The theoretical power conversion efficiency of a silicon solar cell with a fluorescent solar collector is believed to reach 90% of the maximum efficiency of an ideal silicon solar given by the Shockley-Queisser detailed balance limit, but the practical efficiencies are significantly lower due to several loss mechanisms. This work presents an analytic model which take the non-ideal coupling between the collector and the solar cell mounted at the edge into consideration and it is shown in good agreement with experimental results

Danos, Lefteris

Fang, Liping

Markvart, Tom

Parel, Thomas

English

Southampton (e-Prints Soton)

Modelling the Performance of Fluorescent Solar Collectors  Liping Fang*, Thomas Parel, Lefteris Danos, Tomas Markvart Solar Energy Laboratory, School of Engineering Sciences, University of Southampton, Southampton, SO17 1BJ, UK * Corresponding Author, Email: L.Fang@soton.ac.uk   1. Introduction Fluorescent  solar  collector  (FSC)  (also known  as  fluorescent  solar  concentrator) bears  the  promise  to  reduce  the  size  of solar  cells  in  solar  systems  and  thereby reduce the overall cost [1-3]. A typical FSC consists of a transparent sheet doped with fluorescent  dyes.  Save  for  one  edge coupled  with  a  solar  cell,  the  remained edges and the bottom surface of this sheet are  usually  covered  with  mirrors.  Photon flux  (including  both  direct  and  diffuse radiation) incident onto the front surface of the  FSC  can  be  absorbed  by  the  dyes inside the collector. The excited dyes will re-emit photons at higher wavelength with a  lower  energy.  The  photons  trapped  in the collector can be finally collected by the solar  cell  mounted  at  the  edge  by  total internal reflection.  Although the theoretical energy conversion efficiency of a silicon solar cell with a FSC is believed to reach 90% of the maximum efficiency of an ideal silicon solar given by the  Shockley-Queisser  detailed  balance  limit  [4],  the  practical  efficiencies  are significantly  lower  due  to  several  loss mechanisms, and further study is needed to  understand  these  losses  in  detail  and build up a more realistic model. Following previous  approaches  [5-7],  this  work  will present a modified model based on Weber and Lambe’s theory [1], which will take the non-ideal  coupling  between  the  collector and the solar cell mounted at the edge into consideration.  2. Model Description The structure of the collector considered in this model is shown in Figure 1. Except for the  edge  mounted  solar  cell,  the  other three  edges  are  assumed  to  be  covered with  mirrors.  The  bottom  surface  of  the collector is not coupled with mirror. An air gap  is  assumed  to  exist  between  the collector and the solar cell, and this is the real situation when the edge fluorescence is  detected  by  an  optical  fibre  in  the experiments.  The  optical  index  (PMMA, n=1.5)  of  the  collector  is  higher  than  air (n=1),  so  there  will  be  an  escape  cone (Fig.1,  cone  1  and  cone  3)  at  the  edge. Only photons emitted in these two cones can be collected by the solar cell, if there is not any reabsorption. Photons emitted in the other two cones (Fig.1, Cone 2, 4) are considered as loss.  Mirror Solar Cell123qc4Air Gap Figure 1. Collector struct ure and escape cones.  c q  is  the  critical  angle  for  total internal reflection. The  fluorescence  photon  collection efficiency of the collector, defined to be the number of photons collected by the solar cell  divided  by  the  number  of  photons emitted by the dye inside the collector. In Weber and Lambe’s theory, the collection efficiency only counts the first generation photons  [5].  However,  their  theory  could be employed to calculate the reabsorption probability  [7].  Assuming  the  fluorescent photons emitted to be isotropic and using the  Beer-Lambert  Law,  the  spectral reabsorption  probabilities  () r   are  given by  WL011 ( )4 (1 )sin sin()2 ( )1 exp sinsin sincc eerPdLLd  qqqqqq                           (1)  MWL011 ( )2 ( )2 ( )1 exp cos sincosceerPLLdqq q qq             (2) where   WL 1 ( ) r   is  the  spectral reabsorption  probability  for  Weber  and Lambe’s  theory,   MWL 1 ( ) r    is  for Modified  Weber  and  Lambe’s theory, 1 cos c P q   is  the  fraction  of photons emitted inside cone 1 and cone 3 or  cone  2  and  cone  4,    c q  is  the  critical angle for total internal reflection,  () e  is the attenuation coefficient at the emission wavelength,    and  L  is  the  length  of  the collector. Figure 2 compares the numerical calculation  results  of  the  two  spectral reabsorption probabilities.  Figure  2.  Comparison  of  reabsorpiton probabilities  calculated  from  Weber  and Lambe’s    theory  (dashed  line)  and Modified Weber and Lambe’s theory (solid line). Figure  2  shows  that  the  spectral reabsorption  probability  for  Weber  and Lambe’s  theory  is  higher  than  that  for Modified Weber and Lambe’s theory. The apparent  reason  is  that  the  photons considered to be collected in Weber and Lambe’s  theory  propagate  a  longer pathlength  before  reaching  the  edge mounted solar cell.  3. Collector Characterisation Method The reabsorption loss, the main loss of the collector,  is evaluated by scaling the edge fluorescence  to  the  first  generation fluorescence,  which  is  free  from reabsorption.  In  experiments,  the  first generation  fluorescence  is  the  measured top  fluorescence  with  a  low  dye concentration [7].  The probability that photons of the whole fluorescent  emission  spectrum  will  be reabsorbed is  1 ( ) ( ) R r f d                                        (3) where  1() f   is  the  normalised  first gerenation  fluorescence  which  satisfies the equation: 1( ) 1 fd                                               (4) The  reabsorption  p robability  can  be evaluated  from  the  first  generation fluorescence  and  the  edge  fluoresc ence by using the equation: 1ˆ ()1 ( )()e frf                                        (5) where  ˆ () e f  is  a  normalised  edge fluorescence,  which  is  equal  to  the  first generation  fluorescence  at  the  long wavelength range, where the absorbance is low. The edge fluorescence is fitted to the  first  generation  fluorescence  in  this long wavelength range.   4. Experimental The collector samples were made of glass substrates (20×20×1mm) spin-coated with dye (BASF Lumogen F Red 300, Perylene) diluted  in  PMMA  (MICRO  CHEM  950 PMMA  C  10).  The  spin-coating  process lasts for 3 minutes at 1000 RPM [7].  For absorbance measurments, a halogen lamp was used as the excitation source to illuminate the top surface of the collector samples.  The  transmitted  light  was detected  by  a  detector  connected  with  a Bentham  spectrometer  (equipmed  with  a  TM300  monochromator).  For  high  dye concentration  collector  samples,  the detector  was  a  PMT  tube  (Hamamatsu R446),  while  for  low  concentration  ones, the  detector  was  a  calibrated  silicon detector (dh_Si, Bentham Instruments Ltd).  The excitation source for top fluorescence was  a  metal  halide  lamp,  the  collector samples  were  excited  at  440  nm  by  an optical  fibre  connected  with  the monochromator.  An  optical  fiber connected with the Benthem spectrometer, fixed normal to the surface of the collector samples,  was  used  to  detect  the  top fluorescence.  The  excitation  optical  fiber was fixed 60◦ to the detection direction.  The edge fluorescence was measured by an Avantes spectrometer (AvaSpec-2048, grating  UA(200-1100  nm),  slit-25  m based  on  the  symmetrical  Czerny-Tuner design  with  a  2048  pixel  CCD  dector array).  The  top  surface  of  the  collector samples  were  uniformly  illuminated  by  a Xenon  lamp  approximating  AM1.5 spectrum with a 10 nm band pass filter at 440  nm.  The  edge  fluorescence  was detected from one edge by an optical fiber with  a  convergent  lens  and  then conducted  to  the  Avantes  spectrometer. The  remaining  edges  were  coupled  to  a highly reflecting film (3M Vikuiti Enhanced Specular  Reflector)  with  specular reflectance  over  98%  across  the  visible spectrum.  5. Results and Analysis  Figure  3.  Normalised  absorption,  normalised  first  generation  fluorescence (f1) and fitted edge fluorescence. In Fig. 3, the first generation fluorescence was normalised from the top fluorescence of  the  collector  sample  with  dye concentration of 40 mg/L by using Eq. (4). The  edge  fluorescence  of  the  collector samples with dye concentration of 40 mg/L, 60mg/L,  400mg/L  and  600  mg/L  were fitted  to  the  normalised  first  generation fluorescence in the wavelength range from 650 nm to 850 nm, where the absorbance of  all  the  collector  samples  were  below 0.0012. Fig. 3 shows that higher dye concentration collector samples have larger reabsorption probability in the wavelengh interval where the  absorption  overlaps  with  the fluorescence.  It  is  seen  that,  in  the  long wavelength range, the edge fluorescence of  all  the  samples  fits  well  with  the  first generation fluorescence.  Figure  4.  Experimental  reabsorption probability  of  different  dye  concentration collector samples compared with theory.  By using Eq. (5), the experimental spectral reabsorption  probability  of  different  dye concentration  collector  samples  can  be obtained  and  they  were  compared  with Weber and Lambe’s theory  and  Modified Weber and Lambe’s theory in Fig. 4.  In  Fig.  4,  eff  is  an  effective  absorption coefficient.  The  collector  samples  were assumed  to  be  homogeneous  since  the optical  index  of  the  PMMA  films  were similar  to  that  of  the  glass  substrates. Then the effective absorption coefficient is given by: filmeff filmfilm glassddd     (6) where  film   is  the  true  absorption coefficient  of  the  PMMA  film  diluted  with dye,  film d  is  the  thickness  of  the  PMMA film and  glass d  is the thickness of the glass substrate.  Fig.  4  shows  that  the  experimental reabsorption  probability  fits  well  with  the Modified Weber and Lambe’s theory.   6. Conclusions This paper presents a model which takes into  consideration  the  non-ideal  coupling between the collector edge and the solar cell.  In  accordance  with  the  edge fluorescence  detection  approach  in  the experiments,  an  air  gap  is  assumed  to exist between the collector edge and the solar cell, thus only photons emitted in the escape  cones  (Cone  1  and  3)  could  be  collected by the solar cell or observed as edge  fluorescence.  This  Modified  Weber and  Lambe’s  theory  fits  well  with experimental results, and could be used to analyse  the  spectrum  of  the  photons emitted out of the collector edges. Based on this analysis, it will be possible to infer the  photon  flux  emitted  by  the  collector onto a fully coupled solar cell which is not easy to measure experimentally.  7. Acknowledgements This work has benefited from the financial support  from  SUPERGEN  PV21,  Faculty of  Engineering  and  Environment, University  of  Southampton,  and  China Scholarship Council.  8. References [1]  W.H.  Weber,  J.  Lambe,  Luminescent greenhouse  collector  for  solar  radiation, Appl. Opt., 15 (1976) 2299-2300. [2] J.A.  Levitt, W.H. Weber, Materials for luminescent  greenhouse  solar  collectors, Appl. Opt., 16 (1977) 2684-2689. [3]  A.  Goetzberger,  W.  Greube,  Solar energy  conversion  with  fluorescent collectors,  Applied  Physics  A:  Materials Science  &amp;  Processing,  14  (1977) 123-139. [4] T. Markvart, Detailed balance method for  ideal  single-stage  fluorescent collectors, Journal of Applied Physics, 99 (2006) 026101. [5] J.S. Batchelder, A.H. Zewail, T. Cole, Luminescent  solar  concentrators.  1: Theory  of  operation  and  techniques  for performance  evaluation,  Appl.  Opt.,  18 (1979) 3090-3110. [6] J.S. Batchelder, A.H. Zewail, T. Cole, Luminescent  solar  concentrators.  2: Experimental  and  theoretical  analysis  of their  possible  efficiencies,  Appl.  Opt.,  20 (1981) 3733-3754. [7]  P.  Kittidachachan,  L.  Danos,  T.J.J. Meyer, N. Alderman, T. Markvart, Photon Collection  Efficiency  of  Fluorescent  Solar Collectors,  CHIMIA  International  Journal for Chemistry, 61 (2007) 780-786.   

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