)

N (day)

Fig. 11. The variation of gioe with N and tS (u = 35�).

0 50 100 150 200 250 300 350-10

0

10

20η

N (day)

Fig. 13. The variation of gioe,d with N in different latitudes.

gioe;y ¼ 0:247uþ 0:00086u2; for L ¼ 10 m; 0� < u < 55� ð25cÞ

gioe;y ¼ 0:316u� 1:319; for L ¼ 10 m; u > 25� ð25dÞ

3.3. Discussion

As mentioned above (see Figs. 11, 13 and 14) that the increasedoptical efficiency occurs negative value sometimes, and it is be-cause that the incident angle h not only can change degree, but alsodirection, i.e., the incident angle (incident ray) sometimes is to the

0 50 100 150 200 250 300 350-10

0

10

20

30

40

50

60

70

80η i

oe,d

(%)

N (day)

MR TR (Guangzhou) MR TR (Beijing) MR TR (Manzhouli)

Fig. 14. The variation of gioe,d with N. Where, ‘‘MR’’ and ‘‘TR’’ stands for ‘‘measure-ment results’’ and ‘‘theoretical results’’, respectively.

6 7 8 9 10 11 12 13 14 15 16 17 1802468

101214161820

θ o

tS (hours)

To the south of normal To the north of normal

Fig. 16. Change of degree and direction of incidence angle (N = 100, u = 25�).

134 C. Xu et al. / Applied Energy 115 (2014) 128–139

south of normal and sometimes to the north side of normal. Asshown in Fig. 16, in the period of about 7 > tS > 17, the incident an-gle is to the south of normal, while in the periods of about tS < 7and tS > 17, the incident angle is to the north of normal. Whenthe incident angle is to the north of normal, the end plane mirrorcannot compensate the end loss, but shelters part light to the par-abolic trough and causes end shade loss effect. The optical endshade loss ratio can be expressed as

goesl ¼ 2f tan h=L ð26Þ

If let the incident angle is negative at the north side of normal,then the increased optical efficiency is negative. Fig. 17 shows thecomparison of goel, goesl and gioe. Evidently, the duration when endshade loss effect occurs is relatively short in the regions of high lat-itude, and it generally occurs in the early morning and late after-noon, during which time, the intensity of solar radiation isrelatively low. Therefore, the end shade loss effect has little effecton the total energy collected all day long, or is even negligible.

Because in low latitudes, the yearly increased optical efficiencyis usually relatively small; on the equator, gioe,y = 0. Therefore, thiscompensation method is not applicable in regions with low lati-tude, but has obvious compensation effect in regions with latitudeover 25� (especially over 30�). To be able to use this compensationmethod in the regions of lower latitudes, the end plane mirror canbe designed with a movable structure, which can be appropriatelytilted backwards during the days when end shade loss effect oc-curred. In addition, in small-scale applications, for example, whenusing PTC systems to drive absorption chiller system, in order to

0 5 10 15 20 25 30 35 40 45 50 550

5

10

15

20

25

30

35

η ioe,

y (%

)

ϕ ( o )

TR MR (L = 5m) TR MR (L = 10m)

Fig. 15. The variation of gioe,y with u. Where, TR and MR stands for theoreticalresults and measurement results, respectively.

obtain a higher compensation effect, a PTC may be divided intotwo or more shorter PTC units, each with an end plane mirror in-stalled at the end. For example, a 10 m PTC is divided into two5 m PTC units; a 15 m PTC is divided into two 7.5 m PTC units, orthree 5 m PTC units. For a long trough collector, it having a not veryhigh increased efficiency though, can obtain as much increasedamount of energy as a short collector. In the region of high latitude,it needs sometimes to tilt PTC system to reduce the cosine loss. Inthe case of small inclination, installation of an end plane mirror canalso be used for compensation.

4. Test experiments

4.1. Experimental setup

The purpose of this experiment is to verify the feasibility of thiscompensation method. Therefore, the increased thermal efficiencyof PTC-HNSA after an end plane mirror is set shall be measured anda comparison of experimental results (increased thermal effi-ciency) and theoretical results (increased optical efficiency) willalso be made.

The experimental setup mainly included concentrating system,sun tracking system, pipeline system and measurement system.Fig. 18 shows the schematic diagrams and photos of experimentalsetup.

The concentrating system mainly consists of parabolic troughreflector and its end plane mirror. Some parameters of the concen-trating system are listed in Table 3. The tracking system adoptsclosed-loop tracking [19] with a tracking precision to about 0.1 de-gree. The pipeline system includes cavity absorber tube, work fluidcirculating pipe, pump, oil tank and valves. The measurement sys-tem consists of thermocouple at both sides of cavity absorber tube,

6 7 8 9 10 11 12 13 14 15 16 17 180

1

2

3

4

5

6

7

Rat

io(%

)

tS (hours)

ηoel

ηoesl

ηioe

Fig. 17. Comparison of goel, goesl and gioe (N = 100, u = 25�).

Table 3Parameters of experimental concentrating system.

Parameters Value

Length of parabolic trough L (m) 5Aperture width of parabolic trough w (m) 2Focal distance of parabolic trough f (m) 1.2Width of end plane mirror (m) 2Reflectivity of end plane mirror qp 0.85

C. Xu et al. / Applied Energy 115 (2014) 128–139 135

solar beam radiation meter, wind velocity sensor, flow meter anddata acquisition system.

4.2. Experimental principle

The best way to finish this test experiment is to adopt twotrough concentrators with almost the same parameters (includinglength, width, focal distance and reflectivity of parabolic trough,and the absorptivity of heat absorber tube, etc.). If there is onlyone PTC, and to realize the experimental purpose, the method ofcovering and uncovering the plane mirror in turns can be adopted,i.e., during experiment, firstly, test relative parameters under thecondition of covering the end plane mirror (equivalent to absenceof end plane mirror), a while later, uncover the plane mirror, andtest the same parameters, and a while later, cover the plane mirroragain to test relative data, and repeat such steps continuously (seeFig. 19). It is better to make the experiment under the conditions ofsunny, cloudless, windless or gentle wind.

Let the fluid’s specific heat capacity at constant pressure as cp

(J/kg/�C), and its mass flow rate as lm (kg/s), when uncovering theend plane mirror, the gained heat (heat transfer rate) and instanta-neous thermal efficiency of PTC at any time are respectively:

Q ¼ lmcPðTo � TiÞ ð27Þ

gt;ins ¼Q

IbwL cos h¼ lmcPðTo � TiÞ

IbwL cos hð28Þ

where Ti and To are respectively the temperatures of heat transferoil at the inlet and outlet of heat absorber tube. Only the solar beamradiation is considered and the diffuse radiation is neglected (sameas below).

In the same way, when covering the plane mirror, the gainedheat (heat transfer rate) and instantaneous thermal efficiency ofPTC at any time are respectively:

Q 0 ¼ lmcPðT 0o � T 0iÞ ð29Þ

Beam radiation meter

Wind velocity sensor

Data acquisition system

Thermocouple

Oil tank

Valve

Pump Flow meter

End plane mirror

Parabolic troughreflector

M

To

Cavity absorber tube

Thermocouple

Ti

Outer shield

Working fluid

Insulation

Aluminium tube

Fins

Selecttive absorbing coatinAperture

Fig. 18. Top: schematic diagram (left) and photo (right) of experimental setup; bottom:

g0t;ins ¼Q 0

I0bwL cos h¼ lmcPðT 0o � T 0iÞ

I0bwL cos h0ð30Þ

where T 0i and T 0o are respectively the temperatures of heat transferoil at the inlet and outlet of heat absorber tube.

It can be gained that the increased thermal collection efficiency(increased thermal efficiency) of PTC set with end plane mirror canbe expressed as:

gite ¼gt;ins � g0t;ins

g0t;ins

¼ ðTo � TiÞI0b cos h0

ðT 0o � T 0iÞIb cos h� 1 ð31Þ

If the interval time for shifting covering and uncovering planemirror is relatively short, the variation of incident angle is smalland can be deemed as cosh0 = cosh, thus, Eq. (31) can be simplifiedas:

gite ¼ðTo � TiÞI0bðT 0o � T 0iÞIb

� 1 ð32Þ

Under the sunny weather condition, if neglect the influence ofslim change of solar beam radiation, i.e., let Ib ¼ I0b, then the aboveequation can be further simplified as

gite ¼To � Ti

T 0o � T 0i� 1 ð33Þ

The above equation can also be combined with Eqs. (27) and(29) and then get

g

the cross section schematic diagram (left) and photo (right) of cavity absorber tube.

Fig. 19. Left: uncover end plane mirror, and right: cover end plane mirror.

11.4 11.5 11.6 11.7 11.8 11.9 12.0 12.1 12.23.0

3.5

4.0

4.5

5.0

5.5

0.0

0.2

0.4

0.6

0.8

1.0

0

1

2

3

4

5

6

Tem

pera

ture

diff

eren

ce ( o C

)

Solar time tS (hours)

Bea

m ra

diat

ion

I b (k

W/m

2 )

Win

d ve

loci

ty v

w (m

/s)

To − Ti To' − Ti ' Ib vw

Fig. 20. Changes of inlet and outlet temperature difference within 5 min whenshifting covering and uncovering the end plane mirror.

136 C. Xu et al. / Applied Energy 115 (2014) 128–139

gite ¼Q � Q 0

Q 0¼ To � Ti

T 0o � T 0i� 1 ð34Þ

As the rated power of pump is constant, the mass flow rate offluid lm and the specific heat capacity at constant pressure cp

can be deemed as constant, therefore, only such parameters asthe wind velocity, solar beam radiation, inlet and outlet tempera-ture of fluid need to be measured during experiments.

4.3. Experimental results

As mentioned above, when measuring the increased thermalefficiency by shifting covering and uncovering end plane mirror,the problem of shifting interval time will be involved. The experi-ments show that after covering end plane mirror, within 2–3 min,the temperature difference T 0o � T 0i decreases rapidly and thenkeeps almost constant; after uncovering the plane mirror, it is alsowithin 2–3 min, the temperature difference To � Ti increases rap-idly and then keeps almost constant (see Fig. 20); therefore, whenmake experiment under a sunny weather condition, it is better tochoose the shifting interval time as about 5 min.

We conducted the experiments in Kunming (102�430E and25�020N), China respectively on January 14, 2012 (N = 14) and Feb-ruary 26, 2012 (N = 57). Table 4 gives the experimental results onJan. 14, 2012 (N = 14). It is obviously that DTn and DT 0n in Table 4is not gained at the same time. As the interval time for measuringDTn and DT 0n is relatively short (about 5 min), they can be deemedas being measured at the same time; the variation of incident angleis small, then the Eq. (32) can be used to calculate the increasedthermal efficiency.

To ensure the calculation results be more accurate, we adoptedaverage method to calculate the increased thermal efficiency gite,and the calculation formulae are as follows:

gite;n ¼I0b;n

2DT 0n

DTn�1

Ib;n�1þ DTnþ1

Ib;nþ1

� �� 1 ð35Þ

gite;n ¼DTn

2Ib;n

I0b;n�1

DT 0n�1

þI0b;nþ1

DT 0nþ1

!� 1 ð36Þ

The Eqs. (35) and (36) were also used to calculate gite in the exper-iment made on February 26, 2012. The experimental results areshown in Fig. 21. The figure also shows the wind speed vw, solarbeam radiation Ib, temperature rise DT (outlet and inlet tempera-ture difference), theoretical value of optical end loss ratio goel, the-oretical value of increased optical efficiency gioe and experimentalerrors gerror. Where, gerror is calculated by the following equation:

gerror ¼gite � gioe

gioeð37Þ

4.4. Analysis and discussion

Fig. 21 shows that the experimental error is less than 0.25. Inthe experiments, the temperature of working fluid was around100 �C. The distribution of experimental data show that the exper-imental results are agreed with the theoretical results, and it isindicated that this compensation method is feasible. The sourcesof the experimental errors come from many aspects, such asweather conditions (such as beam radiation and wind speed), sen-sitivity of measuring instruments, precision of concentrating sys-tem and the stability of supporting structure [20,21]. The mostimportant factor is the weather conditions. Fig. 21 shows that Jan-uary 14 was a cloudy day, the value of solar beam radiation duringthe experiment changed a lot, so did the temperature rise DT ofworking fluid through heat absorber tube. February 26 was a sunnyday, and solar beam radiation changed little during the experi-ment. During the experiment dated January 14, we chose onlythe data during the time periods when small changes occurred tosolar beam radiation (tS = 11.1–12.05 h and 13.25–13.8 h) to calcu-late increased thermal efficiency gite. In other time periods, due tothe great change in solar beam radiation value, the change in tem-perature rise of working fluid was very large and very complicated.Fig. 22 shows the comparison between the measured temperaturerise DT of working fluid and change in solar beam radiation Ib in acloudy day and the time interval of adjacent data was 1 min. Thefigure shows that the change in temperature rise DT is not syn-chronized with that in solar beam radiation Ib, and DT is generally1–3 min later than Ib, which may be the result of complexity ofheat transfer process, sensitivity of data acquisition recording,and other factors. Thus, such situation is inevitable during the

Table 4Experimental results (January 14, 2012. N = 14).

n tS (h) Ib (W/m2) Uncover end plane mirror Cover end plane mirror gite,n

DTn = (To � Ti)n (�C) DT 0n ¼ ðT0o � T 0iÞn (�C)

1 11.11 831 4.7 – –2 11.21 935 – 2.7 0.6753 11.28 945 3.8 – 0.4864 11.38 864 – 2.2 0.5265 11.48 853 3.2 – 0.6076 11.65 882 – 1.9 0.6647 11.78 906 3.1 – 0.5478 11.90 881 – 2 –9 13.21 957 – 1.8 –10 13.38 961 2.9 – 0.58711 13.53 989 – 1.9 0.56012 13.65 907 2.7 – 0.57113 13.73 963 – 1.8 –

01234567

0.0

0.2

0.4

0.6

0.8

1.0

1.20

1

2

3

4

5

6

0

10

20

30

40

50

60

70

9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5-0.3

-0.2

-0.1

0.0

0.1

0.2

Tem

pera

ture

diff

eren

ce (o C

)

To− Ti , To' − Ti' (N=14); To− Ti , To' − Ti' (N=57)

Beam

radi

atio

n I b

(kW

/m2 )

N=14N=57

Win

d ve

loci

ty v

w (m

/s)

N=14 N=57

Rat

io (%

)

ηoel, ηioe, ηite (N = 14); ηoel, ηioe, ηite (N = 57)

ηer

ror

Solar time tS (hours)

N = 14 N = 57

Fig. 21. The experimental results of increased thermal efficiency gite.

C. Xu et al. / Applied Energy 115 (2014) 128–139 137

11.6 11.8 12.0 12.2 12.4 12.6 12.8 13.0-1

0

1

2

3

4

5 T Ib

Solar time tS (hours)

Tem

pera

ture

diff

eren

ce Δ

T(o C

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Sola

r bea

m ra

diat

iom

I b (k

W/m

2 )

Fig. 22. Comparison between the temperature difference DT of working fluid andthe solar beam radiation Ib in a cloudy day.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-1

0

1

2

3

4

5 Data point Linear fit

Tem

pera

ture

diff

eren

ce Δ

T (o C

)

Solar beam radiation Ib (kW/m2)

Fig. 23. Temperature difference DT vs. solar beam radiation Ib.

138 C. Xu et al. / Applied Energy 115 (2014) 128–139

calculation: at the same solar beam radiation value, temperaturedifference values may be large or small, or even have a big differ-ence, as shown in Fig. 23. However, the shapes of changes in DTand Ib are substantially the same and substantially in a linear var-iation. The linear fitting results of Fig. 23 is

DT ¼ 5:64Ib � 0:69 ð38Þ

According to this fit results, when Ib < 122 W/m2, DT < 0. It is be-cause that the heat loss of absorber tube is greater than its availableenergy. As we used triangular cavity absorber with non-transparentcover plate (see Fig. 18) in the experiments, and this type of absor-ber tube has a simple structure, relatively low cost, but high heatloss and relatively low collection efficiency. There are many factorsthat influence the heat loss of cavity absorber tube, the theoreticalmodel of the heat loss is very complicated (especially in the cases ofgreat change in solar beam radiation) and need to conduct manytesting experiments [22]. Since the main purpose of this work isto conduct optical analysis and discussion of the end loss effect ofPTC system and its compensation method, a more detailed on theheat loss analysis is not given here due to the limited of the lengthof article. Thus, in the deal with of experimental data dated January14, only the data during the time periods when small changes oc-curred to solar beam radiation value were used.

In a word, the parabolic trough solar collector has the most ma-ture technology, it not only applied in concentrated solar power inabout 400 �C, but also can be used in industrial process heat andother fields around 100–250 �C. For places and factories with lim-ited areas, roof trough heat collector can be built. As the roof width

is always relatively small, it is proper to adopt end plane mirror tocompensate the optical end loss effect to improve the collectionefficiency, especially in regions with relatively high latitude, thelength of trough collectors are relatively short, and the small rimangle of parabolic trough (especially trough system with cavity ab-sorber tubes was used, and because of its unique structure, the rimangle of the parabolic trough is relatively small, about 45�). Thiscompensation method is also applicable for other types of troughconcentrators, such as compound parabolic solar concentrators[1,23,24], and even can be applied to trough concentrating photo-voltaic/thermal systems [25].

5. Conclusions

The end loss effect decreases with the increase of trough lengthL, and varies significantly in the range of L < 15 m (for PTC withw = 3 m and f = 1.2 m). The yearly optical end loss ratio changes lit-tle in the range of about u < 20� and significantly in the range ofabout u > 20�. For daily and yearly optical end loss ratio, the mea-surement results are substantially agreed with the theoreticalresults.

As the end plane mirror can not only reflect the light reflectedout from the trough’s end by parabolic trough to the absorber tube,but also can reflect the light directly to the mirror to the parabolictrough and the trough will reflect the light to the absorber tube,therefore, the available increased energy of PTC system is largerthan the end loss.

When the incident rays are to the north of normal, as the endplane mirror shelters partial sun light to the parabolic troughand causes end shade loss effect, the increased collection efficiencyis negative. The end shade loss effect decreases with the increasingof latitude.

The increased collection efficiency decreases with the increaseof trough length L, and varies significantly in the range ofL < 15 m (for PTC with w = 3 m and f = 1.2 m), therefore, this com-pensation method is specially applicable for short troughcollectors.

The yearly increased optical efficiency changes almost in a lin-ear with latitude. The calculation shown that, for daily and yearlyincreased optical efficiency, the measurement results are also sub-stantially agreed with the theoretical results.

When u > 33�, gioe,y > 0 all over the year. According to the vari-ation of daily and yearly increased optical efficiency with latitude,this compensation method is not applicable for regions with lowlatitude, especially regions around the equator; and it is applicablefor regions with the latitude over 25�, especially regions with thelatitude over 30�.

In the operation temperature range of about 100 �C, the exper-imental results (increased thermal efficiency) are substantiallyagreed with the theoretical results (increased optical efficiency).

Furthermore, the increased collection efficiency is in direct ratioto qp, so the end plane mirror shall be made from materials withrelatively high reflectivity.

Acknowledgements

The present study was supported by National Natural ScienceFoundation, China (Grant No.: U1137605 and 51106134), and theProgram of Changjiang Scholars and Innovative Research Teamin Ministry of Education, China (Grant No.: IRT0979).

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