## Thermodynamic efficiency of solar concentrators

Optics Express, Vol. 18, Issue S1, pp. A5-A16 (2010)

http://dx.doi.org/10.1364/OE.18.0000A5

Acrobat PDF (962 KB)

### Abstract

The optical thermodynamic efficiency is a comprehensive metric that takes into account all loss mechanisms associated with transferring flux from the source to the target phase space, which may include losses due to inadequate design, non-ideal materials, fabrication errors, and less than maximal concentration. We discuss consequences of Fermat’s principle of geometrical optics and review étendue dilution and optical loss mechanisms associated with nonimaging concentrators. We develop an expression for the optical thermodynamic efficiency which combines the first and second laws of thermodynamics. As such, this metric is a gold standard for evaluating the performance of nonimaging concentrators. We provide examples illustrating the use of this new metric for concentrating photovoltaic systems for solar power applications, and in particular show how skewness mismatch limits the attainable optical thermodynamic efficiency.

© 2010 OSA

## 1. Introduction

*α*is defined as the incidence angle on the entrance pupil for which the concentrator collects 90% of the on-axis power.[2] Although it is a useful merit function in many cases of interest, CAP has a number of limitations. One drawback is that it is only applicable in cases where the desired acceptance solid-angular region is axisymmetric. Another problem is that it is not normalized to unity for ideal concentrators operating under étendue-matched conditions. In fact, it can be shown that the CAP value of such an ideal concentrator is given by the formulawhere

*α*provides a measure of the off-axis flux-transfer relative to the on-axis flux transfer, but does not provide an absolute measure of flux transfer. Consider, for example, two concentrators that are identical except that one has optical coatings with significantly higher efficiency. The two concentrators could have the same CAP value, even though the concentrator with the better coatings transfers significantly more flux from the source to the target.

## 2. Geometrical optics and thermodynamics

*S*is an even-dimensional piecewise differentiable manifold and

*n*( = 2) is the number of generalized coordinates. The starting point for this formulation is the generalization of Fermat's variational principle, which states that a ray of light propagates through an optical system in such a manner that the time required for it to travel from one point to another is stationary. This mapping is purely geometrical and is independent of thermodynamic quantities such as heat or temperature.

*g*be a differentiable mapping. The mapping

*g*is called canonical[3,4] if

*g*preserves the differential 2-form

*w*

^{2}=

*dp*^

_{i}*dq*

_{i}*i*=1..

*n*, where

*q*is the generalized coordinate and

*p*is the generalized momentum. Applying the Euler-Lagrange necessary condition to Fermat's principle and then the Legendre transformation, we obtain a canonical Hamiltonian system, which defines a vector field on a symplectic manifold (a closed nondegenerate differential 2-form). Now, a vector field on a manifold determines a phase flow, i.e., a one-parameter group of diffeomorphisms (transformations which are differentiable and also possess a differentiable inverse). The phase flow of a Hamiltonian vector field on a symplectic manifold preserves the symplectic structure of phase space and consequently is canonical. The properties of these mappings can be summarized as follows:

- 1) The mappings from input phase space to output phase space are piecewise diffeomorphic. Consequently they are one-to-one and onto.
- 2) The transformation of phase space induced by the phase flow is canonical, i.e., it preserves the differential 2-form.
- 3) The mappings preserve the integral invariants, known as the Poincaré-Cartan invariants. Geometrically, these invariants are the sums of the oriented volumes of the projections onto the coordinate planes.
- 4) One of the Poincaré-Cartan invariants preserved by the mappings is the phase-space volume element (i.e., étendue). The volume of
*gD*is equal to the volume of*D*, for any region*D*.

6. J. Bortz, N. Shatz, and R. Winston, “Performance limitations of translationally symmetric nonimaging devices,” Proc. SPIE **4446**, 201–220 (2001). [CrossRef]

8. E. Yablonovitch, “Thermodynamics of the fluorescent planar concentrator,” J. Opt. Soc. Am. **70**(11), 1362–1363 (1980). [CrossRef]

*S*, and the étendue,

*E*, which contains both geometrical and thermal terms:where k is a constant. Setting aside the thermal term, which applies only in the case of a wavelength shift, we see that when considered from the statistical viewpoint, étendue conservation along the path of a beam in transparent media implies the conservation of entropy. The thermal term in the equation extends the case beyond geometrical optics.

## 3. Optical thermodynamic loss mechanisms in PV systems

*thermodynamic*efficiencies, and b) the optical thermodynamic efficiency is just as influential as the cell thermodynamic efficiency when it comes to achieving total system thermodynamic efficiency. There are three primary mechanisms that cause loss of optical thermodynamic performance:

- 1. Losses that occur along the ray paths: coatings, bulk attenuation, scattering, etc. These are energy losses.
- 2. Losses that occur due to ray rejection. This is light that could have been collected by the solar cell but, due to an inadequate optical design, ends up either rejected outside the optical system or absorbed within the optical system at a location other than the solar cell. This type of loss is often caused by inappropriate use of optical components (e.g., utilizing imaging spherical optical surfaces).
- 3. Losses due to étendue dilution. Given the required active area of the solar cell and acceptance angle at the entrance aperture of the optical system, dilution provides a measure of how much additional flux could have been transferred to the cell by utilizing optics of larger entrance-aperture area than the candidate optics. This loss is an indirect loss since it represents the lost opportunity of failing to use a larger entrance aperture and/or a smaller target (i.e., a higher concentration ratio). An associated loss here is that lower concentration ratios are generally associated with lower cell efficiencies. However, it should be noted that cell efficiency does not increase indefinitely with concentration; at present, the maximum conversion efficiencies typically occur for concentration ratios between 300 and 600.

## 4. The optical thermodynamic efficiency

*is not necessarily equal to the target étendue ε*

_{src}*. The source radiance is a function of position within the source’s phase-space volume and is zero outside this volume. For simplicity, we define radiance as the flux per unit phase-space volume, which is sometimes referred to as the generalized radiance. The advantage of using generalized radiance is that it is conserved along any loss-free ray path, even when the refractive index varies along the ray path.*

_{trg}*as the total flux emitted by the source. Similarly, we define the total flux transferred by the optical system from the source to the target as Φ*

_{src}*. We also define the maximum dilution-free target flux Φ*

_{trg}*as the total flux that would be contained by the target phase space if it were to be completely filled with a constant radiance equal to the average source radiance. From the definition of generalized radiance, we find that the average source radiance equals the source flux divided by the source étendue:Based on the definition of the dilution-free target flux, we then find that its value is given by the formulaCombining Eqs. (5) and (6), we find thatIn other words, the maximum dilution-free target flux equals the source flux times the target-to-source étendue ratio. When the target étendue is less than or equal to the source étendue, we define the optical thermodynamic efficiency asDue to étendue conservation, this quantity will always be less than 100% when the target étendue is less than the source étendue. For example, when the source radiance is constant and the target étendue is one fourth of the source étendue, the optical thermodynamic efficiency defined in Eq. (8) will always be less than or equal to 25%. When the target étendue is greater than the source étendue, the thermodynamic efficiency is defined asUsing Eq. (7) we can rewrite this formula asBy inspection we see immediately that this quantity is always less than or equal to the source-to-target étendue ratio. In this case, it is possible for up to 100% of the source flux to be transferred to the target. However, since the target étendue is greater than that of the source, it is not possible to completely fill the target phase space with radiance equal to that of the source. This phenomenon is referred to as étendue dilution. The definition of Eq. (9) accounts for this inability to completely fill the target phase space by dividing the actual flux transferred to the target by the maximum flux that would have been transferred to the target if the target phase space were to be completely filled.*

^{max}_{trg}*a*,

*b*) is equal to

*a*when

*a*≤

*b*and

*b*when

*b*<

*a*. Equation (11) provides a convenient formula for computing the optical thermodynamic efficiency.

## 5. Examples

*= 5° centered on the optical axis. This type of acceptance-angle requirement could be derived, for example, from known alignment and tracking errors. The source and target étendue areandwhere*

_{src}*A*is the source (i.e., entrance-aperture) area and

_{src}*A*is the target area. We also consider a hypothetical ideal concentrator that transfers 100% of the flux from the 40-mm-diameter, 5°-half-angle source to a flat on-axis étendue-matched target. Like the actual target, the étendue-matched target used in defining the ideal concentrator is assumed to absorb flux at all incidence angles relative to its surface normal.

_{trg}*/ε*

_{src}*relative to the source-to-target flux-transfer efficiency.*

_{trg}^{6}on East–west-oriented non-tracking translationally-symmetric solar concentrators. It should be emphasized that we are not here analyzing the performance of any specific concentrator. Instead, we are computing an upper limit on performance for a specific class of concentrator.

*T*(≤ 12 hr), where the time interval

*T*is centered on solar noon. We neglect latitude-dependent shadowing effects by the earth’s horizon, which will in practice limit the operation time at certain times of the year for concentrators operating at high latitudes.

^{6}, it can be shown that 100% flux transfer using a translationally-symmetric concentrator is impossible when the source and target have equal étendue. Instead, the source-to-target étendue ratio has to be less than unity to meet the 100%-flux-transfer-limit requirement. This, in turn, means the optical thermodynamic efficiency must be less than 100%, as is apparent from Eq. (11).

*k*and

_{y}*k*are referred to as the vertical and horizontal direction cosines, respectively. The

_{z}*k*direction cosine is parallel to the symmetry axis. Both of these direction-cosine coordinates are parallel to the plane containing the entrance pupil. The quantities

_{z}*T*= 6 hr. The projected solid angle of the source equals the area of its direction-cosine region. Similarly, the projected solid angle of the target equals the area of its direction-cosine region. To compute the étendue we multiply the projected solid angle by the surface area. For the source, the result iswhere

*A*is the source area (i.e., the entrance-pupil area). The target étendue iswhere

_{src}*A*is the target area. Using the methods described in Ref. 6

_{trg}6. J. Bortz, N. Shatz, and R. Winston, “Performance limitations of translationally symmetric nonimaging devices,” Proc. SPIE **4446**, 201–220 (2001). [CrossRef]

*S*is the translational skew-invariant. Similarly, the translational-skewness distribution of the target isPlots of these two skewness distributions are provided in Fig. 4 for the case of a unit-area source and an étendue-matched target. Since the source’s skewness distribution extends beyond the edge of the target’s skewness distribution, we conclude that it is impossible for an East–west-oriented translationally-symmetric concentrator to transfer 100% of the source flux to the target under étendue-matched conditions. However, by increasing the target area

_{z}*A*we can rescale the target’s skewness distribution of Eq. (25) until it completely contains the skewness distribution of the source, as shown in Fig. 5 . In this case, the translational-skewness limit on the flux-transfer efficiency becomes 100%. Examination of Fig. 5 shows that the desired condition occurs when the two corners (i.e., slope discontinuities) of the source’s skewness distribution touch the target’s distribution. From Eqs. (24) and (25) we find that this condition is expressed by the formulawhich reduces towhere

_{trg}*A*is the minimum required target area for which the translational-symmetry limit on the flux-transfer efficiency is 100%.

_{trg,req}*S*-value for which the source’s skewness distribution is non-zero be less than or equal to the maximum

_{z}*S*-value for which the target’s distribution is non-zero. Using Eqs. (24) and (25), this leads to the requirement thatWhen the daily operation time exceeds this value, the translational-skewness limit on flux-transfer efficiency cannot reach 100% no matter how large the target area. By substituting the values

_{z}## 6. Conclusions

## References and links

1. | W. Spirkl, H. Ries, J. Muschaweck, and R. Winston, “Nontracking solar concentrators,” Sol. Energy |

2. | P. Benitez, and J. C. Miñano, “Concentrator Optics for the next generation photovoltaics,” Chap. 13 of A. Marti and A. Luque, |

3. | V. I. Arnold, |

4. | R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, |

5. | H. Ries, N. Shatz, J. Bortz, and W. Spirkl, “Performance limitations of rotationally symmetric nonimaging devices,” J. Opt. Soc. Am. A |

6. | J. Bortz, N. Shatz, and R. Winston, “Performance limitations of translationally symmetric nonimaging devices,” Proc. SPIE |

7. | L. D. Landau, and E. M. Lifshitz, |

8. | E. Yablonovitch, “Thermodynamics of the fluorescent planar concentrator,” J. Opt. Soc. Am. |

**OCIS Codes**

(000.6850) General : Thermodynamics

(080.0080) Geometric optics : Geometric optics

(350.6050) Other areas of optics : Solar energy

(220.2945) Optical design and fabrication : Illumination design

(080.4298) Geometric optics : Nonimaging optics

**ToC Category:**

Solar Concentrators

**History**

Original Manuscript: December 15, 2009

Revised Manuscript: March 13, 2010

Manuscript Accepted: March 13, 2010

Published: April 26, 2010

**Virtual Issues**

Focus Issue: Solar Concentrators (2010) *Optics Express*

**Citation**

Narkis Shatz, John Bortz, and Roland Winston, "Thermodynamic efficiency of solar concentrators," Opt. Express **18**, A5-A16 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-S1-A5

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### References

- W. Spirkl, H. Ries, J. Muschaweck, and R. Winston, “Nontracking solar concentrators,” Sol. Energy 62(2), 113–120 (1998). [CrossRef]
- P. Benitez, and J. C. Miñano, “Concentrator Optics for the next generation photovoltaics,” Chap. 13 of A. Marti and A. Luque, Next Generation Photovoltaics: High Efficiency through Full Spectrum Utilization, Taylor & Francis, CRC Press, London (2004).
- V. I. Arnold, Mathematical Methods of Classical Mechanics, 88–91 & 161–270, Springer Verlag (1989).
- R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. Bortz, Nonimaging Optics, Elsevier Academic Press, New York (2005).
- H. Ries, N. Shatz, J. Bortz, and W. Spirkl, “Performance limitations of rotationally symmetric nonimaging devices,” J. Opt. Soc. Am. A 14(10), 2855–2862 (1997). [CrossRef]
- J. Bortz, N. Shatz, and R. Winston, “Performance limitations of translationally symmetric nonimaging devices,” Proc. SPIE 4446, 201–220 (2001). [CrossRef]
- L. D. Landau, and E. M. Lifshitz, Statistical Physics, Pergamon, London (1958).
- E. Yablonovitch, “Thermodynamics of the fluorescent planar concentrator,” J. Opt. Soc. Am. 70(11), 1362–1363 (1980). [CrossRef]

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