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Performance simulation of a parabolic trough solar collector
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Solar Energy 86 (2012) 746–755 www.elsevier.com/locate/solener
Performance simulation of a parabolic trough solar collector Weidong Huang a,⇑, Peng Hu b, Zeshao Chen b b
a Department of Earth and Space Science, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui 230026, China 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 2011 Available 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 performance of a parabolic trough solar collector with vacuum tube receiver. The analytical equation for optical efficiency of each point at reflector is derived first, then the optical efficiency of the system is simulated by numerical integration algorithm. The cosine factor, receiver 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 from installation of receiver, optical properties of reflector, transmittance and absorptivity of vacuum tube receiver on efficiencies of the trough system are simulated and analyzed as well as optical parameter. Ó 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 energy absorbed by receiver to the incidence solar energy in solar energy utilization. It is one of key parameters in optical design of concentrated solar energy system. The optical efficiency of concentrated solar energy system is affected by the absorptivity of receiver, the transmittance of glass envelope of vacuum tube receiver and the reflectivity of mirror as well as optical parameter and optical error. In order to calculate the optical efficiency, the energy flux distribution on the receiver is usually calculated first, the total absorbed energy in a receiver is calculated by integration, the optical efficiency is obtained as the ratio of the absorbed energy to the incidence energy. For the calculation of energy flux distribution on the receiver, three methods are often applied: the cone optics method (Bendt and ⇑ Corresponding author. Tel.: +86 551 3606631; fax: +86 551 3607386.
E-mail address: [email protected] (W. Huang). 0038-092X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2011.11.018
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 the fact that the incidence ray from sun to a point in mirror and the reflected ray from the point at mirror to the receiver is also an optical cone. The flux of any point at the receiver is obtained by integrating solar ray from the mirror. The ray tracing method needs to trace a large number of rays from any point of mirror. Both of the methods consume great computer resources. The semifinite integration formulation has concise physical concept, but has complicated formulation and need many computation resources. In order to optimize the receiver geometry, Bennett (2008) applied the following formula to calculate the optical efficiency go of each point at parabolic trough reflectors: x go ¼ a erf pffiffiffi ð1Þ r 8 here a is the product of mirror reflectivity and receiver absorbance, x is the angular width of the heat collection
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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 receiver Beff(h) the energy distribution function of reflected ray in radial direction (W/m2/rad) Blinear(h\) the linear brightness distribution function at transverse 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 Earth’s atmosphere (W/m2) K(d) absorptivity, transmittance or reflectivity calculated from incidence angle L the length of the parabolic mirror (m) m the air mass nx the transverse section of the normal vector at the reflection point p the atmospheric transparency qloss the heat loss of receiver (W/m2) 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)
element from a point of mirror, r is the Gaussian function parameter of the reflected ray when the brightness of sun is regarded 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 several simplifications are applied, such as, the energy distribution of reflected ray is different with the assumed Gaussian distribution (Nicolas, 1987), the reflectivity and absorptivity is not constant but related to the incidence angle (Jeter, 1987; Grena, 2010). In this paper, we first calculate optical efficiency of each point at parabolic solar trough reflector, and then integrate them to obtain the optical efficiency of the whole concentrated solar trough system. We further consider the optical
s S t T w a b c d gc gd gh go gP gt h h0 h\ hjj h0 k q roptic rtracking r\ rjj s u
the whole surface of reflection mirror (W/m2) the projected area of mirror under Direct Normal Incidence (W/m2) time (day) temperature of receiver (K) the half width of the trough mirror (m) absorptivity of vacuum receiver the rim angle to the focus point at point P (rad) azimuth of sun (rad) the incidence angle (rad) cosine factor the efficiency related to the discard loss receiver efficiency the optical efficiency of parabolic trough solar collector the optical efficiency at point P the annual average efficiency radial angular displacement or angular displacement in transverse direction (rad) the maximum angle of ray to the receiver (rad) angular displacement in transverse direction (rad) angular displacement in longitudinal direction (rad) the variable in convolution calculation parameter for calculating optical error is the reflectivity of mirror at point P the total optical error (rad) tracking error (rad) optical error in transverse direction (rad) optical error in longitudinal direction (rad) and are the transmittance of glass envelope and one coordinate in cylindrical coordinate system (rad)
error, the tracing error and displacement error, the incidence angle effect to the absorptivity of receiver, transmittance of glass envelope and the reflectivity of the mirror, apply the actual sun shape data to simulate the energy distribution of solar radiation. The abnormal incidence, the shadow effect of the receiver and the end loss are also considered to simulate the actual system. A quick algorithm is developed specially for the integration computation. So it is rather quick to compute the discard energy loss, the cosine factor, the optical efficiency, the receiver efficiency and the efficiency of conversion of solar energy into net heat energy at a moment or a time span. We apply it to simulate the effects of main optical defects and properties of materials in parabolic trough collector.
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2. Computation method 2.1. Fundamental of computation As shown in Fig. 1, a reflected ray from a point P at mirror is absorbed by receiver at point Q, assumed point P is the origin of the coordinates, z axis is the reflected ray from P of center ray of the sun. At cylindrical coordinate, the coordinate of the point Q is (r, u, fP), the absorbed energy of reflected solar from point P can be calculated as following: Z Z Ip ¼ qsaBeff ðhÞduhdh ð2Þ s
Fig. 2. Radial distribution of sun brightness.
where h = atan (r/fP), q is the reflectivity of mirror at point P, s and a are the transmittance of glass envelope and absorptivity of vacuum receiver which are all related to the incidence angle, s represents the integration to the whole surface of the receiver, Beff(h) is the energy distribution function of reflected ray from point Q. So the optical efficiency gP at point P is calculated as following: gp ¼ I p =I in
ð3Þ
where Iin is the incidence solar energy. The average optical efficiency go of the whole solar trough system is calculated as following: RR g dS s P go ¼ ð4Þ S where S represents the projected area of mirror under Direct Normal Incidence. We apply Eqs. (1)–(3) to calculate the optical efficiency of 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 distribution Bradial(h) in W/(m2 sr), h being measured from the
center ray of the solar disk. Here we use polynomial function to simulate the radial brightness function of sun disk and part of circumsolar region (Neumann et al., 2002) as shown in Fig. 2 and apply exponent decreasing function of Buie et al. (2003) to simulate other circumsolar region. Detailed formula is given in Appendix A. For line focus systems, it is convenient to transform the radial distribution Bradial(h) to a linear one (Bendt and Rabi, 1979) according to Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ Blinear ðh? Þ ¼ dhjj Bradial ðhÞ; h ¼ h2? þ h2jj \ and || in subscript represent longitudinal and transverse direction. In the remainder of this paper, only the linear brightness function (in W/m2 rad) is considered for trough concentrator. The bright distribution from a point reflected is a convolution of Gaussians with sun brightness function when the optical error of concentrated mirror is approximately described by a Gaussian distribution G, which is as following (Bendt and Rabi, 1979): ! Z 1 h02 0 pffiffiffiffiffiffi dh exp Beff ðh? Þ ¼ Blinear ðh? h0 Þ ð6Þ 2r2optic roptic 2p where roptic is the total optical error, it can be calculate as following: r2optic ¼ 4r2contour? þ r2specular? þ k 4r2contourjj þ r2specularjj þ r2tracking þ r2displacement ¼ r2? þ k r2jj
ð7Þ
k is related to the position of reflection at mirror which is calculated as following: k ¼ n2x tan2 h
Fig. 1. Calculation of the absorbed energy by receiver from a point of a mirror.
ð8Þ
nx is the transverse section of the normal vector at the reflection point. When we study the relationship between optical efficiency and tracking error or displacement error and considering optical error simultaneously, we should calculate the total optical error by deduct their error from Eq. (6). For example, we calculate the total optic error with following formula when the displacement error is studied:
W. Huang et al. / Solar Energy 86 (2012) 746–755
r2optic ¼ 4r2contour? þ r2specular? þ k 4r2contourjj þ r2specularjj þ r2tracking
ð9Þ
2.3. Optical efficiency of solar trough system Fig. 3 is the transverse cross section of parabolic trough solar collector with tube receiver when the incidence ray is perpendicular to the parabolic mirror. The top of parabolic curve “O” is the origin of the coordinate, the axis of parabolic curve is the x axis. For a point P at mirror, when the angle between a ray to the ray from center of sun is h\, the brightness of the reflected ray is Beff(h\)dh\, the absorbed part is psaBeff(h\)dh\, so the optical efficiency from the point P of parabolic mirror is: R h0 qtaBeff ðh? Þ dh? h0 gP ðyÞ ¼ I in f0 tanðhjj Þ ð1 þ tan2 ðb=2ÞÞ 1 ð10Þ L The later item is the end loss, it gives the part of ray that does not reach the receiver when the incidence ray is not normal to the parabolic mirror. L is the length of the parabolic mirror (assumed the receiver has the same length with mirror), f0 is the focal length, b is the rim angle to the focus point at point P. Iin is the incidence solar energy flux, h0 is the maximum angle of ray to the receiver, it is calculated as following: h0 ¼ tanðr0 =fp Þ
when a tracking error rtracking is considered, the ray from sun center is reflected rtracking away from the ray to the focus point, then the up and low limit of integration is from (h0 + rtracking) to h0–rtracking. When the incidence ray is not normal to the parabolic mirror, the limit of integration and the incidence angle for qsa calculation can be calculated from 3-dimensional analysis, a detailed analysis is given in Appendix B. So the optical efficiency of the whole system is calculated from integration of each point of mirror as following: Rr Rw g ðyÞ I in dy þ 0 0 taI in cosðuÞdy R P go ¼ I in w R r0 Rw g ðyÞ dy þ ta cosðdÞdy 0 R P ¼ ð13Þ w where first part of the equation is the contribution of the reflected ray, and the later part of the equation is the contribution of the ray irradiated on the collector directly, R is the envelope tube radius of vacuum receiver to consider the shadow of receiver in the calculation, w is the width of the trough mirror, d is the incidence angle of the radiation on the absorption tube which can be calculate from the geometry. 2.4. Efficiency of parabolic trough solar collector Efficiency of parabolic trough solar collector is the ratio of the net heat collected to the Direct Normal Incidence (DNI) solar energy. It is related to the following factors (Shaner and Duff, 1978):
ð11Þ
where r0 is the radius of the tube receiver, fP is the distance for point P to the focus point which is calculated as following (Duffie and Beckman, 1991): fp ¼ f0 = cos2 ðb=2Þ
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ð12Þ
a. Cosine factor (gc) : when the incidence is not normal to the reflection mirror, the energy reflected by the mirror is DNI multiplied with cosine of incidence angle, so the cosine factor is give as following (Chen and Li, 2003): gc ¼ cosðhÞ
ð14Þ
b. Optical efficiency (go): it is related to the end loss, optical error, interceptance, reflectivity of mirror, transmittance of glass envelope, absorptivity of receiver, 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 heat to the absorbed energy by the receiver. The energy loss comes from heat radiation, convection and conduction of receiver. The receiver efficiency is calculated as following: gh ¼ 1 qloss =ðDNI cosðhÞ go Þ
Fig. 3. Transverse cross section of parabolic trough solar collector with tube receiver.
ð15Þ
where the energy loss qloss is calculated according to the Patnode’s equation ( Patnode, 2006) from SEGS data which is for vacuum tube receiver: qloss ¼ a0 þ a1 T þ a2 T 2 þ a3 T 3 þ DNI ðb0 þ b1 T 2 Þ
ð16Þ
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2.5. Incidence angle effects where T is the temperature of the receiver, The parameter is given in Table C1 of Appendix C. d. Discard loss: when the DNI radiation from sun is rather low in early morning or later afternoon, the absorbed energy cannot increase the temperature of the receiver to the demanded temperature and thereby can not supply available energy, thus, this part of the solar energy is discarded. The efficiency related to the discard loss gd is calculated as following: R DNId dt gd ¼ R ð17Þ DNI dt
The reflectivity of mirror, transmittance of glass envelope and absorptivity of receiver will decrease when the incidence angle increases. The optical efficiency is changed when the incidence angle effect is considered according the experimental data in Jeter’s calculation (Jeter, 1987). Grena considers the incidence angle effect to simulate the flux distribution in receiver according to the Fresnel Law (Grena, 2010). Here we apply experimental data from Tesfamichael and Wackelgard (2000) to simulate the incidence angle effects to the absorptivity of receiver with following equation: c
where DNId represents the DNI radiation at the time which outputs net heat. According to the definition, the annual average efficiency of parabolic trough solar collector is given as (Duffie and Beckman, 1991): gt ¼ Qnet =E
ð18Þ
E is the total DNI energy in a year, it is calculated (Ge, 1988) as: Z 365 E¼ DNIðtÞdt ð19Þ 0
We need DNI data in a specific area for calculation. considered that the data is not enough for calculation in many areas, here a sunny day model is applied for DNI calculation. In sunny day model, DNI is related to the atmospheric transparency and solar altitude which can be calculated as (Chen and Li, 2003): DNIðtÞ ¼ I s ðtÞpm ¼0
h>0
ð20Þ
h60
where h is the solar altitude, Is(t) = 1367 (1 + 0.034 cos (2pt/365)), is the solar irradiance on the outer surface of Earth’s atmosphere (W/m2), p is the atmospheric transparency which is approximately as a constant, m is the air mass, m = [1229 + (614sin (h))2]1/2614sin (h). The net energy is calculated as following (Ge, 1988): Z 365 Qnet ¼ 86400qnet ðtÞdt ð21Þ 0
here the scattering radiation is ignored, qnet is the net energy power in any time which is calculated as following: qnet ðtÞ ¼ I in ðtÞ cosðhjj Þ go qloss ðtÞ ¼0
qnet ðtÞ > 0 qnet ðtÞ 6 0
ð22Þ
for north–south axis tracking system, cosðhjj Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 cos2 ðhÞ cos2 ðcÞ , where c is azimuth; for east–west qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi axis tracking system, cosðhjj Þ ¼ 1 cos2 ðhÞ sin2 ðcÞ, the altitude h and azimuth c is calculated from the time and latitude of the site (Chen and Li, 2003).
KðdÞ ¼ a½1 bð1= cos d 1Þ
ð23Þ
here d is the incidence angle, a, b and c is the parameter from simulation to the experimental data, a = 0.96, b = 0.057, c = 1.2. Using Helgesson’s glass transmission data (Helgesson et al., 2000), and applying the above equation to simulate relationship between transmittance and incidence angle, the parameter is a = 0.925; b = 0.2, c = 1. If the simulation result is less than 0, then the transmittance of glass or absorptivity of receiver is set to 0. For reflectivity, we apply Chin’s data (Chin, 1978) and above equation to simulate, the parameter for new mirror is a = 0.915, b = 0.01079, c = 0.31985; for using mirror is a = 0.875, b = 0.05103, c = 0.44747. The new mirror is the mirror that does not used in the environment before. The using mirror is the mirror which has been used for some times and being used before test. The test data shows that the mirror has been using for 8 months. 2.6. Numerical method The ray tracing is often used in present optical simulation, however, we need to trace millions of rays, it will spend rather long time to obtain the calculation. Du et al. (2006)spend 84 h to simulate the point focus parabolic system, Grena (2010) spend 300 s to get optical efficiency of parabolic trough collector after tracing 2.3 million rays. After getting the flux distribution at receiver, we need to integrate it to obtain the optical efficiency which will spend more time. In the method to calculate optical efficiency directly developed here, the four times integration is also needed, including integration of radial distribution of solar radiation to the transverse linear distribution, then through convolution of Gaussians for integration to the effective distribution of reflected ray, then the third integration of different ray from reflection point to obtain the optical efficiency of the reflection point, and the fourth integration of point optical efficiency to the whole mirror. For present integration algorithms, assumed an integral needs 100 times calculation of the function, four times integration needs about 100 million times function calculation.
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Here we introduce a quick algorithm for integration. We use function to calculate about 11–50 points, and apply polynomial function to simulate the function for integration. Then the later calculation is based on the polynomial function. The integration of polynomial function is also a polynomial function calculation. Not only the computation of integration is reduced, but also all integrations are changed to an algebra calculation. The optical efficiency of a point at mirror is integrated from each ray which the brightness decrease gradually as well as optical efficiency itself, so a small number of point is needed for simulation with polynomial function, it greatly decrease the computation. When the tracing error and displacement error is really a rather small part of the total optical error, the Gaussian function is applied for optical error, then the optical efficiency of point at mirror is symmetrical to the axis of the parabolic mirror, and half simulation is need for optical efficiency which reduces the computation. By using one CPU of an i3 processor in a notebook computer to calculate the annual net heat efficiency of parabolic trough solar collector, only 0.11 h is needed with the time step of 0.024 h or 0.18 h is needed with Intel celeron 2003 CPU, the numerical error is less than 0.01% on the optical efficiency. 3. Results We calculate a typical parabolic trough solar collector with vacuum tube receiver. The central line of the tube receiver is installed at the focus line of the parabolic mirror. The parameters for the collector are shown at Table 1. We simulate the efficiency under various parameters of the parabolic trough solar collector with a vacuum tube receiver. Fig. 4 shows the optical efficiency at different incidence angle with new mirror and using mirror. The reflectivity for new and using mirror decrease as described with Eq. (21) when the incidence angle increases, but the new one has higher reflectivity than using mirror. The optical efficiency of the trough collector decreases with the increase of the incidence angle for both new mirror and using mirror. The different position at parabolic mirror has different optical efficiency as shown in Fig. 5. The reflection position near the axis of the parabolic reflector has shorter distance
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Fig. 4. The optical efficiency at different incidence angle with new mirror and using mirror in a parabolic trough solar collector with vacuum tube receiver, the other parameters are shown in Table 1.
than those away from axis. When the distance of the reflection point to the focus point increases, the angle at which they reach the receiver tube tangentially decreases, the energy absorbed by receiver decreases. So the optical efficiency decreases gradually and obviously at point away from the axis, and the average optical efficiency for the whole parabolic trough solar collector decreases when the width of the mirror increases as shown in Fig. 6. When the optical error increases, the reflected solar image on the receiver diffuses, and the ray reaches the receiver decreases with high optical error as shown in Fig. 7. The tracking error leads to the focus line deviated from the center of the receiver tube, and the reflected ray which does not reach to the receiver tube increases when the tracking error increases as shown in Fig. 8. The results show that tracking error leads to low optical efficiency. Under normal incidence, the optical efficiency decrease from 71% to 53% with tracking error increasing from 0 to 12 m rad. We can simulate optical efficiency, cosine factor, receiver efficiency and total net heat efficiency at any time, and
Table 1 Parameter of the typical parabolic trough solar collector with vacuum tube receiver. Parameter
Data
Focus length f0 Half trough width w Radius of envelope of receiver R Radius of receiver tube r0 Transverse optical error h\ Longitudinal optical error hjj Tracing error rtracking Operation temperature of receiver T
1.7 m 3.0 m 0.035 m 0.055 m 6.0 m rad 6.0 m rad 0 m rad 400 °C
Fig. 5. Optical efficiency at different point of mirror in a parabolic trough solar collector with vacuum tube receiver, the other parameters are shown in Table 1.
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Fig. 6. Optical efficiency of a parabolic trough solar collector with vacuum tube receiver in different geometrical concentration, the other parameters are shown in Table 1. GC: geometrical concentration.
Fig. 7. Optical efficiency of a parabolic trough solar collector with vacuum tube receiver in different optical error, the other parameters are shown in Table 1.
Fig. 8. Optical efficiency of a parabolic trough solar collector with vacuum tube receiver in different tracking error, the other parameters are shown in Table 1.
annual average efficiency with further considering the discard efficiency. Figs. 9 and 10 shows the cosine factor, opti-
cal efficiency, receiver efficiency and total efficiency at any time in the summer solstice and the Spring or Autumnal Equinox with north–south axis tracking and ease–west axis tracking system. For north–south axis tracking system, the incidence angle is rather small at early morning or later afternoon, and the optical efficiency will be high, so more collecting time and less discard energy is for north–south axis tracking system than east–west axis tracking system. However, at the winter solstice, the incidence angle is larger for north–south axis tracking system than for east–west one at most time, and the optical efficiency and total efficiency is lower for north–south axis tracking system than for east–west one at most time of the day as shown in Fig. 11. When the absorbed energy cannot increase the temperature of the receiver to the demanded temperature, then the solar energy is discarded. At early morning or later afternoon, the incidence angle is smaller for north–south axis tracking system than east–west axis tracking system for more than half year. So less solar energy is discarded, and the year average optical efficiency is larger for north– south axis tracking system with less discard energy and higher optical efficiency. As the receiver has the same heat loss for both tracking system, the absorbed energy is more for north–south axis tracking system than east–west axis tracking system, so the receiver efficiency for north–south axis tracking system is larger than east–west axis tracking system for the same receiver. The annual average efficiency is higher for north–south axis tracking system than east–west axis tracking system as shown in Fig. 12. For both tracking system, the optical efficiency decreases but the receiver efficiency increases with the increase of trough width, so an optimum trough width with the maximum total efficiency is shown in the Fig. 12. It can be seen that the half width of SEGS trough (Patnode, 2006) is rather near to the optimum trough. The optical error and tracking error will decrease the efficiency of the solar trough collector as shown in Figs. 13 and 14 as well as optical efficiency.
Fig. 9. The efficiency at different time in a parabolic trough solar collector with vacuum tube receiver under two kinds of tracking system in summer solstice, the other parameters are shown in Table 1. NS: north–south axis tracking system, EW: ease–west axis tracking system.
W. Huang et al. / Solar Energy 86 (2012) 746–755
Fig. 10. The efficiency at different time in a parabolic trough solar collector with vacuum tube receiver under two kinds of tracking system in the spring or Autumnal Equinox, the other parameters are shown in Table 1, NS: north–south axis tracking system, EW: ease–west axis tracking system.
Fig. 11. The efficiency at different time in a parabolic trough solar collector with vacuum tube receiver under two kinds of tracking system in winter solstice, the other parameters are shown in Table 1. NS: north– south axis tracking system, EW: ease–west axis tracking system.
Fig. 12. The year average efficiency of a parabolic trough solar collector with vacuum tube receiver in different trough width under two kinds of tracking system, the other parameters are shown in Table 1. Up: north– south axis tracking system, down: east–west axis tracking system.
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Fig. 13. The year average efficiency of a parabolic trough solar collector with vacuum tube receiver in different optical error under two kinds of tracking system, the other parameters are shown in Table 1. Up: north– south axis tracking system, down: east–west axis tracking system.
Fig. 14. The year average efficiency of a parabolic trough solar collector with vacuum tube receiver in different tracking error under two kinds of tracking system, the other parameters are shown in Table 1, Up: north– south axis tracking system, down: east–west 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: experimental validation to the model is rather difficult, since the efficiency is related to many factors and it is hard to test so many related factors precisely and simultaneously. However, one of the key result of our model is the intercept factor (assumed sa = 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 intercept factor is constant up to large incidence angles and decrease gradually when the incidence angle increases in both test and model prediction. It can be seen that the prediction agrees the test result rather well. Recent experimental solar to net heat efficiency for about spring Equinox indicates that the solar trough collector with north–south axis tracking system has lower total efficiency at noon than 4 h early or later (Price, 2002). It agrees rather well with the present simulation.
integration can be applied to other concentrated solar system. We are developing a program with the methods for solar tower system. Acknowledgement This work is partially supported by the National Natural Science Foundation of China (No. 50736005) The numerical calculations in this paper have partly been done on the supercomputing system in the Supercomputing Center of University of Science and Technology of China. Appendix A. Brightness distribution of sun Bradial ðhÞ ¼ 03047222e 15 h30 0:35757701839e 13h28 þ 0:30769982398e 11h26 0:15267102168e 9h24 þ 0:48824953590e 8h22 0:10627588060e
4. Conclusion In the paper, we proposed a new analytical method to calculate the optical efficiency of solar concentrator, it is based on the effective light distribution from reflected point to calculate the optical efficiency of each point at mirror. A quick numerical method for integration is developed for optical efficiency simulation, it applies polynomial function to simulate function, and then the integration of function can be replaced by integration of polynomial function which is also a polynomial calculation. We apply the two methods to develop a program to simulate a parabolic trough solar collector to obtain the cosine factor, the optical efficiency, the receiver efficiency and total efficiency at any time as well as yearly or daily average efficiency including the discard efficiency. Trough collectors are today the most widely used solar power generating systems and are assuming a great importance in solar energy development strategies. Since the main aim in solar plant building is cost reduction, defects and imperfections of various kinds are always present. These aspects have been considered in the program for an accurate simulation which allows one to calculate the efficiencies of an actual 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 to net heat efficiency at different tracking error or optical error. b. evaluating the optical efficiency and total solar to net heat efficiency at different displacement error. c. effects of the cluster and aging of the reflection mirror on the efficiency of the system. d. effects of the property of receiver and mirror to the efficiency of the system. e. optimizing the optical parameter of the solar trough system. The analytical method to calculate the optical efficiency of solar concentrator and quick numerical method for
6h20 þ 0:16177926059e 5h18 0:17405417057e 4h16 þ 0:13212195269e 3h14 0:69876231477e 3h12 þ 0:25133685240e 2h10 0:59138793819e 2h8 þ 0:85647334392e 2h6 0:70829038680e 2h4 0:55636572263e 2h2 þ 1:00000; h 6 0:0049 ^
k ðhÞ c
h > 0:0049
k ¼ expð0:9 logð13:5 aÞ=a:^ 0:3Þ; c ¼ ð2:2 logð0:52 aÞ a:^ 0:43 0:1Þ;
ðA:1Þ
where a is the circumsolar ratio (CSR) which is defined as the radiant flux contained within the circumsolar region of the sky, divided by the incident radiant flux from the direct beam 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 of the coordinate, the axis of parabolic curve is the x axis, the length of the trough is the z axis. Assumed that the tracking error is r, that is to say that the normal ray will be reflected away from the focus ray with angle r; assumed that the displacement error at x direction is Dx, at y direction is Dy, then the coordinate of the central line of receiver for a cross section is (f0 + Dx, Dy). For any ray with incidence angel h||, transverse angle between the incidence ray and central line of sun is h. To facilitate the analysis, it is assumed that the incidence ray includes the diffusion caused by optical error, and the mirror has perfect face and does not increase diffusion to the solar ray. If the tracking error is 0, then the vector of incidence ray i is (cos h cos h||,sin h,cos h sin h||), when the tracking error is r, it is equal to rotate coordinate with angle r anticlockwise about z coordinate axis, so the vector of
W. Huang et al. / Solar Energy 86 (2012) 746–755 Table C1 Heat loss coefficient for vacuum tube receiver. Parameter
Value
Std.
a0 a1 a2 a3 b0 b1
9.463033e+00 3.029616e01 1.386833e03 6.929243e06 7.649610e02 1.128818e07
8.463850e01 1.454877e02 7.305717e05 1.070953e07 5.293835e04 6.394787e09
incidence ray i is ((cos h|| cos h cos r + sin h sin r), (cos h cos h|| sin r + sin hcos r), cos h|| sin h) in new coordinates. For a point P in the mirror, normal vector n is (cos (b/ 2), sin (b/2), 0), so the reflected ray r: r ¼ i þ 2ði nÞn
ðB:1Þ
we obtain: rx ¼ cos h cos h cosðb rÞ sin h sinðb rÞ
ðB:2Þ
ry ¼ cos h cos h sinðb rÞ sin h cosðb rÞ
ðB:3Þ
then we can calculate the incidence angle at the absorbed point. Assumed the coordinates of P is (x0, y0, 0), then the equation of the reflected ray is: ðx x0 Þ=rx ¼ ðy y 0 Þ=ry ¼ ðz z0 Þ=rz ;
ðB:4Þ
the equation for the receiver is: 2
2
ðx f0 DxÞ þ ðy DyÞ ¼ r2
ðB:5Þ
from Eqs. (4) and (5), then coordinates of the absorbed position (x1, y1, z1) can be obtained. The normal vector at the absorbed position should be ((x1 f0 Dx)/r, (y1 Dy)/ r,0), then the incidence angle can be calculated: cos d ¼ ½rx ðx1 f0 DxÞ þ ry ðy 1 DyÞ=r
ðB:6Þ
so the absorptivity can be calculated at the position, the transmittance can be obtained with the same methods. From Eqs. (4) and (5), we can also obtain the integration limit of h when the reflected ray reaches the receiver tube tangentially Appendix C See Table C1. References Bendt, P., Rabi, A., et al., 1979. Optical Analysis and Optimization of Line Focus Solar Collectors. Solar Energy Research Institute Report No. TR-34-092, p. 11–15.
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Bendt, P., Rabl, A., 1981. Optical analysis of point parabolic radiation. Applied Optics 20 (4), 674–683. Bennett, C.L., 2008. Optimal heat collection element shapes for parabolic trough concentrators. Journal of Solar Energy Engineering – Transactions of the ASME 130 (2). Buie, D., Monger, A.G., et al., 2003. Sunshape distributions for terrestrial solar simulations. Solar energy 74 (2), 113–122. Chen, W., Li, J., 2003. Optical performance of analysis for parabolictrough focusing collector with several tracking modes. Acta Energiae Solarris Sinica 24 (4), 477–482. Chin, S., 1978. Solar Reflectivity and Absorptivity Studies. Texas Tech University, MS. Daly, J.C., 1979. Solar concentrator flux distribution using backward tracing. Applied Optics 18 (15), 2696–2700. Du, S., Xia, X., et al., 2006. Numerical investigation on effects of nonparallelism of solar rays on concentrating solar power. Acta Energiae Solaris Sinica (Chinese) 27 (4), 388–395. Duffie, J.A., Beckman, W.A., 1991. Solar Engineering of Thermal Processes. Wiley-Interscience, New York. Ge, X.S., 1988. Solar Engineering – Fundamental and Application. Academic Journal Publishment Co., Beijing. Grena, R., 2010. Optical simulation of a parabolic solar trough collector. International Journal of Sustainable Energy 29 (1), 19–36. Helgesson, A., Karlsson, B., et al., 2000. Angular dependent optical properties from outdoor measurements of solar glazings. Solar Energy 69 (1–6), 93–102. Jeter, S.M., 1986. The distribution of concentrated solar radiation in paraboloidal collectors. Journal of Energy Engineering 108 (1), 219– 225. Jeter, S.M., 1987. Analytical determination of the optical performance of practical parabolic trough collectors from design data. Solar Energy 39 (1), 11–21. Jiang, S., Hu, P., et al., 2010. Optical modeling for a two-stage parabolic trough concentrating photovoltaic/thermal system using spectral beam splitting technology. Solar Energy Materials & Solar Cells 94, 1686– 1696. Neumann, A., Witzke, A., et al., 2002. Representative terrestrial solar brightness profiles. Journal of Solar Energy Engineering – Transactions of the ASME 124 (2), 198–204. Nicolas, R.O., 1987. Optical analysis of cylindrical-parabolic concentrators: validity limits for models of solar disk intensity. Applied Optics 26 (18), 3866–3870. Patnode, A.M., 2006. Simulation and Performance Evaluation of Parabolic Trough Solar Power Plants. University of Wisconsin-Madison. Price, H., 2002. Concentrating Solar Power Systems Analysis and Implications, slide 39.. Riffelmann, K.J., Neumann, A., et al., 2006. Performance enhancement of parabolic trough collectors by solar flux measurement in the focal region. Solar Energy 80, 1303–1313. Shaner, W.W., Duff, W.S., 1978. Solar thermal electric power systems: comparison of line-focus collectors. Solar Energy 22, 49–61. Tesfamichael, T., Wackelgard, E., 2000. Angular solar absorptance and incident angle modifier of selective absorbers for solar thermal collectors. Solar Energy 68 (4), 335–341. Zhao, J., Ge, X., et al., 1994. Theoretical and experimental study of the radiation flux distribution in the focal plane of a parabolic trough concentrator. Acta Energiae Solaris Sinica (Chinese) 15 (3), 262–267.
Solar Energy 86 (2012) 746–755 www.elsevier.com/locate/solener
Performance simulation of a parabolic trough solar collector Weidong Huang a,⇑, Peng Hu b, Zeshao Chen b b
a Department of Earth and Space Science, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui 230026, China 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 2011 Available 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 performance of a parabolic trough solar collector with vacuum tube receiver. The analytical equation for optical efficiency of each point at reflector is derived first, then the optical efficiency of the system is simulated by numerical integration algorithm. The cosine factor, receiver 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 from installation of receiver, optical properties of reflector, transmittance and absorptivity of vacuum tube receiver on efficiencies of the trough system are simulated and analyzed as well as optical parameter. Ó 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 energy absorbed by receiver to the incidence solar energy in solar energy utilization. It is one of key parameters in optical design of concentrated solar energy system. The optical efficiency of concentrated solar energy system is affected by the absorptivity of receiver, the transmittance of glass envelope of vacuum tube receiver and the reflectivity of mirror as well as optical parameter and optical error. In order to calculate the optical efficiency, the energy flux distribution on the receiver is usually calculated first, the total absorbed energy in a receiver is calculated by integration, the optical efficiency is obtained as the ratio of the absorbed energy to the incidence energy. For the calculation of energy flux distribution on the receiver, three methods are often applied: the cone optics method (Bendt and ⇑ Corresponding author. Tel.: +86 551 3606631; fax: +86 551 3607386.
E-mail address: [email protected] (W. Huang). 0038-092X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2011.11.018
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 the fact that the incidence ray from sun to a point in mirror and the reflected ray from the point at mirror to the receiver is also an optical cone. The flux of any point at the receiver is obtained by integrating solar ray from the mirror. The ray tracing method needs to trace a large number of rays from any point of mirror. Both of the methods consume great computer resources. The semifinite integration formulation has concise physical concept, but has complicated formulation and need many computation resources. In order to optimize the receiver geometry, Bennett (2008) applied the following formula to calculate the optical efficiency go of each point at parabolic trough reflectors: x go ¼ a erf pffiffiffi ð1Þ r 8 here a is the product of mirror reflectivity and receiver absorbance, x is the angular width of the heat collection
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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 receiver Beff(h) the energy distribution function of reflected ray in radial direction (W/m2/rad) Blinear(h\) the linear brightness distribution function at transverse 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 Earth’s atmosphere (W/m2) K(d) absorptivity, transmittance or reflectivity calculated from incidence angle L the length of the parabolic mirror (m) m the air mass nx the transverse section of the normal vector at the reflection point p the atmospheric transparency qloss the heat loss of receiver (W/m2) 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)
element from a point of mirror, r is the Gaussian function parameter of the reflected ray when the brightness of sun is regarded 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 several simplifications are applied, such as, the energy distribution of reflected ray is different with the assumed Gaussian distribution (Nicolas, 1987), the reflectivity and absorptivity is not constant but related to the incidence angle (Jeter, 1987; Grena, 2010). In this paper, we first calculate optical efficiency of each point at parabolic solar trough reflector, and then integrate them to obtain the optical efficiency of the whole concentrated solar trough system. We further consider the optical
s S t T w a b c d gc gd gh go gP gt h h0 h\ hjj h0 k q roptic rtracking r\ rjj s u
the whole surface of reflection mirror (W/m2) the projected area of mirror under Direct Normal Incidence (W/m2) time (day) temperature of receiver (K) the half width of the trough mirror (m) absorptivity of vacuum receiver the rim angle to the focus point at point P (rad) azimuth of sun (rad) the incidence angle (rad) cosine factor the efficiency related to the discard loss receiver efficiency the optical efficiency of parabolic trough solar collector the optical efficiency at point P the annual average efficiency radial angular displacement or angular displacement in transverse direction (rad) the maximum angle of ray to the receiver (rad) angular displacement in transverse direction (rad) angular displacement in longitudinal direction (rad) the variable in convolution calculation parameter for calculating optical error is the reflectivity of mirror at point P the total optical error (rad) tracking error (rad) optical error in transverse direction (rad) optical error in longitudinal direction (rad) and are the transmittance of glass envelope and one coordinate in cylindrical coordinate system (rad)
error, the tracing error and displacement error, the incidence angle effect to the absorptivity of receiver, transmittance of glass envelope and the reflectivity of the mirror, apply the actual sun shape data to simulate the energy distribution of solar radiation. The abnormal incidence, the shadow effect of the receiver and the end loss are also considered to simulate the actual system. A quick algorithm is developed specially for the integration computation. So it is rather quick to compute the discard energy loss, the cosine factor, the optical efficiency, the receiver efficiency and the efficiency of conversion of solar energy into net heat energy at a moment or a time span. We apply it to simulate the effects of main optical defects and properties of materials in parabolic trough collector.
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2. Computation method 2.1. Fundamental of computation As shown in Fig. 1, a reflected ray from a point P at mirror is absorbed by receiver at point Q, assumed point P is the origin of the coordinates, z axis is the reflected ray from P of center ray of the sun. At cylindrical coordinate, the coordinate of the point Q is (r, u, fP), the absorbed energy of reflected solar from point P can be calculated as following: Z Z Ip ¼ qsaBeff ðhÞduhdh ð2Þ s
Fig. 2. Radial distribution of sun brightness.
where h = atan (r/fP), q is the reflectivity of mirror at point P, s and a are the transmittance of glass envelope and absorptivity of vacuum receiver which are all related to the incidence angle, s represents the integration to the whole surface of the receiver, Beff(h) is the energy distribution function of reflected ray from point Q. So the optical efficiency gP at point P is calculated as following: gp ¼ I p =I in
ð3Þ
where Iin is the incidence solar energy. The average optical efficiency go of the whole solar trough system is calculated as following: RR g dS s P go ¼ ð4Þ S where S represents the projected area of mirror under Direct Normal Incidence. We apply Eqs. (1)–(3) to calculate the optical efficiency of 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 distribution Bradial(h) in W/(m2 sr), h being measured from the
center ray of the solar disk. Here we use polynomial function to simulate the radial brightness function of sun disk and part of circumsolar region (Neumann et al., 2002) as shown in Fig. 2 and apply exponent decreasing function of Buie et al. (2003) to simulate other circumsolar region. Detailed formula is given in Appendix A. For line focus systems, it is convenient to transform the radial distribution Bradial(h) to a linear one (Bendt and Rabi, 1979) according to Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ Blinear ðh? Þ ¼ dhjj Bradial ðhÞ; h ¼ h2? þ h2jj \ and || in subscript represent longitudinal and transverse direction. In the remainder of this paper, only the linear brightness function (in W/m2 rad) is considered for trough concentrator. The bright distribution from a point reflected is a convolution of Gaussians with sun brightness function when the optical error of concentrated mirror is approximately described by a Gaussian distribution G, which is as following (Bendt and Rabi, 1979): ! Z 1 h02 0 pffiffiffiffiffiffi dh exp Beff ðh? Þ ¼ Blinear ðh? h0 Þ ð6Þ 2r2optic roptic 2p where roptic is the total optical error, it can be calculate as following: r2optic ¼ 4r2contour? þ r2specular? þ k 4r2contourjj þ r2specularjj þ r2tracking þ r2displacement ¼ r2? þ k r2jj
ð7Þ
k is related to the position of reflection at mirror which is calculated as following: k ¼ n2x tan2 h
Fig. 1. Calculation of the absorbed energy by receiver from a point of a mirror.
ð8Þ
nx is the transverse section of the normal vector at the reflection point. When we study the relationship between optical efficiency and tracking error or displacement error and considering optical error simultaneously, we should calculate the total optical error by deduct their error from Eq. (6). For example, we calculate the total optic error with following formula when the displacement error is studied:
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r2optic ¼ 4r2contour? þ r2specular? þ k 4r2contourjj þ r2specularjj þ r2tracking
ð9Þ
2.3. Optical efficiency of solar trough system Fig. 3 is the transverse cross section of parabolic trough solar collector with tube receiver when the incidence ray is perpendicular to the parabolic mirror. The top of parabolic curve “O” is the origin of the coordinate, the axis of parabolic curve is the x axis. For a point P at mirror, when the angle between a ray to the ray from center of sun is h\, the brightness of the reflected ray is Beff(h\)dh\, the absorbed part is psaBeff(h\)dh\, so the optical efficiency from the point P of parabolic mirror is: R h0 qtaBeff ðh? Þ dh? h0 gP ðyÞ ¼ I in f0 tanðhjj Þ ð1 þ tan2 ðb=2ÞÞ 1 ð10Þ L The later item is the end loss, it gives the part of ray that does not reach the receiver when the incidence ray is not normal to the parabolic mirror. L is the length of the parabolic mirror (assumed the receiver has the same length with mirror), f0 is the focal length, b is the rim angle to the focus point at point P. Iin is the incidence solar energy flux, h0 is the maximum angle of ray to the receiver, it is calculated as following: h0 ¼ tanðr0 =fp Þ
when a tracking error rtracking is considered, the ray from sun center is reflected rtracking away from the ray to the focus point, then the up and low limit of integration is from (h0 + rtracking) to h0–rtracking. When the incidence ray is not normal to the parabolic mirror, the limit of integration and the incidence angle for qsa calculation can be calculated from 3-dimensional analysis, a detailed analysis is given in Appendix B. So the optical efficiency of the whole system is calculated from integration of each point of mirror as following: Rr Rw g ðyÞ I in dy þ 0 0 taI in cosðuÞdy R P go ¼ I in w R r0 Rw g ðyÞ dy þ ta cosðdÞdy 0 R P ¼ ð13Þ w where first part of the equation is the contribution of the reflected ray, and the later part of the equation is the contribution of the ray irradiated on the collector directly, R is the envelope tube radius of vacuum receiver to consider the shadow of receiver in the calculation, w is the width of the trough mirror, d is the incidence angle of the radiation on the absorption tube which can be calculate from the geometry. 2.4. Efficiency of parabolic trough solar collector Efficiency of parabolic trough solar collector is the ratio of the net heat collected to the Direct Normal Incidence (DNI) solar energy. It is related to the following factors (Shaner and Duff, 1978):
ð11Þ
where r0 is the radius of the tube receiver, fP is the distance for point P to the focus point which is calculated as following (Duffie and Beckman, 1991): fp ¼ f0 = cos2 ðb=2Þ
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ð12Þ
a. Cosine factor (gc) : when the incidence is not normal to the reflection mirror, the energy reflected by the mirror is DNI multiplied with cosine of incidence angle, so the cosine factor is give as following (Chen and Li, 2003): gc ¼ cosðhÞ
ð14Þ
b. Optical efficiency (go): it is related to the end loss, optical error, interceptance, reflectivity of mirror, transmittance of glass envelope, absorptivity of receiver, 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 heat to the absorbed energy by the receiver. The energy loss comes from heat radiation, convection and conduction of receiver. The receiver efficiency is calculated as following: gh ¼ 1 qloss =ðDNI cosðhÞ go Þ
Fig. 3. Transverse cross section of parabolic trough solar collector with tube receiver.
ð15Þ
where the energy loss qloss is calculated according to the Patnode’s equation ( Patnode, 2006) from SEGS data which is for vacuum tube receiver: qloss ¼ a0 þ a1 T þ a2 T 2 þ a3 T 3 þ DNI ðb0 þ b1 T 2 Þ
ð16Þ
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2.5. Incidence angle effects where T is the temperature of the receiver, The parameter is given in Table C1 of Appendix C. d. Discard loss: when the DNI radiation from sun is rather low in early morning or later afternoon, the absorbed energy cannot increase the temperature of the receiver to the demanded temperature and thereby can not supply available energy, thus, this part of the solar energy is discarded. The efficiency related to the discard loss gd is calculated as following: R DNId dt gd ¼ R ð17Þ DNI dt
The reflectivity of mirror, transmittance of glass envelope and absorptivity of receiver will decrease when the incidence angle increases. The optical efficiency is changed when the incidence angle effect is considered according the experimental data in Jeter’s calculation (Jeter, 1987). Grena considers the incidence angle effect to simulate the flux distribution in receiver according to the Fresnel Law (Grena, 2010). Here we apply experimental data from Tesfamichael and Wackelgard (2000) to simulate the incidence angle effects to the absorptivity of receiver with following equation: c
where DNId represents the DNI radiation at the time which outputs net heat. According to the definition, the annual average efficiency of parabolic trough solar collector is given as (Duffie and Beckman, 1991): gt ¼ Qnet =E
ð18Þ
E is the total DNI energy in a year, it is calculated (Ge, 1988) as: Z 365 E¼ DNIðtÞdt ð19Þ 0
We need DNI data in a specific area for calculation. considered that the data is not enough for calculation in many areas, here a sunny day model is applied for DNI calculation. In sunny day model, DNI is related to the atmospheric transparency and solar altitude which can be calculated as (Chen and Li, 2003): DNIðtÞ ¼ I s ðtÞpm ¼0
h>0
ð20Þ
h60
where h is the solar altitude, Is(t) = 1367 (1 + 0.034 cos (2pt/365)), is the solar irradiance on the outer surface of Earth’s atmosphere (W/m2), p is the atmospheric transparency which is approximately as a constant, m is the air mass, m = [1229 + (614sin (h))2]1/2614sin (h). The net energy is calculated as following (Ge, 1988): Z 365 Qnet ¼ 86400qnet ðtÞdt ð21Þ 0
here the scattering radiation is ignored, qnet is the net energy power in any time which is calculated as following: qnet ðtÞ ¼ I in ðtÞ cosðhjj Þ go qloss ðtÞ ¼0
qnet ðtÞ > 0 qnet ðtÞ 6 0
ð22Þ
for north–south axis tracking system, cosðhjj Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 cos2 ðhÞ cos2 ðcÞ , where c is azimuth; for east–west qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi axis tracking system, cosðhjj Þ ¼ 1 cos2 ðhÞ sin2 ðcÞ, the altitude h and azimuth c is calculated from the time and latitude of the site (Chen and Li, 2003).
KðdÞ ¼ a½1 bð1= cos d 1Þ
ð23Þ
here d is the incidence angle, a, b and c is the parameter from simulation to the experimental data, a = 0.96, b = 0.057, c = 1.2. Using Helgesson’s glass transmission data (Helgesson et al., 2000), and applying the above equation to simulate relationship between transmittance and incidence angle, the parameter is a = 0.925; b = 0.2, c = 1. If the simulation result is less than 0, then the transmittance of glass or absorptivity of receiver is set to 0. For reflectivity, we apply Chin’s data (Chin, 1978) and above equation to simulate, the parameter for new mirror is a = 0.915, b = 0.01079, c = 0.31985; for using mirror is a = 0.875, b = 0.05103, c = 0.44747. The new mirror is the mirror that does not used in the environment before. The using mirror is the mirror which has been used for some times and being used before test. The test data shows that the mirror has been using for 8 months. 2.6. Numerical method The ray tracing is often used in present optical simulation, however, we need to trace millions of rays, it will spend rather long time to obtain the calculation. Du et al. (2006)spend 84 h to simulate the point focus parabolic system, Grena (2010) spend 300 s to get optical efficiency of parabolic trough collector after tracing 2.3 million rays. After getting the flux distribution at receiver, we need to integrate it to obtain the optical efficiency which will spend more time. In the method to calculate optical efficiency directly developed here, the four times integration is also needed, including integration of radial distribution of solar radiation to the transverse linear distribution, then through convolution of Gaussians for integration to the effective distribution of reflected ray, then the third integration of different ray from reflection point to obtain the optical efficiency of the reflection point, and the fourth integration of point optical efficiency to the whole mirror. For present integration algorithms, assumed an integral needs 100 times calculation of the function, four times integration needs about 100 million times function calculation.
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Here we introduce a quick algorithm for integration. We use function to calculate about 11–50 points, and apply polynomial function to simulate the function for integration. Then the later calculation is based on the polynomial function. The integration of polynomial function is also a polynomial function calculation. Not only the computation of integration is reduced, but also all integrations are changed to an algebra calculation. The optical efficiency of a point at mirror is integrated from each ray which the brightness decrease gradually as well as optical efficiency itself, so a small number of point is needed for simulation with polynomial function, it greatly decrease the computation. When the tracing error and displacement error is really a rather small part of the total optical error, the Gaussian function is applied for optical error, then the optical efficiency of point at mirror is symmetrical to the axis of the parabolic mirror, and half simulation is need for optical efficiency which reduces the computation. By using one CPU of an i3 processor in a notebook computer to calculate the annual net heat efficiency of parabolic trough solar collector, only 0.11 h is needed with the time step of 0.024 h or 0.18 h is needed with Intel celeron 2003 CPU, the numerical error is less than 0.01% on the optical efficiency. 3. Results We calculate a typical parabolic trough solar collector with vacuum tube receiver. The central line of the tube receiver is installed at the focus line of the parabolic mirror. The parameters for the collector are shown at Table 1. We simulate the efficiency under various parameters of the parabolic trough solar collector with a vacuum tube receiver. Fig. 4 shows the optical efficiency at different incidence angle with new mirror and using mirror. The reflectivity for new and using mirror decrease as described with Eq. (21) when the incidence angle increases, but the new one has higher reflectivity than using mirror. The optical efficiency of the trough collector decreases with the increase of the incidence angle for both new mirror and using mirror. The different position at parabolic mirror has different optical efficiency as shown in Fig. 5. The reflection position near the axis of the parabolic reflector has shorter distance
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Fig. 4. The optical efficiency at different incidence angle with new mirror and using mirror in a parabolic trough solar collector with vacuum tube receiver, the other parameters are shown in Table 1.
than those away from axis. When the distance of the reflection point to the focus point increases, the angle at which they reach the receiver tube tangentially decreases, the energy absorbed by receiver decreases. So the optical efficiency decreases gradually and obviously at point away from the axis, and the average optical efficiency for the whole parabolic trough solar collector decreases when the width of the mirror increases as shown in Fig. 6. When the optical error increases, the reflected solar image on the receiver diffuses, and the ray reaches the receiver decreases with high optical error as shown in Fig. 7. The tracking error leads to the focus line deviated from the center of the receiver tube, and the reflected ray which does not reach to the receiver tube increases when the tracking error increases as shown in Fig. 8. The results show that tracking error leads to low optical efficiency. Under normal incidence, the optical efficiency decrease from 71% to 53% with tracking error increasing from 0 to 12 m rad. We can simulate optical efficiency, cosine factor, receiver efficiency and total net heat efficiency at any time, and
Table 1 Parameter of the typical parabolic trough solar collector with vacuum tube receiver. Parameter
Data
Focus length f0 Half trough width w Radius of envelope of receiver R Radius of receiver tube r0 Transverse optical error h\ Longitudinal optical error hjj Tracing error rtracking Operation temperature of receiver T
1.7 m 3.0 m 0.035 m 0.055 m 6.0 m rad 6.0 m rad 0 m rad 400 °C
Fig. 5. Optical efficiency at different point of mirror in a parabolic trough solar collector with vacuum tube receiver, the other parameters are shown in Table 1.
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Fig. 6. Optical efficiency of a parabolic trough solar collector with vacuum tube receiver in different geometrical concentration, the other parameters are shown in Table 1. GC: geometrical concentration.
Fig. 7. Optical efficiency of a parabolic trough solar collector with vacuum tube receiver in different optical error, the other parameters are shown in Table 1.
Fig. 8. Optical efficiency of a parabolic trough solar collector with vacuum tube receiver in different tracking error, the other parameters are shown in Table 1.
annual average efficiency with further considering the discard efficiency. Figs. 9 and 10 shows the cosine factor, opti-
cal efficiency, receiver efficiency and total efficiency at any time in the summer solstice and the Spring or Autumnal Equinox with north–south axis tracking and ease–west axis tracking system. For north–south axis tracking system, the incidence angle is rather small at early morning or later afternoon, and the optical efficiency will be high, so more collecting time and less discard energy is for north–south axis tracking system than east–west axis tracking system. However, at the winter solstice, the incidence angle is larger for north–south axis tracking system than for east–west one at most time, and the optical efficiency and total efficiency is lower for north–south axis tracking system than for east–west one at most time of the day as shown in Fig. 11. When the absorbed energy cannot increase the temperature of the receiver to the demanded temperature, then the solar energy is discarded. At early morning or later afternoon, the incidence angle is smaller for north–south axis tracking system than east–west axis tracking system for more than half year. So less solar energy is discarded, and the year average optical efficiency is larger for north– south axis tracking system with less discard energy and higher optical efficiency. As the receiver has the same heat loss for both tracking system, the absorbed energy is more for north–south axis tracking system than east–west axis tracking system, so the receiver efficiency for north–south axis tracking system is larger than east–west axis tracking system for the same receiver. The annual average efficiency is higher for north–south axis tracking system than east–west axis tracking system as shown in Fig. 12. For both tracking system, the optical efficiency decreases but the receiver efficiency increases with the increase of trough width, so an optimum trough width with the maximum total efficiency is shown in the Fig. 12. It can be seen that the half width of SEGS trough (Patnode, 2006) is rather near to the optimum trough. The optical error and tracking error will decrease the efficiency of the solar trough collector as shown in Figs. 13 and 14 as well as optical efficiency.
Fig. 9. The efficiency at different time in a parabolic trough solar collector with vacuum tube receiver under two kinds of tracking system in summer solstice, the other parameters are shown in Table 1. NS: north–south axis tracking system, EW: ease–west axis tracking system.
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Fig. 10. The efficiency at different time in a parabolic trough solar collector with vacuum tube receiver under two kinds of tracking system in the spring or Autumnal Equinox, the other parameters are shown in Table 1, NS: north–south axis tracking system, EW: ease–west axis tracking system.
Fig. 11. The efficiency at different time in a parabolic trough solar collector with vacuum tube receiver under two kinds of tracking system in winter solstice, the other parameters are shown in Table 1. NS: north– south axis tracking system, EW: ease–west axis tracking system.
Fig. 12. The year average efficiency of a parabolic trough solar collector with vacuum tube receiver in different trough width under two kinds of tracking system, the other parameters are shown in Table 1. Up: north– south axis tracking system, down: east–west axis tracking system.
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Fig. 13. The year average efficiency of a parabolic trough solar collector with vacuum tube receiver in different optical error under two kinds of tracking system, the other parameters are shown in Table 1. Up: north– south axis tracking system, down: east–west axis tracking system.
Fig. 14. The year average efficiency of a parabolic trough solar collector with vacuum tube receiver in different tracking error under two kinds of tracking system, the other parameters are shown in Table 1, Up: north– south axis tracking system, down: east–west 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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W. Huang et al. / Solar Energy 86 (2012) 746–755
Comparison with experimental results: experimental validation to the model is rather difficult, since the efficiency is related to many factors and it is hard to test so many related factors precisely and simultaneously. However, one of the key result of our model is the intercept factor (assumed sa = 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 intercept factor is constant up to large incidence angles and decrease gradually when the incidence angle increases in both test and model prediction. It can be seen that the prediction agrees the test result rather well. Recent experimental solar to net heat efficiency for about spring Equinox indicates that the solar trough collector with north–south axis tracking system has lower total efficiency at noon than 4 h early or later (Price, 2002). It agrees rather well with the present simulation.
integration can be applied to other concentrated solar system. We are developing a program with the methods for solar tower system. Acknowledgement This work is partially supported by the National Natural Science Foundation of China (No. 50736005) The numerical calculations in this paper have partly been done on the supercomputing system in the Supercomputing Center of University of Science and Technology of China. Appendix A. Brightness distribution of sun Bradial ðhÞ ¼ 03047222e 15 h30 0:35757701839e 13h28 þ 0:30769982398e 11h26 0:15267102168e 9h24 þ 0:48824953590e 8h22 0:10627588060e
4. Conclusion In the paper, we proposed a new analytical method to calculate the optical efficiency of solar concentrator, it is based on the effective light distribution from reflected point to calculate the optical efficiency of each point at mirror. A quick numerical method for integration is developed for optical efficiency simulation, it applies polynomial function to simulate function, and then the integration of function can be replaced by integration of polynomial function which is also a polynomial calculation. We apply the two methods to develop a program to simulate a parabolic trough solar collector to obtain the cosine factor, the optical efficiency, the receiver efficiency and total efficiency at any time as well as yearly or daily average efficiency including the discard efficiency. Trough collectors are today the most widely used solar power generating systems and are assuming a great importance in solar energy development strategies. Since the main aim in solar plant building is cost reduction, defects and imperfections of various kinds are always present. These aspects have been considered in the program for an accurate simulation which allows one to calculate the efficiencies of an actual 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 to net heat efficiency at different tracking error or optical error. b. evaluating the optical efficiency and total solar to net heat efficiency at different displacement error. c. effects of the cluster and aging of the reflection mirror on the efficiency of the system. d. effects of the property of receiver and mirror to the efficiency of the system. e. optimizing the optical parameter of the solar trough system. The analytical method to calculate the optical efficiency of solar concentrator and quick numerical method for
6h20 þ 0:16177926059e 5h18 0:17405417057e 4h16 þ 0:13212195269e 3h14 0:69876231477e 3h12 þ 0:25133685240e 2h10 0:59138793819e 2h8 þ 0:85647334392e 2h6 0:70829038680e 2h4 0:55636572263e 2h2 þ 1:00000; h 6 0:0049 ^
k ðhÞ c
h > 0:0049
k ¼ expð0:9 logð13:5 aÞ=a:^ 0:3Þ; c ¼ ð2:2 logð0:52 aÞ a:^ 0:43 0:1Þ;
ðA:1Þ
where a is the circumsolar ratio (CSR) which is defined as the radiant flux contained within the circumsolar region of the sky, divided by the incident radiant flux from the direct beam 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 of the coordinate, the axis of parabolic curve is the x axis, the length of the trough is the z axis. Assumed that the tracking error is r, that is to say that the normal ray will be reflected away from the focus ray with angle r; assumed that the displacement error at x direction is Dx, at y direction is Dy, then the coordinate of the central line of receiver for a cross section is (f0 + Dx, Dy). For any ray with incidence angel h||, transverse angle between the incidence ray and central line of sun is h. To facilitate the analysis, it is assumed that the incidence ray includes the diffusion caused by optical error, and the mirror has perfect face and does not increase diffusion to the solar ray. If the tracking error is 0, then the vector of incidence ray i is (cos h cos h||,sin h,cos h sin h||), when the tracking error is r, it is equal to rotate coordinate with angle r anticlockwise about z coordinate axis, so the vector of
W. Huang et al. / Solar Energy 86 (2012) 746–755 Table C1 Heat loss coefficient for vacuum tube receiver. Parameter
Value
Std.
a0 a1 a2 a3 b0 b1
9.463033e+00 3.029616e01 1.386833e03 6.929243e06 7.649610e02 1.128818e07
8.463850e01 1.454877e02 7.305717e05 1.070953e07 5.293835e04 6.394787e09
incidence ray i is ((cos h|| cos h cos r + sin h sin r), (cos h cos h|| sin r + sin hcos r), cos h|| sin h) in new coordinates. For a point P in the mirror, normal vector n is (cos (b/ 2), sin (b/2), 0), so the reflected ray r: r ¼ i þ 2ði nÞn
ðB:1Þ
we obtain: rx ¼ cos h cos h cosðb rÞ sin h sinðb rÞ
ðB:2Þ
ry ¼ cos h cos h sinðb rÞ sin h cosðb rÞ
ðB:3Þ
then we can calculate the incidence angle at the absorbed point. Assumed the coordinates of P is (x0, y0, 0), then the equation of the reflected ray is: ðx x0 Þ=rx ¼ ðy y 0 Þ=ry ¼ ðz z0 Þ=rz ;
ðB:4Þ
the equation for the receiver is: 2
2
ðx f0 DxÞ þ ðy DyÞ ¼ r2
ðB:5Þ
from Eqs. (4) and (5), then coordinates of the absorbed position (x1, y1, z1) can be obtained. The normal vector at the absorbed position should be ((x1 f0 Dx)/r, (y1 Dy)/ r,0), then the incidence angle can be calculated: cos d ¼ ½rx ðx1 f0 DxÞ þ ry ðy 1 DyÞ=r
ðB:6Þ
so the absorptivity can be calculated at the position, the transmittance can be obtained with the same methods. From Eqs. (4) and (5), we can also obtain the integration limit of h when the reflected ray reaches the receiver tube tangentially Appendix C See Table C1. References Bendt, P., Rabi, A., et al., 1979. Optical Analysis and Optimization of Line Focus Solar Collectors. Solar Energy Research Institute Report No. TR-34-092, p. 11–15.
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