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Performance simulation of a parabolic trough solar collector

Weidong Huang a,, Peng Hu b, Zeshao Chen b

a Department of Earth and Space Science, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui 230026, Chinab Department of Thermal Science and Energy Engineering, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui 230026, China

Received 11 March 2011; received in revised form 8 November 2011; accepted 30 November 2011Available online 23 December 2011

Communicated by: Associate Editor Bibek Bandyopadhyay

Abstract

A new analytical model for optical performance and a modified integration algorithm are proposed and applied to simulate the per-formance of a parabolic trough solar collector with vacuum tube receiver. The analytical equation for optical efficiency of each point atreflector is derived first, then the optical efficiency of the system is simulated by numerical integration algorithm. The cosine factor, recei-ver efficiency, heat loss and efficiency of conversion of solar energy into net heat energy at any time can be calculated with the program.The annual average efficiency is also simulated considering discard loss. The effects of optical error, tracking error, position error frominstallation of receiver, optical properties of reflector, transmittance and absorptivity of vacuum tube receiver on efficiencies of the troughsystem are simulated and analyzed as well as optical parameter. 2011 Elsevier Ltd. All rights reserved.

Keywords: Optical simulation; Parabolic trough collector; Optical efficiency; Photothermal conversion efficiency

1. Introduction

The optical efficiency is defined as the ratio of the energyabsorbed by receiver to the incidence solar energy in solarenergy utilization. It is one of key parameters in opticaldesign of concentrated solar energy system. The opticalefficiency of concentrated solar energy system is affectedby the absorptivity of receiver, the transmittance of glassenvelope of vacuum tube receiver and the reflectivity ofmirror as well as optical parameter and optical error.

In order to calculate the optical efficiency, the energyflux distribution on the receiver is usually calculated first,the total absorbed energy in a receiver is calculated by inte-gration, the optical efficiency is obtained as the ratio of theabsorbed energy to the incidence energy. For the calcula-tion of energy flux distribution on the receiver, three meth-ods are often applied: the cone optics method (Bendt and

Rabl, 1981), ray tracing method (Daly, 1979; Jiang et al.,2010) and semifinite integration formulation (Jeter, 1986;Zhao et al., 1994). The cone optics method is base on thefact that the incidence ray from sun to a point in mirrorand the reflected ray from the point at mirror to the recei-ver is also an optical cone. The flux of any point at thereceiver is obtained by integrating solar ray from the mir-ror. The ray tracing method needs to trace a large numberof rays from any point of mirror. Both of the methods con-sume great computer resources. The semifinite integration

formulation has concise physical concept, but has compli-cated formulation and need many computation resources.

In order to optimize the receiver geometry, Bennett(2008)applied the following formula to calculate the opti-cal efficiencygoof each point at parabolic trough reflectors:

goaerf x

rffiffiffi

8p

1

here a is the product of mirror reflectivity and receiverabsorbance, x is the angular width of the heat collection

0038-092X/$ - see front matter 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.solener.2011.11.018

Corresponding author. Tel.: +86 551 3606631; fax: +86 551 3607386.E-mail address:[email protected](W. Huang).

www.elsevier.com/locate/solener

Available online at www.sciencedirect.com

Solar Energy 86 (2012) 746755
http://dx.doi.org/10.1016/j.solener.2011.11.018mailto:[email protected]://dx.doi.org/10.1016/j.solener.2011.11.018http://dx.doi.org/10.1016/j.solener.2011.11.018mailto:[email protected]://dx.doi.org/10.1016/j.solener.2011.11.018

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element from a point of mirror, r is the Gaussian function

parameter of the reflected ray when the brightness of sun isregarded as Gaussian distribution as well as optical error.It is rather fast to calculate the optical efficiency, however,it is different with the actual trough system because severalsimplifications are applied, such as, the energy distributionof reflected ray is different with the assumed Gaussian dis-tribution (Nicolas, 1987), the reflectivity and absorptivity isnot constant but related to the incidence angle(Jeter, 1987;Grena, 2010).

In this paper, we first calculate optical efficiency of eachpoint at parabolic solar trough reflector, and then integratethem to obtain the optical efficiency of the whole concen-trated solar trough system. We further consider the optical

error, the tracing error and displacement error, the inci-

dence angle effect to the absorptivity of receiver, transmit-tance of glass envelope and the reflectivity of the mirror,apply the actual sun shape data to simulate the energy dis-tribution of solar radiation. The abnormal incidence, theshadow effect of the receiver and the end loss are also con-sidered to simulate the actual system. A quick algorithm isdeveloped specially for the integration computation. So itis rather quick to compute the discard energy loss, thecosine factor, the optical efficiency, the receiver efficiencyand the efficiency of conversion of solar energy into netheat energy at a moment or a time span. We apply it tosimulate the effects of main optical defects and propertiesof materials in parabolic trough collector.

Nomenclature

a, b and c the parameter for calculating absorptivity,transmittance or reflectivity

a0, a1, a2, a3, b0, b1 parameter for calculating the heat

loss of receiverBeff(h) the energy distribution function of reflected rayin radial direction (W/m2/rad)

Blinear(h\) the linear brightness distribution function attransverse direction (W/m2/rad)

DNI Direct Normal Incidence (W/m2)DNId the DNI radiation at the time which outputs net

heat (W/m2)E the total DNI energy in a year (J/m2)f0 the focal length (m)fP distance form the reflection point to the focus

point (m)h the solar altitude (rad)

Iin is the incidence solar energy (W/m2)IP the absorbed energy of reflected solar from

point P (W/m2)Is the solar irradiance on the outer surface of

Earths atmosphere (W/m2)K(d) absorptivity, transmittance or reflectivity calcu-

lated from incidence angleL the length of the parabolic mirror (m)m the air massnx the transverse section of the normal vector at the

reflection pointp the atmospheric transparency

qloss the heat loss of receiver (W/m

2

)qnet the net energy power in any time (W/m2)

Qnet net energy obtained in a year (J/m2)

r one coordinate in cylindrical coordinate system(m)

r0 the radius of the tube receiver (m)R the radius of envelope for vacuum tube receiver

(m)

s the whole surface of reflection mirror (W/m2)S the projected area of mirror under Direct Nor-

mal Incidence (W/m2)

t time (day)T temperature of receiver (K)w the half width of the trough mirror (m)a absorptivity of vacuum receiverb the rim angle to the focus point at point P (rad)c azimuth of sun (rad)d the incidence angle (rad)gc cosine factorgd the efficiency related to the discard lossgh receiver efficiencygo the optical efficiency of parabolic trough solar

collectorgP the optical efficiency at point P

gt the annual average efficiencyh radial angular displacement or angular displace-

ment in transverse direction (rad)h0 the maximum angle of ray to the receiver (rad)h\ angular displacement in transverse direction

(rad)hjj angular displacement in longitudinal direction

(rad)h0 the variable in convolution calculationk parameter for calculating optical errorq is the reflectivity of mirror at point Proptic the total optical error (rad)

rtracking tracking error (rad)r\ optical error in transverse direction (rad)rjj optical error in longitudinal direction (rad)s and are the transmittance of glass envelope andu one coordinate in cylindrical coordinate system

(rad)

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2. Computation method

2.1. Fundamental of computation

As shown inFig. 1, a reflected ray from a point P at mir-ror is absorbed by receiver at point Q, assumed point P is

the origin of the coordinates,zaxis is the reflected ray fromP of center ray of the sun. At cylindrical coordinate, thecoordinate of the point Q is (r, u,fP), the absorbed energyof reflected solar from point P can be calculated asfollowing:

IpZ Z

s

qsaBeffhduhdh 2

where h = atan(r/fP),q is the reflectivity of mirror at pointP, s and a are the transmittance of glass envelope andabsorptivity of vacuum receiver which are all related tothe incidence angle, s represents the integration to thewhole surface of the receiver, Beff(h) is the energy distribu-

tion function of reflected ray from point Q. So the opticalefficiency gPat point P is calculated as following:

gpIp=Iin 3where Iin is the incidence solar energy. The average opticalefficiency go of the whole solar trough system is calculatedas following:

goR R

sgPdS

S 4

where S represents the projected area of mirror underDirect Normal Incidence.

We apply Eqs.(1)(3)to calculate the optical efficiencyof solar trough system with vacuum tube receiver.

2.2. Energy distribution function of solar ray from sun and

reflected from mirror

Solar brightness data is usually reported as radial distri-bution Bradial(h) in W/(m

2 sr), h being measured from the

center ray of the solar disk. Here we use polynomial func-tion to simulate the radial brightness function of sun diskand part of circumsolar region (Neumann et al., 2002) as

shown in Fig. 2 and apply exponent decreasing functionofBuie et al. (2003)to simulate other circumsolar region.Detailed formula is given inAppendix A.

For line focus systems, it is convenient to transform theradial distribution Bradial(h) to a linear one (Bendt andRabi, 1979) according to

Blinearh? Z dhjjBradialh; h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih2?h2jj

q 5

\and || in subscript represent longitudinal and transversedirection. In the remainder of this paper, only the linearbrightness function (in W/m2 rad) is considered for troughconcentrator.

The bright distribution from a point reflected is a convo-lution of Gaussians with sun brightness function when theoptical error of concentrated mirror is approximatelydescribed by a Gaussian distribution G, which is as follow-ing (Bendt and Rabi, 1979):

Beffh? 1roptic

ffiffiffiffiffiffi2p

pZ dh

0 exp h02

2r2optic

!Blinearh?h0 6

where roptic is the total optical error, it can be calculate asfollowing:

r2optic

4r2contour?

r2specular?

k 4r2contourjj

r2specularjj r2trackingr2displacementr2?kr2jj 7

k is related to the position of reflection at mirror which iscalculated as following:

kn2x tan2 h 8nx is the transverse section of the normal vector at thereflection point. When we study the relationship betweenoptical efficiency and tracking error or displacement errorand considering optical error simultaneously, we shouldcalculate the total optical error by deduct their error fromEq. (6). For example, we calculate the total optic error with

following formula when the displacement error is studied:

Fig. 1. Calculation of the absorbed energy by receiver from a point of a

mirror.

Fig. 2. Radial distribution of sun brightness.

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r2optic4r2contour?r2specular?k 4r2contourjjr2specularjj

r2tracking 9

2.3. Optical efficiency of solar trough system

Fig. 3is the transverse cross section of parabolic troughsolar collector with tube receiver when the incidence ray isperpendicular to the parabolic mirror. The top of paraboliccurve Ois the origin of the coordinate, the axis of para-bolic curve is thex axis. For a point P at mirror, when theangle between a ray to the ray from center of sun is h\, thebrightness of the reflected ray is Beff(h\)dh\, the absorbedpart is psaBeff(h\)dh\, so the optical efficiency from thepoint P of parabolic mirror is:

gPy Rh0

h0 qtaBeffh? dh?Iin

1f0tanhjj 1tan2b=2

L

10

The later item is the end loss, it gives the part of ray thatdoes not reach the receiver when the incidence ray is notnormal to the parabolic mirror. L is the length of the par-abolic mirror (assumed the receiver has the same lengthwith mirror), f0 is the focal length, b is the rim angle tothe focus point at point P. Iin is the incidence solar energyflux, h0 is the maximum angle of ray to the receiver, it iscalculated as following:

h0tanr0=fp 11wherer0is the radius of the tube receiver, fPis the distancefor point P to the focus point which is calculated as follow-ing (Duffie and Beckman, 1991):

fpf0= cos2b=2 12

when a tracking error rtracking is considered, the ray fromsun center is reflected rtracking away from the ray to the fo-cus point, then the up and low limit of integration is from(h0+ rtracking) to h0rtracking. When the incidence ray isnot normal to the parabolic mirror, the limit of integrationand the incidence angle for qsa calculation can be calcu-

lated from 3-dimensional analysis, a detailed analysis is gi-ven inAppendix B.So the optical efficiency of the whole system is calculated

from integration of each point of mirror as following:

goRwR

gPy IindyRr0

0 taIincosudy

Iinw

RwR

gPy dyRr0

0 tacosddy

w 13

where first part of the equation is the contribution of thereflected ray, and the later part of the equation is the con-tribution of the ray irradiated on the collector directly, R is

the envelope tube radius of vacuum receiver to consider theshadow of receiver in the calculation, w is the width of thetrough mirror, d is the incidence angle of the radiation onthe absorption tube which can be calculate from thegeometry.

2.4. Efficiency of parabolic trough solar collector

Efficiency of parabolic trough solar collector is the ratioof the net heat collected to the Direct Normal Incidence(DNI) solar energy. It is related to the following factors(Shaner and Duff, 1978):

a. Cosine factor (gc) : when the incidence is not normalto the reflection mirror, the energy reflected by themirror is DNI multiplied with cosine of incidenceangle, so the cosine factor is give as following (Chenand Li, 2003):

gccosh 14b. Optical efficiency (go): it is related to the end loss,

optical error, interceptance, reflectivity of mirror,transmittance of glass envelope, absorptivity of recei-ver, tracing error and displacement error as well as

optical parameter which have been described.c. Receiver efficiency (gh) : it is the ratio of the net heatto the absorbed energy by the receiver. The energyloss comes from heat radiation, convection and con-duction of receiver. The receiver efficiency is calcu-lated as following:

gh1qloss=DNI cosh go 15where the energy loss qlossis calculated according to thePatnodes equation ( Patnode, 2006) from SEGS datawhich is for vacuum tube receiver:

qlossa0a1Ta2T2 a3T3 DNI b0b1T2 16

Fig. 3. Transverse cross section of parabolic trough solar collector with

tube receiver.


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whereTis the temperature of the receiver, The parame-ter is given inTable C1ofAppendix C.

d. Discard loss: when the DNI radiation from sun israther low in early morning or later afternoon, the

absorbed energy cannot increase the temperature ofthe receiver to the demanded temperature and therebycan not supply available energy, thus, this part of thesolar energy is discarded. The efficiency related to thediscard loss gd is calculated as following:

gdR

DNIddtRDNIdt 17

where DNId represents the DNI radiation at the timewhich outputs net heat.

According to the definition, the annual average effi-ciency of parabolic trough solar collector is given as (Duffie

and Beckman, 1991):gtQnet=E 18E is the total DNI energy in a year, it is calculated(Ge, 1988) as:

EZ 365

0

DNItdt 19

We need DNI data in a specific area for calculation. con-sidered that the data is not enough for calculation in manyareas, here a sunny day model is applied for DNI calcula-tion. In sunny day model, DNI is related to the atmo-spheric transparency and solar altitude which can be

calculated as (Chen and Li, 2003):

DNIt Istpm h> 00 h 6 0 20

where h is the solar altitude, Is(t) = 1367(1 + 0.034cos(2pt/365)), is the solar irradiance on the outer surface ofEarths atmosphere (W/m2),p is the atmospheric transpar-ency which is approximately as a constant, m is the airmass, m = [1229 + (614sin(h))2]1/2614sin(h).

The net energy is calculated as following (Ge, 1988):

Qnet Z 365

0

86400qnettdt 21

here the scattering radiation is ignored, qnet is the netenergy power in any time which is calculated as following:

qnett Iint coshjj goqlosst qnett> 00 qnett 6 0

22

for northsouth axis tracking system, coshjj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1cos2h cos2c

p , where c is azimuth; for eastwest

axis tracking system, coshjj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1cos2h sin2cq

, the

altitudehand azimuthcis calculated from the time and lat-itude of the site (Chen and Li, 2003).

2.5. Incidence angle effects

The reflectivity of mirror, transmittance of glass enve-lope and absorptivity of receiver will decrease when theincidence angle increases. The optical efficiency is changedwhen the incidence angle effect is considered according the

experimental data in Jeters calculation(Jeter, 1987). Grenaconsiders the incidence angle effect to simulate the flux dis-tribution in receiver according to the Fresnel Law (Grena,2010). Here we apply experimental data fromTesfamichaeland Wackelgard (2000) to simulate the incidence angleeffects to the absorptivity of receiver with followingequation:

Kd a1b1= cos d1c 23

here d is the incidence angle, a, b and c is the parameterfrom simulation to the experimental data, a= 0.96,b= 0.057, c = 1.2.

Using Helgessons glass transmission data (Helgessonet al., 2000), and applying the above equation to simulaterelationship between transmittance and incidence angle,the parameter isa = 0.925;b = 0.2,c = 1. If the simulationresult is less than 0, then the transmittance of glass orabsorptivity of receiver is set to 0.

For reflectivity, we apply Chins data (Chin, 1978) andabove equation to simulate, the parameter for new mirroris a= 0.915, b= 0.01079, c= 0.31985; for using mirror isa= 0.875, b= 0.05103, c= 0.44747. The new mirror isthe mirror that does not used in the environment before.The using mirror is the mirror which has been used forsome times and being used before test. The test data showsthat the mirror has been using for 8 months.

2.6. Numerical method

The ray tracing is often used in present optical simula-tion, however, we need to trace millions of rays, it willspend rather long time to obtain the calculation. Du etal. (2006)spend 84 h to simulate the point focus parabolicsystem, Grena (2010) spend 300 s to get optical efficiencyof parabolic trough collector after tracing 2.3 million rays.After getting the flux distribution at receiver, we need tointegrate it to obtain the optical efficiency which will spend

more time. In the method to calculate optical efficiencydirectly developed here, the four times integration is alsoneeded, including integration of radial distribution of solarradiation to the transverse linear distribution, then throughconvolution of Gaussians for integration to the effectivedistribution of reflected ray, then the third integration ofdifferent ray from reflection point to obtain the optical effi-ciency of the reflection point, and the fourth integration ofpoint optical efficiency to the whole mirror. For presentintegration algorithms, assumed an integral needs 100times calculation of the function, four times integrationneeds about 100 million times function calculation.


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Here we introduce a quick algorithm for integration. Weuse function to calculate about 1150 points, and applypolynomial function to simulate the function for integra-tion. Then the later calculation is based on the polynomialfunction. The integration of polynomial function is also apolynomial function calculation. Not only the computation

of integration is reduced, but also all integrations are chan-ged to an algebra calculation.The optical efficiency of a point at mirror is integrated

from each ray which the brightness decrease gradually aswell as optical efficiency itself, so a small number of pointis needed for simulation with polynomial function, itgreatly decrease the computation. When the tracing errorand displacement error is really a rather small part of thetotal optical error, the Gaussian function is applied foroptical error, then the optical efficiency of point at mirroris symmetrical to the axis of the parabolic mirror, and halfsimulation is need for optical efficiency which reduces thecomputation.

By using one CPU of an i3 processor in a notebookcomputer to calculate the annual net heat efficiency of par-abolic trough solar collector, only 0.11 h is needed with thetime step of 0.024 h or 0.18 h is needed with Intel celeron2003 CPU, the numerical error is less than 0.01% on theoptical efficiency.

3. Results

We calculate a typical parabolic trough solar collectorwith vacuum tube receiver. The central line of the tubereceiver is installed at the focus line of the parabolic mirror.

The parameters for the collector are shown atTable 1.We simulate the efficiency under various parameters of

the parabolic trough solar collector with a vacuum tubereceiver. Fig. 4 shows the optical efficiency at differentincidence angle with new mirror and using mirror. Thereflectivity for new and using mirror decrease as describedwith Eq. (21) when the incidence angle increases, but thenew one has higher reflectivity than using mirror. The opticalefficiency of the trough collector decreases with the increaseof the incidence angle for both new mirror and using mirror.

The different position at parabolic mirror has differentoptical efficiency as shown inFig. 5. The reflection position

near the axis of the parabolic reflector has shorter distance

than those away from axis. When the distance of the reflec-tion point to the focus point increases, the angle at whichthey reach the receiver tube tangentially decreases, theenergy absorbed by receiver decreases. So the optical effi-ciency decreases gradually and obviously at point awayfrom the axis, and the average optical efficiency for thewhole parabolic trough solar collector decreases when thewidth of the mirror increases as shown in Fig. 6.

When the optical error increases, the reflected solarimage on the receiver diffuses, and the ray reaches the recei-ver decreases with high optical error as shown in Fig. 7.

The tracking error leads to the focus line deviated fromthe center of the receiver tube, and the reflected ray whichdoes not reach to the receiver tube increases when thetracking error increases as shown in Fig. 8. The results

show that tracking error leads to low optical efficiency.Under normal incidence, the optical efficiency decreasefrom 71% to 53% with tracking error increasing from 0to 12 m rad.

We can simulate optical efficiency, cosine factor, receiverefficiency and total net heat efficiency at any time, and

Table 1Parameter of the typical parabolic trough solar collector with vacuumtube receiver.

Parameter Data

Focus length f0 1.7 mHalf trough width w 3.0 mRadius of envelope of receiverR 0.035 mRadius of receiver tube r0 0.055 mTransverse optical error h\ 6.0 m radLongitudinal optical error h jj 6.0 m radTracing error r tracking 0 m rad

Operation temperature of receiver T 400

C

Fig. 4. The optical efficiency at different incidence angle with new mirrorand using mirror in a parabolic trough solar collector with vacuum tubereceiver, the other parameters are shown inTable 1.

Fig. 5. Optical efficiency at different point of mirror in a parabolic troughsolar collector with vacuum tube receiver, the other parameters are shown

inTable 1.


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annual average efficiency with further considering the dis-

card efficiency.Figs. 9 and 10shows the cosine factor, opti-

cal efficiency, receiver efficiency and total efficiency at anytime in the summer solstice and the Spring or AutumnalEquinox with northsouth axis tracking and easewest axistracking system. For northsouth axis tracking system, theincidence angle is rather small at early morning or laterafternoon, and the optical efficiency will be high, so more

collecting time and less discard energy is for northsouthaxis tracking system than eastwest axis tracking system.However, at the winter solstice, the incidence angle is largerfor northsouth axis tracking system than for eastwestone at most time, and the optical efficiency and total effi-ciency is lower for northsouth axis tracking system thanfor eastwest one at most time of the day as shown inFig. 11.

When the absorbed energy cannot increase the tempera-ture of the receiver to the demanded temperature, then thesolar energy is discarded. At early morning or later after-noon, the incidence angle is smaller for northsouth axistracking system than eastwest axis tracking system for

more than half year. So less solar energy is discarded,and the year average optical efficiency is larger for northsouth axis tracking system with less discard energy andhigher optical efficiency. As the receiver has the same heatloss for both tracking system, the absorbed energy is morefor northsouth axis tracking system than eastwest axistracking system, so the receiver efficiency for northsouthaxis tracking system is larger than eastwest axis trackingsystem for the same receiver. The annual average efficiencyis higher for northsouth axis tracking system thaneastwest axis tracking system as shown in Fig. 12. Forboth tracking system, the optical efficiency decreases but

the receiver efficiency increases with the increase of troughwidth, so an optimum trough width with the maximumtotal efficiency is shown in the Fig. 12. It can be seen thatthe half width of SEGS trough ( Patnode, 2006) is rathernear to the optimum trough.

The optical error and tracking error will decrease theefficiency of the solar trough collector as shown inFigs. 13 and 14as well as optical efficiency.

Fig. 6. Optical efficiency of a parabolic trough solar collector withvacuum tube receiver in different geometrical concentration, the otherparameters are shown inTable 1. GC: geometrical concentration.

Fig. 7. Optical efficiency of a parabolic trough solar collector withvacuum tube receiver in different optical error, the other parameters areshown inTable 1.

Fig. 8. Optical efficiency of a parabolic trough solar collector withvacuum tube receiver in different tracking error, the other parameters areshown inTable 1. Fig. 9. The efficiency at different time in a parabolic trough solar collector

with vacuum tube receiver under two kinds of tracking system in summersolstice, the other parameters are shown in Table 1. NS: northsouth axis

tracking system, EW: easewest axis tracking system.


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Fig. 10. The efficiency at different time in a parabolic trough solar collectorwith vacuum tube receiver under two kinds of tracking system in the springor Autumnal Equinox, the other parameters are shown in Table 1, NS:northsouth axis tracking system, EW: easewest axis tracking system.

Fig. 11. The efficiency at different time in a parabolic trough solarcollector with vacuum tube receiver under two kinds of tracking system inwinter solstice, the other parameters are shown in Table 1. NS: northsouth axis tracking system, EW: easewest axis tracking system.

Fig. 12. The year average efficiency of a parabolic trough solar collectorwith vacuum tube receiver in different trough width under two kinds oftracking system, the other parameters are shown in Table 1. Up: north

south axis tracking system, down: eastwest axis tracking system.

Fig. 13. The year average efficiency of a parabolic trough solar collectorwith vacuum tube receiver in different optical error under two kinds oftracking system, the other parameters are shown in Table 1. Up: northsouth axis tracking system, down: eastwest axis tracking system.

Fig. 14. The year average efficiency of a parabolic trough solar collectorwith vacuum tube receiver in different tracking error under two kinds oftracking system, the other parameters are shown in Table 1, Up: northsouth axis tracking system, down: eastwest axis tracking system.

Fig. 15. Comparison of test data and model prediction of intercept factor

in different incidence angles. Test data is from Riffelmann et al. (2006).


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Comparison with experimental results: experimentalvalidation to the model is rather difficult, since the effi-ciency is related to many factors and it is hard to test somany related factors precisely and simultaneously. How-ever, one of the key result of our model is the intercept fac-tor (assumedsa= 1). Here the intercept factor for different

incidence angle is calculated and compared with test data(Riffelmann et al., 2006) as shown in Fig. 15. The interceptfactor is constant up to large incidence angles and decreasegradually when the incidence angle increases in both testand model prediction. It can be seen that the predictionagrees the test result rather well.

Recent experimental solar to net heat efficiency forabout spring Equinox indicates that the solar trough collec-tor with northsouth axis tracking system has lower totalefficiency at noon than 4 h early or later (Price, 2002). Itagrees rather well with the present simulation.

4. Conclusion

In the paper, we proposed a new analytical method to cal-culate the optical efficiency of solar concentrator, it is basedon the effective light distribution from reflected point to cal-culate the optical efficiency of each point at mirror. A quicknumerical method for integration is developed for opticalefficiency simulation, it applies polynomial function to sim-ulate function, and then the integration of function can bereplaced by integration of polynomial function which is alsoa polynomial calculation. We apply the two methods todevelop a program to simulate a parabolic trough solar col-lector to obtain the cosine factor, the optical efficiency, the

receiver efficiency and total efficiency at any time as well asyearly or daily average efficiency including the discard effi-ciency. Trough collectors are today the most widely usedsolar power generating systems and are assuming a greatimportance in solar energy development strategies. Sincethe main aim in solar plant building is cost reduction, defectsand imperfections of various kinds are always present. Theseaspects have been considered in the program for an accuratesimulation which allows one to calculate the efficiencies of anactual parabolic trough solar collector. It is rather quick,and the method can be applied as following:

a. The effects to the optical efficiency and total solar tonet heat efficiency at different tracking error or opti-cal error.

b. evaluating the optical efficiency and total solar to netheat efficiency at different displacement error.

c. effects of the cluster and aging of the reflection mirroron the efficiency of the system.

d. effects of the property of receiver and mirror to theefficiency of the system.

e. optimizing the optical parameter of the solar troughsystem.

The analytical method to calculate the optical efficiency

of solar concentrator and quick numerical method for

integration can be applied to other concentrated solar sys-tem. We are developing a program with the methods forsolar tower system.

Acknowledgement

This work is partially supported by the National Natu-ral Science Foundation of China (No. 50736005) Thenumerical calculations in this paper have partly been doneon the supercomputing system in the Supercomputing Cen-ter of University of Science and Technology of China.

Appendix A. Brightness distribution of sun

Bradialh 03047222e15h30 0:35757701839e13h280:30769982398e11h26 0:15267102168e

9h24

0:48824953590e

8h22

0:10627588060e

6h20 0:16177926059e5h18 0:17405417057e4h16 0:13212195269e3h14 0:69876231477e3h12 0:25133685240e2h10 0:59138793819e2h8 0:85647334392e2h6 0:70829038680e2h4 0:55636572263e2h2 1:00000;

h 6 0:0049

k h^c h> 0:0049kexp0:9log13:5a=a:^0:3; c

2:2

log

0:52

a

a:^0:43

0:1

;

A:1

where a is the circumsolar ratio (CSR) which is defined asthe radiant flux contained within the circumsolar region ofthe sky, divided by the incident radiant flux from the directbeam and aureole. Here a= 0.05.

Appendix B. Analysis of optical efficiency at any point of

mirror at non-normal incidence

Assumed that the top of parabolic curve is the origin ofthe coordinate, the axis of parabolic curve is thex axis, thelength of the trough is thezaxis. Assumed that the trackingerror isr, that is to say that the normal ray will be reflectedaway from the focus ray with angle r; assumed that the dis-placement error at x direction is Dx, at y direction is Dy,then the coordinate of the central line of receiver for a crosssection is (f0 + Dx, Dy). For any ray with incidence angelh||, transverse angle between the incidence ray and centralline of sun ish. To facilitate the analysis, it is assumed thatthe incidence ray includes the diffusion caused by opticalerror, and the mirror has perfect face and does not increasediffusion to the solar ray.

If the tracking error is 0, then the vector of incidence rayi is (coshcos h||,sin h,cos hsin h||), when the trackingerror is r, it is equal to rotate coordinate with angle r

anticlockwise about z coordinate axis, so the vector of


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incidence ray i is ((cos h||cos hcos r+ sin hsin r),(cos hcos h||sin r+ sin hcos r), cos h||sin h) in newcoordinates.

For a point P in the mirror, normal vector n is (cos(b/2), sin (b/2), 0), so the reflected ray r:

r i2inn B:1we obtain:

rxcos hcos hcosbr sin hsinbr B:2ry cos hcos hsinbr sin hcosbr B:3then we can calculate the incidence angle at the absorbedpoint. Assumed the coordinates of P is (x0,y0,0), then theequation of the reflected ray is:

xx0=rx yy0=ry zz0=rz; B:4the equation for the receiver is:

xf0Dx2 yDy2 r2 B:5from Eqs.(4)and (5), then coordinates of the absorbed po-

sition (x1,y1, z1) can be obtained. The normal vector at theabsorbed position should be ((x1f0 Dx)/r, (y1 Dy)/r,0), then the incidence angle can be calculated:

cos d rx x1f0 Dx ry y1 Dy=r B:6so the absorptivity can be calculated at the position, thetransmittance can be obtained with the same methods.

From Eqs. (4) and (5), we can also obtain the integra-tion limit of h when the reflected ray reaches the receivertube tangentially

Appendix C

SeeTable C1.

References

Bendt, P., Rabi, A., et al., 1979. Optical Analysis and Optimization ofLine Focus Solar Collectors. Solar Energy Research Institute ReportNo. TR-34-092, p. 1115.

Bendt, P., Rabl, A., 1981. Optical analysis of point parabolic radiation.Applied Optics 20 (4), 674683.

Bennett, C.L., 2008. Optimal heat collection element shapes for parabolictrough concentrators. Journal of Solar Energy Engineering Trans-actions of the ASME 130 (2).

Buie, D., Monger, A.G., et al., 2003. Sunshape distributions for terrestrialsolar simulations. Solar energy 74 (2), 113122.

Chen, W., Li, J., 2003. Optical performance of analysis for parabolic-trough focusing collector with several tracking modes. Acta EnergiaeSolarris Sinica 24 (4), 477482.

Chin, S., 1978. Solar Reflectivity and Absorptivity Studies. Texas TechUniversity, MS.

Daly, J.C., 1979. Solar concentrator flux distribution using backwardtracing. Applied Optics 18 (15), 26962700.

Du, S., Xia, X., et al., 2006. Numerical investigation on effects of non-parallelism of solar rays on concentrating solar power. Acta EnergiaeSolaris Sinica (Chinese) 27 (4), 388395.

Duffie, J.A., Beckman, W.A., 1991. Solar Engineering of ThermalProcesses. Wiley-Interscience, New York.

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Grena, R., 2010. Optical simulation of a parabolic solar trough collector.International Journal of Sustainable Energy 29 (1), 1936.

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Jeter, S.M., 1986. The distribution of concentrated solar radiation inparaboloidal collectors. Journal of Energy Engineering 108 (1), 219225.

Jeter, S.M., 1987. Analytical determination of the optical performance ofpractical parabolic trough collectors from design data. Solar Energy 39(1), 1121.

Jiang, S., Hu, P., et al., 2010. Optical modeling for a two-stage parabolictrough concentrating photovoltaic/thermal system using spectral beamsplitting technology. Solar Energy Materials & Solar Cells 94, 16861696.

Neumann, A., Witzke, A., et al., 2002. Representative terrestrial solarbrightness profiles. Journal of Solar Energy Engineering Transac-tions of the ASME 124 (2), 198204.

Nicolas, R.O., 1987. Optical analysis of cylindrical-parabolic concentra-tors: validity limits for models of solar disk intensity. Applied Optics26 (18), 38663870.

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Price, H., 2002. Concentrating Solar Power Systems Analysis andImplications, slide 39. .

Riffelmann, K.J., Neumann, A., et al., 2006. Performance enhancement ofparabolic trough collectors by solar flux measurement in the focalregion. Solar Energy 80, 13031313.

Shaner, W.W., Duff, W.S., 1978. Solar thermal electric power systems:comparison of line-focus collectors. Solar Energy 22, 4961.

Tesfamichael, T., Wackelgard, E., 2000. Angular solar absorptance andincident angle modifier of selective absorbers for solar thermalcollectors. Solar Energy 68 (4), 335341.

Zhao, J., Ge, X., et al., 1994. Theoretical and experimental study of theradiation flux distribution in the focal plane of a parabolic troughconcentrator. Acta Energiae Solaris Sinica (Chinese) 15 (3), 262267.

Table C1Heat loss coefficient for vacuum tube receiver.

Parameter Value Std.

a0 9.463033e+00 8.463850e01a1 3.029616e01 1.454877e02a2 1.386833e03 7.305717e05a3 6.929243e

06 1.070953e

07

b0 7.649610e02 5.293835e04b1 1.128818e07 6.394787e09

http://www1.eere.energy.gov/solar/pdfs/3solar_henryprice.pdfhttp://www1.eere.energy.gov/solar/pdfs/3solar_henryprice.pdfhttp://www1.eere.energy.gov/solar/pdfs/3solar_henryprice.pdfhttp://www1.eere.energy.gov/solar/pdfs/3solar_henryprice.pdf

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