Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

A fractal concentrator of direct and indirect sunlight

A fractal concentrator of direct and indirect sunlight Abstract A novel concept for concentrating both direct and indirect sunlight employing a combination of waveguiding and refractive optics is presented. More specifically, it is based on both the individual and the collective focusing effects of an array of refractive optical elements of specific geometry. An analytical framework and design principles regarding the geometry of the optical elements taking into account Fresnel reflection and refraction are presented. The principle of operation along with loss mechanisms and acceptance angles are discussed in detail. Ray-tracing simulations of a light concentrator designed accordingly indicate transmittance >90% with acceptance angles of ≤65° at 3 suns for moderate refractive indices. It is further numerically demonstrated that the new approach allows the utilization of the dominant fraction of the solar radiation by harvesting the waste heat in addition to photovoltaic conversion with an operating energy efficiency for diffuse light >70%. Graphical Abstract Open in new tabDownload slide solar concentrator, diffuse light, non-imaging optics, solar harvesting Introduction The flat panel photovoltaic (PV) solar-cell technology is by far the fastest-growing renewable-energy technology today with 584 GW installed cumulative PV capacity by the end of 2019 [1]. Amongst its notable disadvantages is the massive use of semiconductor material in addition to its inability to harvest the waste heat. Concentrating photovoltaics (CPV) holds the promise of reducing the use of semiconductor material in solar panels by 1–3 orders of magnitude in addition to having higher photoelectric-conversion efficiencies. Another promising renewable technology is concentrating solar power (CSP), which employs standard thermomechanical engines to convert sunlight into usable energy. Nevertheless, both the CPV and the CSP technologies have not had significant commercial success despite their obvious advantages. One major obstacle that prevents CPV and CSP from being viable technologies lies in the design and cost of the optical systems used to concentrate solar radiation. Furthermore, optical systems for the most part employ imaging optics that limits the radiation harvested to well-collimated sunlight resulting in the need for accurate tracking of the Sun. Consequently, the extra and considerable cost of a precision tracking system makes current CPV and the CSP techniques economically uncompetitive. For recent reviews on the subject, the reader is referred to references [2–7]. Thus, one pressing challenge to address is the very limited acceptance angle of existing solutions. A common figure of merit used for comparative purposes is the Concentration-Acceptance angle product (CAP) defined as the product between the square root of geometrical concentration and the sine of the acceptance angle. For state-of-the-art concentrators, the value of CAP [8] lies below 1. Thus, it is the objective of this work to suggest a new concept for designing solar concentrators that exhibit large acceptance angles, high transmittance and high CAP values in the low-concentration range. 1 Analytical framework The theoretical framework is based on earlier work [9]. Central to it is the geometry of refractive optical elements (OEs) that have the property of projecting a totally diffuse light flux onto a finite area in the focal plane. Fig. 1 illustrates in 2D one such example representing an optical element in the form of a 2D triangular cone (pyramid) made of an optical material with a refractive index n1 surrounded by a second optical material of optical index n2 < n1, say, glass n1 = 1.5 and air n2 = 1 for specificity. We also denote by θc = asin(n2/n1) the critical angle of total internal reflection (TIR) under these conditions. It is assumed that the front surface is illuminated with a diffuse light flux of uniform angular distribution. Thus, ray r1 is incident at an arbitrary point P1 on the front surface under an arbitrary angle θi with respect to the normal k1. For clarity of the presentation, we initially consider the propagation of the refracted component. Thus, the refracted ray r2 at point P1 traverses the bulk of the cone and collides with the sidewall at point P2. At this stage, ray r2 has two possible trajectories, depending on the magnitude of the incidence angle γi. Thus, if γi < θc, ray r2 is refracted under an angle γr into ray r3, which then is supposedly projected onto the focal plane. Rays that undergo refraction during the first collision with a sidewall are called extrinsic. We are interested in the maximum divergence ψextr of ray r3 with respect to the negative z-direction and which divergence is attained when the value of γi is minimum. The angle γi is a decreasing function of θi and hence its minimum value is attained when θi = π/2, i.e. γi = π/2 – θc – a/2. Thus, from Snell’s law, it follows that: Fig. 1: Open in new tabDownload slide Propagation of extrinsic and intrinsic rays Ψextr =π/2−γr−α/2=π/2−asin[cos(θc+α/2).n1/n2]−α/2(1) where α is the apex angle of the cone. Further, ψextr is an increasing function of α and attains its maximum value of π/2 at α = π. Hence, extrinsic rays have a maximum divergence ψextr < π/2 irrespective of their initial angle of incidence θi and are, therefore, always projected onto a limited area in the focal plane. Considering now the case in which γi > θc, ray r2 undergoes total internal reflection at point P2, propagates as ray r4, which in turn collides with the opposite wall at point P3 under an angle γt. Rays that undergo TIR during the first collision with a sidewall are called intrinsic. From simple geometrical considerations, it follows that γt = γi – α. In other words, the angle of incidence decreases monotonically by an amount equal to the apex angle α with each successive collision. This situation is identical to the so-called escape-cone effect in which rays propagating in a conically shaped mirror lose forward component until it is reversed and they get rejected out of the cone back to the source. In our case, however, γt eventually becomes smaller than θc and hence ray r4 gets refracted as ray r5 out of the cone and is supposedly projected onto the focal plane. Without loss of generality, we assume that this takes place at point P3. In fact, the above observation defines in effect a forbidden zone for intrinsic rays whose magnitude of the vertical component is smaller than |–sin(θc+α/2)|, since such rays are immediately refracted, i.e. their further propagation within the cone through TIR is forbidden. Here, again, we are interested in the maximum divergence ψintr of intrinsic rays. From the above observation, it follows that the smallest possible incidence angle of an intrinsic ray when propagating inside the optical element is γt = θc – α. Thus, from Snell’s law, the maximum divergence ψintr is given by: Ψintr=π/2−γs−α/2=π/2−asin[sin(θc−α).n1/n2]−α/2(2) Here, again, ψintr is an increasing function of α and attains its maximum value of π/2 at α = π. Hence, intrinsic rays have a maximum divergence ψintr < π/2 irrespective of their initial angle of incidence θi and are, therefore, always projected onto a limited area in the focal plane. The immediate observation from Equations (1) and (2) is that the smaller the apex angle, the smaller the divergences of both intrinsic and extrinsic rays. Thus, the apex angle α along with the optical indices n1 and n2 provide the possibility of tailoring ψintr and ψextr at will through Equations (1) and (2). A further useful relation is the condition for the existence of extrinsic rays: β≤θi≤ π/2, whereβ =asin[cos(θc + α/2).n1/n2](3) Thus, extrinsic rays occupy the glancing incidence range and only exist if β is real. Intrinsic rays, however, always exist. Fig. 2 illustrates graphically the results so far. Specifically, the maximum divergence angle of all rays is given by ψmax = max(ψextr,ψintr), i.e. rays r1 and r2 illustrate graphically the maximum divergence range. Thus, the diffusely illuminated front surface is imaged onto the focal plane onto a limited area, called a ‘diffuse image’. Fig. 2: Open in new tabDownload slide Definition of ‘diffuse image’ Further, it is readily shown that Equation (1) (extrinsic case) is equally valid in 3D. The same, however, is not true for Equation (2) (intrinsic case), since the ray’s vertical component and the angle of incidence are now decoupled. The ray’s vertical component still declines monotonically in magnitude as the ray descends through the optical element while a monotonic decrease of the angle of incidence cannot be guaranteed in 3D and hence Equation (2) is not valid. Further, the three-dimensionality of the problem results in that ray exit from the pyramid is a stochastic event, as illustrated in Fig. 3a, which represents the so-called refraction cone having an apex angle of 2θc. The axis a1 of the cone is perpendicular to the sidewall and is always constructed at the impact point of a ray. Rays incident on the inside of the refraction cone (say, ray r1) are refracted, whereas rays incident on the outside (say, ray r2) are internally reflected. Hence, rays are ejected out of the pyramid with a probability proportional to 2θc. Analogously to the 2D, we want to make this probability equal to 1, which is done as follows. Fig. 3b represents an arbitrary cross-section Γ of the wedge formed between two neighbouring sidewalls such that the plane Γ is perpendicular to the edge μ (the latter assumed parallel to the z-axis for simplicity of the treatment). We follow the propagation of all TIR rays in the plane Γ incident from left to right at an arbitrary point P1, lying on the intersection line between sidewall 1 and the plane of incidence Γ. Note that the cone axis a1 is perpendicular to sidewall 1, i.e. a1 lies in Γ and points to the inside of the wedge. Specifically, we construct rays r1 and r2 incident at point P1 and lying in the plane Γ such that r1 is tangential to cone C1 while r2 lies on the intersection line between Γ and sidewall 1. Clearly, r1 and r2 define the range R of all rays incident from left to right at point P1 and lying in Γ that undergo TIR. Their reflections r’ 1 and r’ 2 impact with sidewall 2 at points P2 and P3, respectively. We further assume that the edge angle ζ lies in the range: Fig. 3: Open in new tabDownload slide (a) The refraction cone. (b) TIR in a horizontal plane of incidence. (c) TIR in a rotated plane of incidence. π/2−θc ≤ ζ ≤ 2θc(4) which guarantees that both r’ 1 and r’ 2 are refracted—that is, they both enter their respective refraction cones C2 and C3 on the inside. The rays in the range R are also internally reflected and projected onto the line segment Σ between points P2 and P3. Since r’ 1 and r’ 2 fall within their respective cones, so do all rays in the range R’ between them—that is, they all get refracted. We now start rotating anticlockwise the plane Γ along with all TIR rays considered about axis a1 as illustrated in Fig. 3c. In the process, the entry points of r’ 1 and r’ 2 in the refractive cones C2 and C3, respectively, start drifting towards the edges of the latter until at least one of them, say r’ 1, becomes tangential to its cone C2, as illustrated in Fig. 3c. We denote the angle of rotation in the plane of sidewall 1 when this happens as ω. Clearly, any further incremental increase of ω would result in ray r’ 1 ending up outside cone C2 and hence it would not be refracted. We further denote the z-component of r’ 1 at this point by –sin(θcut-off). Hence, all rays incident at point P1 from left to right and having a z-component in the range (–sin(θcut-off),0) are refracted or, in other words, ejected from the optical element. By virtue of the symmetry of the construction, the same is true for TIR rays incident from left to right at an arbitrary point on sidewall 2. Therefore, all TIR rays entering the edge and having a z-component in the range (–sin(θcut-off),0) are refracted. In other words, the edge angle ζ defines a bandgap that forbids the further propagation of such rays within the optical element, since they all get refracted. Note that the bandgap is associated with the edge. Clearly, the maximum angle of rotation ω and, hence, maximum bandgap width are achieved when r’ 1 and r’ 2 become simultaneously tangential to their respective cones C2 and C3. Thus, from standard geometrical considerations, it follows that the optimal angle ζopt when this occurs is given by: ζopt=acos[1−sin(θc)](5) Further, by the same token, the angle of rotation ω(θc,ζ) is calculated as: ω(θc,ζ)=asin[sin(θc)1−cot2(θc)cot2(ζ)], for π/2−θc≤ζ≤ ζopt(6) and ω(θc,ζ)=acos[cot(θc)1−cos(ζ)sin(ζ)], for ζopt≤ζ≤2θc(7) The right-hand sides of Equations (6) and (7) render an identical function ωopt(θc) at ζ = ζopt but, other than that, they have to be used in their respective ranges of validity above. Further, having calculated ω, the bandgap width θcut-off is readily calculated as: θcut~-off=asin[sin(θc)sin(ω)](8) Clearly, the angles θcut-off and ω define the z-components (r’ 1)z = –sin(θcut-off) and (r’ 2)z = –sin(ω). The choice of θcut-off as the bandgap width (and not the angle of rotation ω) was intentional, since this choice guarantees that all rays entering the edge ζ, irrespective of their horizontal components and having a magnitude of the z-component smaller than sin(θcut-off), are refracted. As an illustration, assuming n1 = 1.5 and n2 = 1, i.e. θc = 41.81o, Equations (5), (6) and (8) yield ζopt = 70.53o, ωopt = 37.76o and θcut-off = 24.09o, respectively. It is also noted that ω(θc,π/2 – θc) ≡ω(θc,2θc) ≡ 0, i.e. the bandgap width is zero at these points. We reiterate the requirement that the area of the cross-section Γ of the optical elements decreases along the negative z-axis (cone-like shape of the optical elements), which condition guarantees that TIR rays continually lose forward (vertical) component until their magnitudes fall within the bandgap of an edge and are consequently refracted. Hence, in analogy with Equation (2), one upper limit for the divergence of intrinsic rays in the 3D case is given by: Ψintr = π/2−γs−α/2 = π/2−asin[sin(θcut~-off−ε).n1/n2]−α/2(9) since the minimum incidence angle at the point of refraction is γt = (θcut-off – ε), where ε(α) defines the maximum angular decrement in the magnitude of the ray’s vertical component between two successive collisions. Clearly, in 3D, ε < α, where α/2 defines the slope of the sidewalls through cot(α/2). Here, again, ψintr < π/2. At this point, a small but important clarification is in order. In the above reasoning, we implicitly assumed that the vertical component of any given TIR ray (which has not refracted stochastically prior to that) will eventually fall within the bandgap. This is only true if the bandgap width θcut-off is larger than ε. Reiterating that ψintr is an increasing function of α and a decreasing function of θcut-off, it is obvious that ψintr is minimal when the difference (θcut-off – ε) is as large as possible. Hence, we arrive at the following relations: 0 < α ≪ θcut~-off(10) In other words, the bandgap width is to be much larger than twice the slope of the sidewall. This is called the apex criterion and its fulfilment guarantees that the vertical component of all unrefracted TIR rays does eventually enter the bandgap and they are all subsequently refracted out of the cone at minimal angles ψintr with respect to the cone axis. Closer inspection of inequalities (10) reveals that they implicitly contain inequalities (4), since, according to inequalities (10), θcut-off must be greater than zero, which is fulfilled if and only if inequalities (4) are satisfied. For practical purposes, the apex criterion for a given combination (n1, n2) can be approximated by the ratio between the height and the average lateral dimension of the front surface of the optical element, which is to be >10 for the cases considered below. Thus, it follows from the above discussion that the apex criterion suffices to guarantee that all TIR rays are eventually refracted out of the optical element in an orderly and optimal manner, and hence the primary escape-cone effect with regard to TIR rays is totally suppressed. In this way, we proved in 3D too that both extrinsic and intrinsic rays can be projected onto a finite area in the focal plane, irrespective of their initial angle of incidence onto the front surface. Fig. 4 represents an experimental confirmation of the above analysis. Specifically, it shows a triangular pyramid made of glass with a front surface 1 × 1 × 1 cm and a height of 15 cm. Under these conditions, all rays entering the pyramid through the front surface are intrinsic (see Equation (3) for very small values of α). The front surface is illuminated by a rastered laser beam under an angle of ~45o. Comparison between Fig. 2 and Fig. 4 identifies the ‘diffuse image’ in the focal plane formed by intrinsic rays. Fig. 4: Open in new tabDownload slide Experimental verification of ‘diffuse image’ 2 Principle of operation For reasons apparent from the above discussion, we consider the case involving intrinsic rays only. This provides the first focusing mechanism since all rays initially travel downwards through TIR through an OE of a decreasing cross-section. The latter, in addition, provides one further focusing mechanism in a diffuse-light concentrator by allowing the OE to be brought in close proximity to each other but still separated by narrow gaps of a lower-refractive-index optical material such that their respective ‘diffuse images’ overlap to a great extent, as illustrated in Fig. 5a. Due to this overlap, the area of the ‘Exit aperture’ is smaller than that of the ‘Input aperture’, which manifests the second confinement mechanism in the concentrator. At the same time, the plurality of the optical elements thus formed gives rise to collective effects, as follows. We present here a qualitative description only. Considering, say, ray r5 in Fig. 1, it is clear that it is ‘discharged’ in the midst of a laterally reflective grating formed by the multiple gap/sidewall interfaces between neighbouring OEs, as illustrated in Fig. 5b. The point at which the ray exits the OE is called a ‘burst point’, since, at this point, the ray bifurcates through a concurrent reflection/refraction event into two fractal rays. Thus, ray r4 splits into one refracted ray r5 and one reflected ray r6. Further, ray r5 collides with the sidewall of a neighbouring optical element and bifurcates into fractal rays r7 and r8, respectively. Analogously, ray r7 bifurcates into rays r9 and r10. The above process takes place at each and every interface, generating in this way an avalanching cascade of fractal rays. Clearly, the lateral components of both rays r8 and r10 are opposite to that of ray r5. Hence, each optical element returns (reflects back) a fraction of the energy emanating from a neighbouring optical element. At the same time, the intensity of ray r9 is smaller than that of r5. Thus, this mechanism eventually results in a skew of the light intensity over the ‘diffuse image’ of a given optical element towards its centre. Figuratively said, the gap/sidewall interfaces act as generators of fractal rays and, in doing so, they define at the same time a lateral fractal reflector that restricts the lateral flow of energy. Consequently, the fractal cascade of rays drifts downwards towards the exit aperture while being at the same time laterally confined by the lateral fractal reflector. Since the final stage of ray propagation is effected through a fractal cascade, such concentrators are called fractal. On the other hand, it can be readily shown that, in the limit of zero gap width, each reflection at any face is equivalent to an escape-cone collision event in terms of loss of the forward (axial) component. In other words, the forward propagation of all fractal rays within the cascade is equivalent to that of a ray propagating through an OE (conical waveguide) with ideally reflecting walls. Thus, all rays eventually lose the forward component and are rejected. Therefore, the focal plane is positioned at a distance from the front surface that is covered by the majority of rays incident onto the front surface within a predefined angular range which in fact defines the acceptance angle. Consequently, the OEs are truncated at the focal plane. It should be pointed out that the gap width is non-zero, which results in a downward drift of refracted rays through the gaps. Thus, assuming that the neighbouring faces are parallel to each other, then ray r4 is parallel to r7 but the latter is somewhat displaced downwards, which is also called gap displacement or gap drift. Hence, the finite gap width results in an additional downward drift of traversing rays. Fig. 5: Open in new tabDownload slide (a) Primary focusing mechanism. (b) Illustration of ray bifurcation. (c) Illustration of fractal cascade. (d) Illustration of TIR trapping. (e) Additional focusing by peripheral mirrors. Fig. 5c illustrates the above description through standard ray tracing. Specifically, it shows the propagation and subsequent fractal cascade formation of a ray incident at an angle of 60° onto the front surface of an optical element in a concentrator consisting of 150 pyramidal OEs having a triangular front surface of 12 × 12 × 12 mm and a height of 250 mm where the focal plane is positioned at a distance of 105 mm from the front surface. To avoid clutter, the vertical edges of the optical elements are not drawn. Their silhouettes, however, are visually outlined by deleting the ray trajectories within the gaps and aligning the viewing angle with the gap planes. In addition, only rays with intensity above a certain limit are drawn for clarity. Thus, the ray enters the top surface and propagates downwards through the optical element through TIR while continually losing the vertical component until, stochastically, the condition for TIR is disobeyed at a particular point (denoted as ‘stochastic burst’ in Fig. 5c), after which the fractal cascade develops according to the description above. In other words, from that point onwards, the fractal rays typically propagate through constant bifurcation within the lateral reflector. One important detail, though, requires a closer look. As noted earlier, the burst point occurs through a probabilistic process and hence its occurrence is not guaranteed. Further, its occurrence does not in any way guarantee that the fractal cascade develops fully. Thus, one or both fractal rays after the point of a stochastic burst may become trapped in neighbouring optical elements and continue their propagation through TIR, and so on. Whatever the sequence of events, however, any ray propagating within the concentrator continually loses vertical direction until the latter falls within the bandgap of an edge, which in turn is guaranteed for all rays by the design chosen, i.e. an adequate apex criterion. Rays may still get TIR trapped by the cooperative action of neighbouring OEs due to the symmetry of the structure. Nevertheless, this type of TIR trapping always involves interface crossing and hence fractal rays always experience the gap drift. Notably, such TIR trapping, if not dominating, is a very common occurrence, as illustrated in Fig. 5d, which shows the propagation of the same ray as in Fig. 5c but with a slightly rotated azimuth. One final flux confinement, however trivial, is readily achieved by the addition of peripheral mirrors, as illustrated in Fig. 5e. These mirrors play a 2-fold role. First, they define the spatial dimensions of the concentrator and thus restrict the sideways drift of the fractal rays by reflecting them back into the core of the concentrator. Second, the lower aperture of the mirrors defines strictly the exit aperture. Thus, the ratio between the input and exit apertures of the mirror structure defines the geometrical concentration coefficient Cg. 3 Major loss mechanisms We now consider typical loss mechanisms in the light concentrator excluding the primary escape-cone effect discussed above, since it has been totally suppressed in the design through inequalities (4). These losses are divided into two categories: propagation and rejection losses, respectively. The propagation losses result in energy deposition in the concentrator, whereas rejection losses represent the light intensity returned back to the ambient. The first category includes mainly absorption losses along the propagation path of the rays and absorption losses in the peripheral mirrors. Noteworthy, the propagation losses are dispersive, i.e. a function of the wavelength, as are the indices of refraction, for that matter. All these are materials- and technology-related issues common to all optical systems and are outside the scope of the current presentation. As for the rejection losses, these include reflection losses from the top surface, light scattering in the bulk and from interfaces due to surface roughness, impurities, particle inclusions, contaminations, etc. One type of rejection loss, i.e. the so-called secondary escape-cone effect, is always present for fractal rays propagating within the concentrator through bifurcation. Referring to Fig. 5b, we consider the fractal ray r6 and its subsequent reflections off the sidewall (not drawn) that remain within the optical element. Clearly, the intensity of these rays decreases exponentially, since it decreases proportionally to the reflection coefficient at each and every collision with the sidewalls. More important for the argument is the fact that the lateral component of successive reflected rays gradually increases due to the geometrical escape-cone effect until it is reversed and the residual intensity at this stage, however small, is rejected. This is a manifestation of the so-called secondary escape-cone effect and refers to a fractal ray propagating exclusively within any given conical structure. Virtual cones can be defined by any arbitrary combination of (partially) reflecting surfaces in the concentrator, including the peripheral mirrors, thus increasing the possible rejection paths. In other words, this rejection mechanism exists within any virtual cone and not within an optical element only. Fortunately, as noted above, its intensity diminishes exponentially and, with careful design, the net effect can be made sufficiently low through strengthening the apex criterion, i.e. by increasing the ratio between the OE’s height and its lateral dimension. Optimal performance is achieved for a ratio for which the secondary escape-cone effect is practically suppressed within the transmission range of the concentrator and is typically of the order of 10. Much higher ratios result in increased absorption losses with little additional benefit. The secondary escape-cone effect, however, becomes dominating around the acceptance angle. In other words, this is the point at which the radial velocity of the fractal rays becomes very significant. Light entering the concentrator through the gaps between the optical elements is obviously not concentrated and thus the net surface area of the gaps needs to be taken into account when calculating or measuring the optical efficiency. 4 Concentrator modelling As an illustration of the performance of the concentrator, we now present selected numerical modelling employing a ray-tracing algorithm based on strict Fresnel reflection/refraction at each and every interface. Reflection off the top surface is initially disregarded but added in the final calculations. We limit the discussion to low concentrations in which the acceptance angle is very large and its decline around the acceptance angle is not very steep, for which reason we refer to the angular dependence of the transmission coefficient as peripheral vision. The concentrator simulated consists of 150 optical elements representing triangular pyramids with a front surface of 12 × 12 × 12 mm, a height of 250 mm and a refractive index of 1.583. Truncation is applied at a height corresponding to the desired geometrical concentration. Thus, their height at Cg = 3 is 105 mm and the gap width is 0.2 mm. The optical-absorption coefficient of the air gaps is assumed to be zero, while that of the optical elements is equal to 0.004 cm–1. The peripheral mirrors are assumed to be made of aluminium. The reflection coefficient for Al is taken from Hébert [10] assuming a refractive index of 1.3 and an extinction coefficient of 7.46 for a wavelength of 619 nm. All calculations have been done with a precision guaranteeing that ≥99% of the energy in a cascade is traced to the end. Fig. 6 shows the calculated performance at a geometrical concentration Cg = 3 for light incident at the centre of the concentrator. Specifically, Fig. 6a shows the transmission, absorption and rejection coefficients as a function of the angle of incidence θ. The calculations are aggregated for 100 rays incident at random points of incidence with a full azimuthal scan. As seen, the transmission coefficient τ(θ) remains well above 90% up to a 65o angle of incidence. Fig. 6b shows the angular dependence of the transmittance η(θ) = τ(θ)[1 – R(θ)]—that is, the corrected transmission by subtracting the reflected fractional intensity R(θ) at the front surface. Assuming a uniform angular distribution of the incident flux, we define the total power density Φo of diffuse light incident onto a unit area parallel to the surface of Earth as: Fig. 6: Open in new tabDownload slide (a) Angular dependence of transmission and loss coefficients at Cg = 3. (b) Transmittance at Cg = 3. Φo=∫π/20A.cos(ϑ)dϑ=A(11a) where A is an arbitrary irradiance constant. Analogously we define the energy (exergy) efficiency Eeff of the concentrator for diffuse light as: Eeff=∫π/20[A.cos(ϑ)η(ϑ) Φ o]dϑ=0,8456(11b) Fig. 6b shows also the transmittance of an OE with ideally reflecting walls. The ideal OE has otherwise the same geometrical dimensions and absorption as the OE in the concentrator at the same Cg. It is seen that the peripheral vision of the ideal OE is seemingly somewhat lower than that of the concentrator. The explanation is that the concentrator has a slightly lower optical concentration coefficient due to the finite gap widths, which effectively reduces the input aperture. On the other hand, the effect of gap drift discussed above to which all fractal rays are subjected through the bandgap ejection results in a wider peripheral vision. Thus, the bandgap ejection makes use of the gap widths to extend the peripheral vision to the limit. Fig. 7a and b show the same calculations for a concentration Cg = 4, where the energy efficiency Eeff = 74%, which reflects the obvious fact that the peripheral vision is a decreasing function of Cg. An interesting case is presented in Fig. 8a and b, which illustrate the performance of the same concentrator with OEs of the refractive index of water, 1.33, at a concentration Cg = 2.5. The energy efficiency in this case Eeff = 82%. Thus, this suggests that OEs made of water in combination with anti-freezing agents may potentially be used in low-end applications. The obvious conclusion, however, is that the higher the refractive index, the higher the performance following standard etendué conservation. More importantly, the above findings demonstrate that the performance of fractal concentrators is practically identical to a conical waveguide with ideally reflecting walls and the same absorption with the same absorption. Fig. 7: Open in new tabDownload slide (a) Angular dependence of transmission and loss coefficients at Cg = 4. (b) Transmittance at Cg = 4. Fig. 8: Open in new tabDownload slide (a) Angular dependence of transmission and loss coefficients at Cg = 2.5 and n = 1.33. (b) Transmittance at Cg = 2.5 and n = 1.33. It should be noted that the overall peripheral vision of the concentrator is notably broader than calculated above, since the concentrator itself occupies a sizeable solid angle, typically 20–30° from normal incidence. In other words, there are always OEs that have a smaller incidence angle than that of the central OE for non-normal incidence up to the angle at which mutual shadowing between neighbouring concentrators occurs. Finally, the gap widths take up a few percent of the input aperture, which needs to be included in the overall performance of the concentrator. The mirror losses have been found to be negligible for all of the above cases. The average mirror losses of rays entering the concentrator through OEs at its very periphery are calculated to be <2%. Thus, the average mirror losses for the whole concentrator are well below 1%, for which reason they are not included in the above calculations. 5 Fabrication and applications The optical elements and other elements may be fabricated of low-cost and recyclable polymer materials of high optical quality such as PMMA, PC, etc, say, by injection moulding and other similar methods. The gaps between the optical elements are defined by appropriate spacers, as illustrated in Fig. 9. The exit aperture, along with the top protective screen, encapsulate the concentrator and provide additional thermal insulation at the same time. The absorber/solar cell is in intimate contact with the exit aperture and is optically matched to the latter. The whole concentrator is assembled in a dust-free environment. Further, since the absorption losses are sufficiently low, fractal solar concentrators may advantageously be thermally insulated from the ambient, as illustrated in Fig. 9. Thus, the concentrated light flux is fed onto a light absorber, which may be a solar cell designed for the intended light concentration or just a light-to-heat converter. In either case, the absorber is thermally insulated from the ambient meaning that most of the heat generated will be transferred to the heat-carrier fluid. In other words, deducting the obvious losses, the remainder of the energy is transformed into usable forms of energy. These types of systems are called concentrating photovoltaic–thermal (CPVT) systems. In practice, arrays of solar concentrators such as that shown in Fig. 9 are to be assembled into panels with a common heat-carrier system thermally insulated from the ambient. Generally, both consumer and industrial applications in the form of CPVT and CSP power-generation systems are feasible. The heat generated in CPVT systems is generally rated as low-grade, since the temperature in the carrier fluid is limited by the maximum operating temperature of the solar cells, typically <90oC, and is suitable for household use, greenhouse heating, etc. Where suitable, a heat reservoir may be used in conjunction for short-term heat storage. Not least, a combination of CPVT and a Sterling engine would increase the electrical efficiency in the warmer months. A heat exchanger is needed in the daytime to dump excess/waste heat to the ambient as well as during the night-time to cool the heat reservoir for daytime cooling purposes. A further possibility is to use the excess heat in the warmer months for absorption cooling. Medium-grade-heat CSP systems may be possible if the thermal insulation is improved. Water desalination is a likely application too. Building integrated solar is another possible application where panels of solar concentrators are to be integrated with the building facades making them an integral part of the building insulation and energy system. Fig. 9: Open in new tabDownload slide Thermally insulated concentrator with active cooling 6 Discussion Certain justification for the applications proposed above is in order. To this end, we consider the example in Fig. 6a. Assuming thermal conductivity of 0.2 W/K/m (typical for plastics) and a height of the OE of 10 cm as well as a temperature difference between the heat-carrier fluid and the ambient of 60oC, we arrive at a thermal loss of 120 W/m2. This loss is then divided by Cg, since the emitting surface now is Cg times smaller. Hence, we have an effective thermal loss of 40 W/m2. Assuming further spring irradiance of 250 W/m2 as well as an energy efficiency Eeff of the concentrator of 80%, we arrive at an operating energy efficiency of 64% under the above conditions. This figure becomes 76% at solar irradiance of 1000 W/m2. In reality, one would vary the temperature difference within certain limits to maintain optimal operating efficiency. The example in Fig. 8 would certainly have an inferior performance due in part to the lower Cg and in part to the higher thermal conductivity of water compounded with convection losses. Nevertheless, this case, perhaps, is not to be dismissed, particularly in smaller geographical latitudes. Further, calculating the CAP FOM for the concentrators in Figs 6 and 7, we arrive at CAP >1.5. One might question the use of CAP FOM in this case but nevertheless it provides a qualitative comparison with the current state of the art. As stated above, the fractal concentrator performs at the limits of what is possible with ideally reflecting concentrating waveguides. This performance is achieved partly by the fractal cascade and to some extent by the bandgap, both of which extend the peripheral vision to the physical limit. Further examples of bandgap use for optimizing performance while minimizing cost will be presented in coming communications. Next, it is thought that spectral uniformity is not an issue with fractal concentrators despite the fact that the refractive index is dispersive. If the latter makes any difference, it would be at the edge of the peripheral vision, which constitutes a negligible fraction of the total energy harvested. The spatial uniformity of the concentrated flux has not been studied, although it is also not thought to be an issue too due to the stochastic nature of concentration and the lack of preferred focusing mechanisms. Concentrator transmission, though, may be spectrally sensitive since both the refractive index and absorption are dispersive. The absorption dependence, however, is readily calculated, since it is additive in nature. Finally, light concentration means that one uses correspondingly smaller (in area) solar cells. In view of the low concentration coefficients, however, it would hardly make any economic sense to use fractal concentrators in PV applications alone. The situation changes radically when bi- and tri-generation systems are considered, since now one can harvest the dominant fraction of the available solar energy and convert it into usable forms of energy. 7 Conclusions A novel approach to concentrating both direct and indirect light based on the individual and collective confinement properties of an array of refractive optical elements has been presented. A theoretical framework relating the geometry of the optical elements to their confinement properties has been developed. Consequently, the principle of operation and major design rules of a light concentrator have been established. The main mechanisms of light propagation through the concentrator as well as associated losses have been described in detail. The concentrators are shown numerically to exhibit acceptance angles for CAP FOMs of the order of 1.5. The main advantage of the proposed concentrator is when it is used in bi- and tri-generation solar systems. Thus, it is numerically demonstrated that such concentrators may be used in a range of applications such as CPVT and CSP with operating energy efficiencies >70%. It is also concluded that materials with higher refractive indices yield higher performance. Current polymer materials in this context have moderate refractive indices and low-cost optical materials with higher refractive indices are needed to increase concentration and, hence, performance as well as to reduce material use. Acknowledgments The publication of this article has been sponsored by Vinnova through contract 2019–04927. The work is done under contracts VK-1931:02 and RK-1932:08. Tribute: In memory of George Carter and Mike J. Nobes. Conflict of Interest None declared. References [1] Fraunhofer Institute for Solar Energy Systems . Photovoltaics Report . https://www.ise.fraunhofer.de/content/dam/ise/de/documents/publications/studies/Photovoltaics-Report.pdf ( 12 August 2020 , date last accessed). [2] Boretti A , Castelletto S, Al-Zubaidy S. Concentrating solar power tower technology: present status and outlook . Nonlinear Engineering 2019 , 8 : 10 - 31 . Google Scholar Crossref Search ADS WorldCat [3] Dixit S . Solar technologies and their implementations: a review . Materials Today: Proceedings 2020 , 28 : 2137 - 48 . Google Scholar Crossref Search ADS WorldCat [4] Gonzalo AP , Marugán AP, Márquez FP. A review of the application performances of concentrated solar power systems . Appl. Energy 2019 , 255 : 113893 . Google Scholar Crossref Search ADS WorldCat [5] Indira SS , Vaithilingam CA, Chong KK, et al. A review on various configurations of hybrid concentrator photovoltaic and thermoelectric generator system . Solar Energy 2020 , 1;201 : 122 - 48 . Google Scholar Crossref Search ADS WorldCat [6] Jia Y , Alva G, Fang G. Development and applications of photovoltaic–thermal systems: a review . Renew Sust Energ Rev 2019 , 102 : 249 - 65 . Google Scholar Crossref Search ADS WorldCat [7] Răboacă MS , Badea G, Enache A, et al. Concentrating solar power technologies . Energies 2019 , 2 : 1048 . Google Scholar Crossref Search ADS WorldCat [8] Buljan M , Mendes-Lopes J, Benítez P, et al. Recent trends in concentrated photovoltaics concentrators’ architecture . J Photonics Energy 2014 , 4 : 040995 . doi:10.1117/1.JPE.4.040995. Google Scholar Crossref Search ADS WorldCat [9] Katardjiev I . Optical system for manipulation and concentration of diffuse light . World Patent WO 2020/139186 A1. 7 July 2020 . Google Scholar OpenURL Placeholder Text WorldCat [10] Hébert M. Reflection and transmission of light by a flat interface, Fresnel’s formulae . Ph.D. Thesis. Paris, France : Institut d’Optique ParisTech , 2013 . http://paristech.institutoptique.fr/site.php?id=797&fileid=11468 ( 12 August 2020 , date last accessed). Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC © The Author(s) 2020. Published by Oxford University Press on behalf of National Institute of Clean-and-Low-Carbon Energy This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com © The Author(s) 2020. Published by Oxford University Press on behalf of National Institute of Clean-and-Low-Carbon Energy http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Clean Energy Oxford University Press

A fractal concentrator of direct and indirect sunlight

Clean Energy , Volume 4 (4) – Dec 31, 2020

A fractal concentrator of direct and indirect sunlight

Clean Energy , Volume 4 (4) – Dec 31, 2020

Abstract

Abstract A novel concept for concentrating both direct and indirect sunlight employing a combination of waveguiding and refractive optics is presented. More specifically, it is based on both the individual and the collective focusing effects of an array of refractive optical elements of specific geometry. An analytical framework and design principles regarding the geometry of the optical elements taking into account Fresnel reflection and refraction are presented. The principle of operation along with loss mechanisms and acceptance angles are discussed in detail. Ray-tracing simulations of a light concentrator designed accordingly indicate transmittance >90% with acceptance angles of ≤65° at 3 suns for moderate refractive indices. It is further numerically demonstrated that the new approach allows the utilization of the dominant fraction of the solar radiation by harvesting the waste heat in addition to photovoltaic conversion with an operating energy efficiency for diffuse light >70%. Graphical Abstract Open in new tabDownload slide solar concentrator, diffuse light, non-imaging optics, solar harvesting Introduction The flat panel photovoltaic (PV) solar-cell technology is by far the fastest-growing renewable-energy technology today with 584 GW installed cumulative PV capacity by the end of 2019 [1]. Amongst its notable disadvantages is the massive use of semiconductor material in addition to its inability to harvest the waste heat. Concentrating photovoltaics (CPV) holds the promise of reducing the use of semiconductor material in solar panels by 1–3 orders of magnitude in addition to having higher photoelectric-conversion efficiencies. Another promising renewable technology is concentrating solar power (CSP), which employs standard thermomechanical engines to convert sunlight into usable energy. Nevertheless, both the CPV and the CSP technologies have not had significant commercial success despite their obvious advantages. One major obstacle that prevents CPV and CSP from being viable technologies lies in the design and cost of the optical systems used to concentrate solar radiation. Furthermore, optical systems for the most part employ imaging optics that limits the radiation harvested to well-collimated sunlight resulting in the need for accurate tracking of the Sun. Consequently, the extra and considerable cost of a precision tracking system makes current CPV and the CSP techniques economically uncompetitive. For recent reviews on the subject, the reader is referred to references [2–7]. Thus, one pressing challenge to address is the very limited acceptance angle of existing solutions. A common figure of merit used for comparative purposes is the Concentration-Acceptance angle product (CAP) defined as the product between the square root of geometrical concentration and the sine of the acceptance angle. For state-of-the-art concentrators, the value of CAP [8] lies below 1. Thus, it is the objective of this work to suggest a new concept for designing solar concentrators that exhibit large acceptance angles, high transmittance and high CAP values in the low-concentration range. 1 Analytical framework The theoretical framework is based on earlier work [9]. Central to it is the geometry of refractive optical elements (OEs) that have the property of projecting a totally diffuse light flux onto a finite area in the focal plane. Fig. 1 illustrates in 2D one such example representing an optical element in the form of a 2D triangular cone (pyramid) made of an optical material with a refractive index n1 surrounded by a second optical material of optical index n2 < n1, say, glass n1 = 1.5 and air n2 = 1 for specificity. We also denote by θc = asin(n2/n1) the critical angle of total internal reflection (TIR) under these conditions. It is assumed that the front surface is illuminated with a diffuse light flux of uniform angular distribution. Thus, ray r1 is incident at an arbitrary point P1 on the front surface under an arbitrary angle θi with respect to the normal k1. For clarity of the presentation, we initially consider the propagation of the refracted component. Thus, the refracted ray r2 at point P1 traverses the bulk of the cone and collides with the sidewall at point P2. At this stage, ray r2 has two possible trajectories, depending on the magnitude of the incidence angle γi. Thus, if γi < θc, ray r2 is refracted under an angle γr into ray r3, which then is supposedly projected onto the focal plane. Rays that undergo refraction during the first collision with a sidewall are called extrinsic. We are interested in the maximum divergence ψextr of ray r3 with respect to the negative z-direction and which divergence is attained when the value of γi is minimum. The angle γi is a decreasing function of θi and hence its minimum value is attained when θi = π/2, i.e. γi = π/2 – θc – a/2. Thus, from Snell’s law, it follows that: Fig. 1: Open in new tabDownload slide Propagation of extrinsic and intrinsic rays Ψextr =π/2−γr−α/2=π/2−asin[cos(θc+α/2).n1/n2]−α/2(1) where α is the apex angle of the cone. Further, ψextr is an increasing function of α and attains its maximum value of π/2 at α = π. Hence, extrinsic rays have a maximum divergence ψextr < π/2 irrespective of their initial angle of incidence θi and are, therefore, always projected onto a limited area in the focal plane. Considering now the case in which γi > θc, ray r2 undergoes total internal reflection at point P2, propagates as ray r4, which in turn collides with the opposite wall at point P3 under an angle γt. Rays that undergo TIR during the first collision with a sidewall are called intrinsic. From simple geometrical considerations, it follows that γt = γi – α. In other words, the angle of incidence decreases monotonically by an amount equal to the apex angle α with each successive collision. This situation is identical to the so-called escape-cone effect in which rays propagating in a conically shaped mirror lose forward component until it is reversed and they get rejected out of the cone back to the source. In our case, however, γt eventually becomes smaller than θc and hence ray r4 gets refracted as ray r5 out of the cone and is supposedly projected onto the focal plane. Without loss of generality, we assume that this takes place at point P3. In fact, the above observation defines in effect a forbidden zone for intrinsic rays whose magnitude of the vertical component is smaller than |–sin(θc+α/2)|, since such rays are immediately refracted, i.e. their further propagation within the cone through TIR is forbidden. Here, again, we are interested in the maximum divergence ψintr of intrinsic rays. From the above observation, it follows that the smallest possible incidence angle of an intrinsic ray when propagating inside the optical element is γt = θc – α. Thus, from Snell’s law, the maximum divergence ψintr is given by: Ψintr=π/2−γs−α/2=π/2−asin[sin(θc−α).n1/n2]−α/2(2) Here, again, ψintr is an increasing function of α and attains its maximum value of π/2 at α = π. Hence, intrinsic rays have a maximum divergence ψintr < π/2 irrespective of their initial angle of incidence θi and are, therefore, always projected onto a limited area in the focal plane. The immediate observation from Equations (1) and (2) is that the smaller the apex angle, the smaller the divergences of both intrinsic and extrinsic rays. Thus, the apex angle α along with the optical indices n1 and n2 provide the possibility of tailoring ψintr and ψextr at will through Equations (1) and (2). A further useful relation is the condition for the existence of extrinsic rays: β≤θi≤ π/2, whereβ =asin[cos(θc + α/2).n1/n2](3) Thus, extrinsic rays occupy the glancing incidence range and only exist if β is real. Intrinsic rays, however, always exist. Fig. 2 illustrates graphically the results so far. Specifically, the maximum divergence angle of all rays is given by ψmax = max(ψextr,ψintr), i.e. rays r1 and r2 illustrate graphically the maximum divergence range. Thus, the diffusely illuminated front surface is imaged onto the focal plane onto a limited area, called a ‘diffuse image’. Fig. 2: Open in new tabDownload slide Definition of ‘diffuse image’ Further, it is readily shown that Equation (1) (extrinsic case) is equally valid in 3D. The same, however, is not true for Equation (2) (intrinsic case), since the ray’s vertical component and the angle of incidence are now decoupled. The ray’s vertical component still declines monotonically in magnitude as the ray descends through the optical element while a monotonic decrease of the angle of incidence cannot be guaranteed in 3D and hence Equation (2) is not valid. Further, the three-dimensionality of the problem results in that ray exit from the pyramid is a stochastic event, as illustrated in Fig. 3a, which represents the so-called refraction cone having an apex angle of 2θc. The axis a1 of the cone is perpendicular to the sidewall and is always constructed at the impact point of a ray. Rays incident on the inside of the refraction cone (say, ray r1) are refracted, whereas rays incident on the outside (say, ray r2) are internally reflected. Hence, rays are ejected out of the pyramid with a probability proportional to 2θc. Analogously to the 2D, we want to make this probability equal to 1, which is done as follows. Fig. 3b represents an arbitrary cross-section Γ of the wedge formed between two neighbouring sidewalls such that the plane Γ is perpendicular to the edge μ (the latter assumed parallel to the z-axis for simplicity of the treatment). We follow the propagation of all TIR rays in the plane Γ incident from left to right at an arbitrary point P1, lying on the intersection line between sidewall 1 and the plane of incidence Γ. Note that the cone axis a1 is perpendicular to sidewall 1, i.e. a1 lies in Γ and points to the inside of the wedge. Specifically, we construct rays r1 and r2 incident at point P1 and lying in the plane Γ such that r1 is tangential to cone C1 while r2 lies on the intersection line between Γ and sidewall 1. Clearly, r1 and r2 define the range R of all rays incident from left to right at point P1 and lying in Γ that undergo TIR. Their reflections r’ 1 and r’ 2 impact with sidewall 2 at points P2 and P3, respectively. We further assume that the edge angle ζ lies in the range: Fig. 3: Open in new tabDownload slide (a) The refraction cone. (b) TIR in a horizontal plane of incidence. (c) TIR in a rotated plane of incidence. π/2−θc ≤ ζ ≤ 2θc(4) which guarantees that both r’ 1 and r’ 2 are refracted—that is, they both enter their respective refraction cones C2 and C3 on the inside. The rays in the range R are also internally reflected and projected onto the line segment Σ between points P2 and P3. Since r’ 1 and r’ 2 fall within their respective cones, so do all rays in the range R’ between them—that is, they all get refracted. We now start rotating anticlockwise the plane Γ along with all TIR rays considered about axis a1 as illustrated in Fig. 3c. In the process, the entry points of r’ 1 and r’ 2 in the refractive cones C2 and C3, respectively, start drifting towards the edges of the latter until at least one of them, say r’ 1, becomes tangential to its cone C2, as illustrated in Fig. 3c. We denote the angle of rotation in the plane of sidewall 1 when this happens as ω. Clearly, any further incremental increase of ω would result in ray r’ 1 ending up outside cone C2 and hence it would not be refracted. We further denote the z-component of r’ 1 at this point by –sin(θcut-off). Hence, all rays incident at point P1 from left to right and having a z-component in the range (–sin(θcut-off),0) are refracted or, in other words, ejected from the optical element. By virtue of the symmetry of the construction, the same is true for TIR rays incident from left to right at an arbitrary point on sidewall 2. Therefore, all TIR rays entering the edge and having a z-component in the range (–sin(θcut-off),0) are refracted. In other words, the edge angle ζ defines a bandgap that forbids the further propagation of such rays within the optical element, since they all get refracted. Note that the bandgap is associated with the edge. Clearly, the maximum angle of rotation ω and, hence, maximum bandgap width are achieved when r’ 1 and r’ 2 become simultaneously tangential to their respective cones C2 and C3. Thus, from standard geometrical considerations, it follows that the optimal angle ζopt when this occurs is given by: ζopt=acos[1−sin(θc)](5) Further, by the same token, the angle of rotation ω(θc,ζ) is calculated as: ω(θc,ζ)=asin[sin(θc)1−cot2(θc)cot2(ζ)], for π/2−θc≤ζ≤ ζopt(6) and ω(θc,ζ)=acos[cot(θc)1−cos(ζ)sin(ζ)], for ζopt≤ζ≤2θc(7) The right-hand sides of Equations (6) and (7) render an identical function ωopt(θc) at ζ = ζopt but, other than that, they have to be used in their respective ranges of validity above. Further, having calculated ω, the bandgap width θcut-off is readily calculated as: θcut~-off=asin[sin(θc)sin(ω)](8) Clearly, the angles θcut-off and ω define the z-components (r’ 1)z = –sin(θcut-off) and (r’ 2)z = –sin(ω). The choice of θcut-off as the bandgap width (and not the angle of rotation ω) was intentional, since this choice guarantees that all rays entering the edge ζ, irrespective of their horizontal components and having a magnitude of the z-component smaller than sin(θcut-off), are refracted. As an illustration, assuming n1 = 1.5 and n2 = 1, i.e. θc = 41.81o, Equations (5), (6) and (8) yield ζopt = 70.53o, ωopt = 37.76o and θcut-off = 24.09o, respectively. It is also noted that ω(θc,π/2 – θc) ≡ω(θc,2θc) ≡ 0, i.e. the bandgap width is zero at these points. We reiterate the requirement that the area of the cross-section Γ of the optical elements decreases along the negative z-axis (cone-like shape of the optical elements), which condition guarantees that TIR rays continually lose forward (vertical) component until their magnitudes fall within the bandgap of an edge and are consequently refracted. Hence, in analogy with Equation (2), one upper limit for the divergence of intrinsic rays in the 3D case is given by: Ψintr = π/2−γs−α/2 = π/2−asin[sin(θcut~-off−ε).n1/n2]−α/2(9) since the minimum incidence angle at the point of refraction is γt = (θcut-off – ε), where ε(α) defines the maximum angular decrement in the magnitude of the ray’s vertical component between two successive collisions. Clearly, in 3D, ε < α, where α/2 defines the slope of the sidewalls through cot(α/2). Here, again, ψintr < π/2. At this point, a small but important clarification is in order. In the above reasoning, we implicitly assumed that the vertical component of any given TIR ray (which has not refracted stochastically prior to that) will eventually fall within the bandgap. This is only true if the bandgap width θcut-off is larger than ε. Reiterating that ψintr is an increasing function of α and a decreasing function of θcut-off, it is obvious that ψintr is minimal when the difference (θcut-off – ε) is as large as possible. Hence, we arrive at the following relations: 0 < α ≪ θcut~-off(10) In other words, the bandgap width is to be much larger than twice the slope of the sidewall. This is called the apex criterion and its fulfilment guarantees that the vertical component of all unrefracted TIR rays does eventually enter the bandgap and they are all subsequently refracted out of the cone at minimal angles ψintr with respect to the cone axis. Closer inspection of inequalities (10) reveals that they implicitly contain inequalities (4), since, according to inequalities (10), θcut-off must be greater than zero, which is fulfilled if and only if inequalities (4) are satisfied. For practical purposes, the apex criterion for a given combination (n1, n2) can be approximated by the ratio between the height and the average lateral dimension of the front surface of the optical element, which is to be >10 for the cases considered below. Thus, it follows from the above discussion that the apex criterion suffices to guarantee that all TIR rays are eventually refracted out of the optical element in an orderly and optimal manner, and hence the primary escape-cone effect with regard to TIR rays is totally suppressed. In this way, we proved in 3D too that both extrinsic and intrinsic rays can be projected onto a finite area in the focal plane, irrespective of their initial angle of incidence onto the front surface. Fig. 4 represents an experimental confirmation of the above analysis. Specifically, it shows a triangular pyramid made of glass with a front surface 1 × 1 × 1 cm and a height of 15 cm. Under these conditions, all rays entering the pyramid through the front surface are intrinsic (see Equation (3) for very small values of α). The front surface is illuminated by a rastered laser beam under an angle of ~45o. Comparison between Fig. 2 and Fig. 4 identifies the ‘diffuse image’ in the focal plane formed by intrinsic rays. Fig. 4: Open in new tabDownload slide Experimental verification of ‘diffuse image’ 2 Principle of operation For reasons apparent from the above discussion, we consider the case involving intrinsic rays only. This provides the first focusing mechanism since all rays initially travel downwards through TIR through an OE of a decreasing cross-section. The latter, in addition, provides one further focusing mechanism in a diffuse-light concentrator by allowing the OE to be brought in close proximity to each other but still separated by narrow gaps of a lower-refractive-index optical material such that their respective ‘diffuse images’ overlap to a great extent, as illustrated in Fig. 5a. Due to this overlap, the area of the ‘Exit aperture’ is smaller than that of the ‘Input aperture’, which manifests the second confinement mechanism in the concentrator. At the same time, the plurality of the optical elements thus formed gives rise to collective effects, as follows. We present here a qualitative description only. Considering, say, ray r5 in Fig. 1, it is clear that it is ‘discharged’ in the midst of a laterally reflective grating formed by the multiple gap/sidewall interfaces between neighbouring OEs, as illustrated in Fig. 5b. The point at which the ray exits the OE is called a ‘burst point’, since, at this point, the ray bifurcates through a concurrent reflection/refraction event into two fractal rays. Thus, ray r4 splits into one refracted ray r5 and one reflected ray r6. Further, ray r5 collides with the sidewall of a neighbouring optical element and bifurcates into fractal rays r7 and r8, respectively. Analogously, ray r7 bifurcates into rays r9 and r10. The above process takes place at each and every interface, generating in this way an avalanching cascade of fractal rays. Clearly, the lateral components of both rays r8 and r10 are opposite to that of ray r5. Hence, each optical element returns (reflects back) a fraction of the energy emanating from a neighbouring optical element. At the same time, the intensity of ray r9 is smaller than that of r5. Thus, this mechanism eventually results in a skew of the light intensity over the ‘diffuse image’ of a given optical element towards its centre. Figuratively said, the gap/sidewall interfaces act as generators of fractal rays and, in doing so, they define at the same time a lateral fractal reflector that restricts the lateral flow of energy. Consequently, the fractal cascade of rays drifts downwards towards the exit aperture while being at the same time laterally confined by the lateral fractal reflector. Since the final stage of ray propagation is effected through a fractal cascade, such concentrators are called fractal. On the other hand, it can be readily shown that, in the limit of zero gap width, each reflection at any face is equivalent to an escape-cone collision event in terms of loss of the forward (axial) component. In other words, the forward propagation of all fractal rays within the cascade is equivalent to that of a ray propagating through an OE (conical waveguide) with ideally reflecting walls. Thus, all rays eventually lose the forward component and are rejected. Therefore, the focal plane is positioned at a distance from the front surface that is covered by the majority of rays incident onto the front surface within a predefined angular range which in fact defines the acceptance angle. Consequently, the OEs are truncated at the focal plane. It should be pointed out that the gap width is non-zero, which results in a downward drift of refracted rays through the gaps. Thus, assuming that the neighbouring faces are parallel to each other, then ray r4 is parallel to r7 but the latter is somewhat displaced downwards, which is also called gap displacement or gap drift. Hence, the finite gap width results in an additional downward drift of traversing rays. Fig. 5: Open in new tabDownload slide (a) Primary focusing mechanism. (b) Illustration of ray bifurcation. (c) Illustration of fractal cascade. (d) Illustration of TIR trapping. (e) Additional focusing by peripheral mirrors. Fig. 5c illustrates the above description through standard ray tracing. Specifically, it shows the propagation and subsequent fractal cascade formation of a ray incident at an angle of 60° onto the front surface of an optical element in a concentrator consisting of 150 pyramidal OEs having a triangular front surface of 12 × 12 × 12 mm and a height of 250 mm where the focal plane is positioned at a distance of 105 mm from the front surface. To avoid clutter, the vertical edges of the optical elements are not drawn. Their silhouettes, however, are visually outlined by deleting the ray trajectories within the gaps and aligning the viewing angle with the gap planes. In addition, only rays with intensity above a certain limit are drawn for clarity. Thus, the ray enters the top surface and propagates downwards through the optical element through TIR while continually losing the vertical component until, stochastically, the condition for TIR is disobeyed at a particular point (denoted as ‘stochastic burst’ in Fig. 5c), after which the fractal cascade develops according to the description above. In other words, from that point onwards, the fractal rays typically propagate through constant bifurcation within the lateral reflector. One important detail, though, requires a closer look. As noted earlier, the burst point occurs through a probabilistic process and hence its occurrence is not guaranteed. Further, its occurrence does not in any way guarantee that the fractal cascade develops fully. Thus, one or both fractal rays after the point of a stochastic burst may become trapped in neighbouring optical elements and continue their propagation through TIR, and so on. Whatever the sequence of events, however, any ray propagating within the concentrator continually loses vertical direction until the latter falls within the bandgap of an edge, which in turn is guaranteed for all rays by the design chosen, i.e. an adequate apex criterion. Rays may still get TIR trapped by the cooperative action of neighbouring OEs due to the symmetry of the structure. Nevertheless, this type of TIR trapping always involves interface crossing and hence fractal rays always experience the gap drift. Notably, such TIR trapping, if not dominating, is a very common occurrence, as illustrated in Fig. 5d, which shows the propagation of the same ray as in Fig. 5c but with a slightly rotated azimuth. One final flux confinement, however trivial, is readily achieved by the addition of peripheral mirrors, as illustrated in Fig. 5e. These mirrors play a 2-fold role. First, they define the spatial dimensions of the concentrator and thus restrict the sideways drift of the fractal rays by reflecting them back into the core of the concentrator. Second, the lower aperture of the mirrors defines strictly the exit aperture. Thus, the ratio between the input and exit apertures of the mirror structure defines the geometrical concentration coefficient Cg. 3 Major loss mechanisms We now consider typical loss mechanisms in the light concentrator excluding the primary escape-cone effect discussed above, since it has been totally suppressed in the design through inequalities (4). These losses are divided into two categories: propagation and rejection losses, respectively. The propagation losses result in energy deposition in the concentrator, whereas rejection losses represent the light intensity returned back to the ambient. The first category includes mainly absorption losses along the propagation path of the rays and absorption losses in the peripheral mirrors. Noteworthy, the propagation losses are dispersive, i.e. a function of the wavelength, as are the indices of refraction, for that matter. All these are materials- and technology-related issues common to all optical systems and are outside the scope of the current presentation. As for the rejection losses, these include reflection losses from the top surface, light scattering in the bulk and from interfaces due to surface roughness, impurities, particle inclusions, contaminations, etc. One type of rejection loss, i.e. the so-called secondary escape-cone effect, is always present for fractal rays propagating within the concentrator through bifurcation. Referring to Fig. 5b, we consider the fractal ray r6 and its subsequent reflections off the sidewall (not drawn) that remain within the optical element. Clearly, the intensity of these rays decreases exponentially, since it decreases proportionally to the reflection coefficient at each and every collision with the sidewalls. More important for the argument is the fact that the lateral component of successive reflected rays gradually increases due to the geometrical escape-cone effect until it is reversed and the residual intensity at this stage, however small, is rejected. This is a manifestation of the so-called secondary escape-cone effect and refers to a fractal ray propagating exclusively within any given conical structure. Virtual cones can be defined by any arbitrary combination of (partially) reflecting surfaces in the concentrator, including the peripheral mirrors, thus increasing the possible rejection paths. In other words, this rejection mechanism exists within any virtual cone and not within an optical element only. Fortunately, as noted above, its intensity diminishes exponentially and, with careful design, the net effect can be made sufficiently low through strengthening the apex criterion, i.e. by increasing the ratio between the OE’s height and its lateral dimension. Optimal performance is achieved for a ratio for which the secondary escape-cone effect is practically suppressed within the transmission range of the concentrator and is typically of the order of 10. Much higher ratios result in increased absorption losses with little additional benefit. The secondary escape-cone effect, however, becomes dominating around the acceptance angle. In other words, this is the point at which the radial velocity of the fractal rays becomes very significant. Light entering the concentrator through the gaps between the optical elements is obviously not concentrated and thus the net surface area of the gaps needs to be taken into account when calculating or measuring the optical efficiency. 4 Concentrator modelling As an illustration of the performance of the concentrator, we now present selected numerical modelling employing a ray-tracing algorithm based on strict Fresnel reflection/refraction at each and every interface. Reflection off the top surface is initially disregarded but added in the final calculations. We limit the discussion to low concentrations in which the acceptance angle is very large and its decline around the acceptance angle is not very steep, for which reason we refer to the angular dependence of the transmission coefficient as peripheral vision. The concentrator simulated consists of 150 optical elements representing triangular pyramids with a front surface of 12 × 12 × 12 mm, a height of 250 mm and a refractive index of 1.583. Truncation is applied at a height corresponding to the desired geometrical concentration. Thus, their height at Cg = 3 is 105 mm and the gap width is 0.2 mm. The optical-absorption coefficient of the air gaps is assumed to be zero, while that of the optical elements is equal to 0.004 cm–1. The peripheral mirrors are assumed to be made of aluminium. The reflection coefficient for Al is taken from Hébert [10] assuming a refractive index of 1.3 and an extinction coefficient of 7.46 for a wavelength of 619 nm. All calculations have been done with a precision guaranteeing that ≥99% of the energy in a cascade is traced to the end. Fig. 6 shows the calculated performance at a geometrical concentration Cg = 3 for light incident at the centre of the concentrator. Specifically, Fig. 6a shows the transmission, absorption and rejection coefficients as a function of the angle of incidence θ. The calculations are aggregated for 100 rays incident at random points of incidence with a full azimuthal scan. As seen, the transmission coefficient τ(θ) remains well above 90% up to a 65o angle of incidence. Fig. 6b shows the angular dependence of the transmittance η(θ) = τ(θ)[1 – R(θ)]—that is, the corrected transmission by subtracting the reflected fractional intensity R(θ) at the front surface. Assuming a uniform angular distribution of the incident flux, we define the total power density Φo of diffuse light incident onto a unit area parallel to the surface of Earth as: Fig. 6: Open in new tabDownload slide (a) Angular dependence of transmission and loss coefficients at Cg = 3. (b) Transmittance at Cg = 3. Φo=∫π/20A.cos(ϑ)dϑ=A(11a) where A is an arbitrary irradiance constant. Analogously we define the energy (exergy) efficiency Eeff of the concentrator for diffuse light as: Eeff=∫π/20[A.cos(ϑ)η(ϑ) Φ o]dϑ=0,8456(11b) Fig. 6b shows also the transmittance of an OE with ideally reflecting walls. The ideal OE has otherwise the same geometrical dimensions and absorption as the OE in the concentrator at the same Cg. It is seen that the peripheral vision of the ideal OE is seemingly somewhat lower than that of the concentrator. The explanation is that the concentrator has a slightly lower optical concentration coefficient due to the finite gap widths, which effectively reduces the input aperture. On the other hand, the effect of gap drift discussed above to which all fractal rays are subjected through the bandgap ejection results in a wider peripheral vision. Thus, the bandgap ejection makes use of the gap widths to extend the peripheral vision to the limit. Fig. 7a and b show the same calculations for a concentration Cg = 4, where the energy efficiency Eeff = 74%, which reflects the obvious fact that the peripheral vision is a decreasing function of Cg. An interesting case is presented in Fig. 8a and b, which illustrate the performance of the same concentrator with OEs of the refractive index of water, 1.33, at a concentration Cg = 2.5. The energy efficiency in this case Eeff = 82%. Thus, this suggests that OEs made of water in combination with anti-freezing agents may potentially be used in low-end applications. The obvious conclusion, however, is that the higher the refractive index, the higher the performance following standard etendué conservation. More importantly, the above findings demonstrate that the performance of fractal concentrators is practically identical to a conical waveguide with ideally reflecting walls and the same absorption with the same absorption. Fig. 7: Open in new tabDownload slide (a) Angular dependence of transmission and loss coefficients at Cg = 4. (b) Transmittance at Cg = 4. Fig. 8: Open in new tabDownload slide (a) Angular dependence of transmission and loss coefficients at Cg = 2.5 and n = 1.33. (b) Transmittance at Cg = 2.5 and n = 1.33. It should be noted that the overall peripheral vision of the concentrator is notably broader than calculated above, since the concentrator itself occupies a sizeable solid angle, typically 20–30° from normal incidence. In other words, there are always OEs that have a smaller incidence angle than that of the central OE for non-normal incidence up to the angle at which mutual shadowing between neighbouring concentrators occurs. Finally, the gap widths take up a few percent of the input aperture, which needs to be included in the overall performance of the concentrator. The mirror losses have been found to be negligible for all of the above cases. The average mirror losses of rays entering the concentrator through OEs at its very periphery are calculated to be <2%. Thus, the average mirror losses for the whole concentrator are well below 1%, for which reason they are not included in the above calculations. 5 Fabrication and applications The optical elements and other elements may be fabricated of low-cost and recyclable polymer materials of high optical quality such as PMMA, PC, etc, say, by injection moulding and other similar methods. The gaps between the optical elements are defined by appropriate spacers, as illustrated in Fig. 9. The exit aperture, along with the top protective screen, encapsulate the concentrator and provide additional thermal insulation at the same time. The absorber/solar cell is in intimate contact with the exit aperture and is optically matched to the latter. The whole concentrator is assembled in a dust-free environment. Further, since the absorption losses are sufficiently low, fractal solar concentrators may advantageously be thermally insulated from the ambient, as illustrated in Fig. 9. Thus, the concentrated light flux is fed onto a light absorber, which may be a solar cell designed for the intended light concentration or just a light-to-heat converter. In either case, the absorber is thermally insulated from the ambient meaning that most of the heat generated will be transferred to the heat-carrier fluid. In other words, deducting the obvious losses, the remainder of the energy is transformed into usable forms of energy. These types of systems are called concentrating photovoltaic–thermal (CPVT) systems. In practice, arrays of solar concentrators such as that shown in Fig. 9 are to be assembled into panels with a common heat-carrier system thermally insulated from the ambient. Generally, both consumer and industrial applications in the form of CPVT and CSP power-generation systems are feasible. The heat generated in CPVT systems is generally rated as low-grade, since the temperature in the carrier fluid is limited by the maximum operating temperature of the solar cells, typically <90oC, and is suitable for household use, greenhouse heating, etc. Where suitable, a heat reservoir may be used in conjunction for short-term heat storage. Not least, a combination of CPVT and a Sterling engine would increase the electrical efficiency in the warmer months. A heat exchanger is needed in the daytime to dump excess/waste heat to the ambient as well as during the night-time to cool the heat reservoir for daytime cooling purposes. A further possibility is to use the excess heat in the warmer months for absorption cooling. Medium-grade-heat CSP systems may be possible if the thermal insulation is improved. Water desalination is a likely application too. Building integrated solar is another possible application where panels of solar concentrators are to be integrated with the building facades making them an integral part of the building insulation and energy system. Fig. 9: Open in new tabDownload slide Thermally insulated concentrator with active cooling 6 Discussion Certain justification for the applications proposed above is in order. To this end, we consider the example in Fig. 6a. Assuming thermal conductivity of 0.2 W/K/m (typical for plastics) and a height of the OE of 10 cm as well as a temperature difference between the heat-carrier fluid and the ambient of 60oC, we arrive at a thermal loss of 120 W/m2. This loss is then divided by Cg, since the emitting surface now is Cg times smaller. Hence, we have an effective thermal loss of 40 W/m2. Assuming further spring irradiance of 250 W/m2 as well as an energy efficiency Eeff of the concentrator of 80%, we arrive at an operating energy efficiency of 64% under the above conditions. This figure becomes 76% at solar irradiance of 1000 W/m2. In reality, one would vary the temperature difference within certain limits to maintain optimal operating efficiency. The example in Fig. 8 would certainly have an inferior performance due in part to the lower Cg and in part to the higher thermal conductivity of water compounded with convection losses. Nevertheless, this case, perhaps, is not to be dismissed, particularly in smaller geographical latitudes. Further, calculating the CAP FOM for the concentrators in Figs 6 and 7, we arrive at CAP >1.5. One might question the use of CAP FOM in this case but nevertheless it provides a qualitative comparison with the current state of the art. As stated above, the fractal concentrator performs at the limits of what is possible with ideally reflecting concentrating waveguides. This performance is achieved partly by the fractal cascade and to some extent by the bandgap, both of which extend the peripheral vision to the physical limit. Further examples of bandgap use for optimizing performance while minimizing cost will be presented in coming communications. Next, it is thought that spectral uniformity is not an issue with fractal concentrators despite the fact that the refractive index is dispersive. If the latter makes any difference, it would be at the edge of the peripheral vision, which constitutes a negligible fraction of the total energy harvested. The spatial uniformity of the concentrated flux has not been studied, although it is also not thought to be an issue too due to the stochastic nature of concentration and the lack of preferred focusing mechanisms. Concentrator transmission, though, may be spectrally sensitive since both the refractive index and absorption are dispersive. The absorption dependence, however, is readily calculated, since it is additive in nature. Finally, light concentration means that one uses correspondingly smaller (in area) solar cells. In view of the low concentration coefficients, however, it would hardly make any economic sense to use fractal concentrators in PV applications alone. The situation changes radically when bi- and tri-generation systems are considered, since now one can harvest the dominant fraction of the available solar energy and convert it into usable forms of energy. 7 Conclusions A novel approach to concentrating both direct and indirect light based on the individual and collective confinement properties of an array of refractive optical elements has been presented. A theoretical framework relating the geometry of the optical elements to their confinement properties has been developed. Consequently, the principle of operation and major design rules of a light concentrator have been established. The main mechanisms of light propagation through the concentrator as well as associated losses have been described in detail. The concentrators are shown numerically to exhibit acceptance angles for CAP FOMs of the order of 1.5. The main advantage of the proposed concentrator is when it is used in bi- and tri-generation solar systems. Thus, it is numerically demonstrated that such concentrators may be used in a range of applications such as CPVT and CSP with operating energy efficiencies >70%. It is also concluded that materials with higher refractive indices yield higher performance. Current polymer materials in this context have moderate refractive indices and low-cost optical materials with higher refractive indices are needed to increase concentration and, hence, performance as well as to reduce material use. Acknowledgments The publication of this article has been sponsored by Vinnova through contract 2019–04927. The work is done under contracts VK-1931:02 and RK-1932:08. Tribute: In memory of George Carter and Mike J. Nobes. Conflict of Interest None declared. References [1] Fraunhofer Institute for Solar Energy Systems . Photovoltaics Report . https://www.ise.fraunhofer.de/content/dam/ise/de/documents/publications/studies/Photovoltaics-Report.pdf ( 12 August 2020 , date last accessed). [2] Boretti A , Castelletto S, Al-Zubaidy S. Concentrating solar power tower technology: present status and outlook . Nonlinear Engineering 2019 , 8 : 10 - 31 . Google Scholar Crossref Search ADS WorldCat [3] Dixit S . Solar technologies and their implementations: a review . Materials Today: Proceedings 2020 , 28 : 2137 - 48 . Google Scholar Crossref Search ADS WorldCat [4] Gonzalo AP , Marugán AP, Márquez FP. A review of the application performances of concentrated solar power systems . Appl. Energy 2019 , 255 : 113893 . Google Scholar Crossref Search ADS WorldCat [5] Indira SS , Vaithilingam CA, Chong KK, et al. A review on various configurations of hybrid concentrator photovoltaic and thermoelectric generator system . Solar Energy 2020 , 1;201 : 122 - 48 . Google Scholar Crossref Search ADS WorldCat [6] Jia Y , Alva G, Fang G. Development and applications of photovoltaic–thermal systems: a review . Renew Sust Energ Rev 2019 , 102 : 249 - 65 . Google Scholar Crossref Search ADS WorldCat [7] Răboacă MS , Badea G, Enache A, et al. Concentrating solar power technologies . Energies 2019 , 2 : 1048 . Google Scholar Crossref Search ADS WorldCat [8] Buljan M , Mendes-Lopes J, Benítez P, et al. Recent trends in concentrated photovoltaics concentrators’ architecture . J Photonics Energy 2014 , 4 : 040995 . doi:10.1117/1.JPE.4.040995. Google Scholar Crossref Search ADS WorldCat [9] Katardjiev I . Optical system for manipulation and concentration of diffuse light . World Patent WO 2020/139186 A1. 7 July 2020 . Google Scholar OpenURL Placeholder Text WorldCat [10] Hébert M. Reflection and transmission of light by a flat interface, Fresnel’s formulae . Ph.D. Thesis. Paris, France : Institut d’Optique ParisTech , 2013 . http://paristech.institutoptique.fr/site.php?id=797&fileid=11468 ( 12 August 2020 , date last accessed). Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC © The Author(s) 2020. Published by Oxford University Press on behalf of National Institute of Clean-and-Low-Carbon Energy This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com © The Author(s) 2020. Published by Oxford University Press on behalf of National Institute of Clean-and-Low-Carbon Energy

Loading next page...
 
/lp/oxford-university-press/a-fractal-concentrator-of-direct-and-indirect-sunlight-pAzcHeJCLi
Publisher
Oxford University Press
Copyright
Copyright © 2022 National Institute of Clean-and-Low-Carbon Energy
ISSN
2515-4230
eISSN
2515-396X
DOI
10.1093/ce/zkaa017
Publisher site
See Article on Publisher Site

Abstract

Abstract A novel concept for concentrating both direct and indirect sunlight employing a combination of waveguiding and refractive optics is presented. More specifically, it is based on both the individual and the collective focusing effects of an array of refractive optical elements of specific geometry. An analytical framework and design principles regarding the geometry of the optical elements taking into account Fresnel reflection and refraction are presented. The principle of operation along with loss mechanisms and acceptance angles are discussed in detail. Ray-tracing simulations of a light concentrator designed accordingly indicate transmittance >90% with acceptance angles of ≤65° at 3 suns for moderate refractive indices. It is further numerically demonstrated that the new approach allows the utilization of the dominant fraction of the solar radiation by harvesting the waste heat in addition to photovoltaic conversion with an operating energy efficiency for diffuse light >70%. Graphical Abstract Open in new tabDownload slide solar concentrator, diffuse light, non-imaging optics, solar harvesting Introduction The flat panel photovoltaic (PV) solar-cell technology is by far the fastest-growing renewable-energy technology today with 584 GW installed cumulative PV capacity by the end of 2019 [1]. Amongst its notable disadvantages is the massive use of semiconductor material in addition to its inability to harvest the waste heat. Concentrating photovoltaics (CPV) holds the promise of reducing the use of semiconductor material in solar panels by 1–3 orders of magnitude in addition to having higher photoelectric-conversion efficiencies. Another promising renewable technology is concentrating solar power (CSP), which employs standard thermomechanical engines to convert sunlight into usable energy. Nevertheless, both the CPV and the CSP technologies have not had significant commercial success despite their obvious advantages. One major obstacle that prevents CPV and CSP from being viable technologies lies in the design and cost of the optical systems used to concentrate solar radiation. Furthermore, optical systems for the most part employ imaging optics that limits the radiation harvested to well-collimated sunlight resulting in the need for accurate tracking of the Sun. Consequently, the extra and considerable cost of a precision tracking system makes current CPV and the CSP techniques economically uncompetitive. For recent reviews on the subject, the reader is referred to references [2–7]. Thus, one pressing challenge to address is the very limited acceptance angle of existing solutions. A common figure of merit used for comparative purposes is the Concentration-Acceptance angle product (CAP) defined as the product between the square root of geometrical concentration and the sine of the acceptance angle. For state-of-the-art concentrators, the value of CAP [8] lies below 1. Thus, it is the objective of this work to suggest a new concept for designing solar concentrators that exhibit large acceptance angles, high transmittance and high CAP values in the low-concentration range. 1 Analytical framework The theoretical framework is based on earlier work [9]. Central to it is the geometry of refractive optical elements (OEs) that have the property of projecting a totally diffuse light flux onto a finite area in the focal plane. Fig. 1 illustrates in 2D one such example representing an optical element in the form of a 2D triangular cone (pyramid) made of an optical material with a refractive index n1 surrounded by a second optical material of optical index n2 < n1, say, glass n1 = 1.5 and air n2 = 1 for specificity. We also denote by θc = asin(n2/n1) the critical angle of total internal reflection (TIR) under these conditions. It is assumed that the front surface is illuminated with a diffuse light flux of uniform angular distribution. Thus, ray r1 is incident at an arbitrary point P1 on the front surface under an arbitrary angle θi with respect to the normal k1. For clarity of the presentation, we initially consider the propagation of the refracted component. Thus, the refracted ray r2 at point P1 traverses the bulk of the cone and collides with the sidewall at point P2. At this stage, ray r2 has two possible trajectories, depending on the magnitude of the incidence angle γi. Thus, if γi < θc, ray r2 is refracted under an angle γr into ray r3, which then is supposedly projected onto the focal plane. Rays that undergo refraction during the first collision with a sidewall are called extrinsic. We are interested in the maximum divergence ψextr of ray r3 with respect to the negative z-direction and which divergence is attained when the value of γi is minimum. The angle γi is a decreasing function of θi and hence its minimum value is attained when θi = π/2, i.e. γi = π/2 – θc – a/2. Thus, from Snell’s law, it follows that: Fig. 1: Open in new tabDownload slide Propagation of extrinsic and intrinsic rays Ψextr =π/2−γr−α/2=π/2−asin[cos(θc+α/2).n1/n2]−α/2(1) where α is the apex angle of the cone. Further, ψextr is an increasing function of α and attains its maximum value of π/2 at α = π. Hence, extrinsic rays have a maximum divergence ψextr < π/2 irrespective of their initial angle of incidence θi and are, therefore, always projected onto a limited area in the focal plane. Considering now the case in which γi > θc, ray r2 undergoes total internal reflection at point P2, propagates as ray r4, which in turn collides with the opposite wall at point P3 under an angle γt. Rays that undergo TIR during the first collision with a sidewall are called intrinsic. From simple geometrical considerations, it follows that γt = γi – α. In other words, the angle of incidence decreases monotonically by an amount equal to the apex angle α with each successive collision. This situation is identical to the so-called escape-cone effect in which rays propagating in a conically shaped mirror lose forward component until it is reversed and they get rejected out of the cone back to the source. In our case, however, γt eventually becomes smaller than θc and hence ray r4 gets refracted as ray r5 out of the cone and is supposedly projected onto the focal plane. Without loss of generality, we assume that this takes place at point P3. In fact, the above observation defines in effect a forbidden zone for intrinsic rays whose magnitude of the vertical component is smaller than |–sin(θc+α/2)|, since such rays are immediately refracted, i.e. their further propagation within the cone through TIR is forbidden. Here, again, we are interested in the maximum divergence ψintr of intrinsic rays. From the above observation, it follows that the smallest possible incidence angle of an intrinsic ray when propagating inside the optical element is γt = θc – α. Thus, from Snell’s law, the maximum divergence ψintr is given by: Ψintr=π/2−γs−α/2=π/2−asin[sin(θc−α).n1/n2]−α/2(2) Here, again, ψintr is an increasing function of α and attains its maximum value of π/2 at α = π. Hence, intrinsic rays have a maximum divergence ψintr < π/2 irrespective of their initial angle of incidence θi and are, therefore, always projected onto a limited area in the focal plane. The immediate observation from Equations (1) and (2) is that the smaller the apex angle, the smaller the divergences of both intrinsic and extrinsic rays. Thus, the apex angle α along with the optical indices n1 and n2 provide the possibility of tailoring ψintr and ψextr at will through Equations (1) and (2). A further useful relation is the condition for the existence of extrinsic rays: β≤θi≤ π/2, whereβ =asin[cos(θc + α/2).n1/n2](3) Thus, extrinsic rays occupy the glancing incidence range and only exist if β is real. Intrinsic rays, however, always exist. Fig. 2 illustrates graphically the results so far. Specifically, the maximum divergence angle of all rays is given by ψmax = max(ψextr,ψintr), i.e. rays r1 and r2 illustrate graphically the maximum divergence range. Thus, the diffusely illuminated front surface is imaged onto the focal plane onto a limited area, called a ‘diffuse image’. Fig. 2: Open in new tabDownload slide Definition of ‘diffuse image’ Further, it is readily shown that Equation (1) (extrinsic case) is equally valid in 3D. The same, however, is not true for Equation (2) (intrinsic case), since the ray’s vertical component and the angle of incidence are now decoupled. The ray’s vertical component still declines monotonically in magnitude as the ray descends through the optical element while a monotonic decrease of the angle of incidence cannot be guaranteed in 3D and hence Equation (2) is not valid. Further, the three-dimensionality of the problem results in that ray exit from the pyramid is a stochastic event, as illustrated in Fig. 3a, which represents the so-called refraction cone having an apex angle of 2θc. The axis a1 of the cone is perpendicular to the sidewall and is always constructed at the impact point of a ray. Rays incident on the inside of the refraction cone (say, ray r1) are refracted, whereas rays incident on the outside (say, ray r2) are internally reflected. Hence, rays are ejected out of the pyramid with a probability proportional to 2θc. Analogously to the 2D, we want to make this probability equal to 1, which is done as follows. Fig. 3b represents an arbitrary cross-section Γ of the wedge formed between two neighbouring sidewalls such that the plane Γ is perpendicular to the edge μ (the latter assumed parallel to the z-axis for simplicity of the treatment). We follow the propagation of all TIR rays in the plane Γ incident from left to right at an arbitrary point P1, lying on the intersection line between sidewall 1 and the plane of incidence Γ. Note that the cone axis a1 is perpendicular to sidewall 1, i.e. a1 lies in Γ and points to the inside of the wedge. Specifically, we construct rays r1 and r2 incident at point P1 and lying in the plane Γ such that r1 is tangential to cone C1 while r2 lies on the intersection line between Γ and sidewall 1. Clearly, r1 and r2 define the range R of all rays incident from left to right at point P1 and lying in Γ that undergo TIR. Their reflections r’ 1 and r’ 2 impact with sidewall 2 at points P2 and P3, respectively. We further assume that the edge angle ζ lies in the range: Fig. 3: Open in new tabDownload slide (a) The refraction cone. (b) TIR in a horizontal plane of incidence. (c) TIR in a rotated plane of incidence. π/2−θc ≤ ζ ≤ 2θc(4) which guarantees that both r’ 1 and r’ 2 are refracted—that is, they both enter their respective refraction cones C2 and C3 on the inside. The rays in the range R are also internally reflected and projected onto the line segment Σ between points P2 and P3. Since r’ 1 and r’ 2 fall within their respective cones, so do all rays in the range R’ between them—that is, they all get refracted. We now start rotating anticlockwise the plane Γ along with all TIR rays considered about axis a1 as illustrated in Fig. 3c. In the process, the entry points of r’ 1 and r’ 2 in the refractive cones C2 and C3, respectively, start drifting towards the edges of the latter until at least one of them, say r’ 1, becomes tangential to its cone C2, as illustrated in Fig. 3c. We denote the angle of rotation in the plane of sidewall 1 when this happens as ω. Clearly, any further incremental increase of ω would result in ray r’ 1 ending up outside cone C2 and hence it would not be refracted. We further denote the z-component of r’ 1 at this point by –sin(θcut-off). Hence, all rays incident at point P1 from left to right and having a z-component in the range (–sin(θcut-off),0) are refracted or, in other words, ejected from the optical element. By virtue of the symmetry of the construction, the same is true for TIR rays incident from left to right at an arbitrary point on sidewall 2. Therefore, all TIR rays entering the edge and having a z-component in the range (–sin(θcut-off),0) are refracted. In other words, the edge angle ζ defines a bandgap that forbids the further propagation of such rays within the optical element, since they all get refracted. Note that the bandgap is associated with the edge. Clearly, the maximum angle of rotation ω and, hence, maximum bandgap width are achieved when r’ 1 and r’ 2 become simultaneously tangential to their respective cones C2 and C3. Thus, from standard geometrical considerations, it follows that the optimal angle ζopt when this occurs is given by: ζopt=acos[1−sin(θc)](5) Further, by the same token, the angle of rotation ω(θc,ζ) is calculated as: ω(θc,ζ)=asin[sin(θc)1−cot2(θc)cot2(ζ)], for π/2−θc≤ζ≤ ζopt(6) and ω(θc,ζ)=acos[cot(θc)1−cos(ζ)sin(ζ)], for ζopt≤ζ≤2θc(7) The right-hand sides of Equations (6) and (7) render an identical function ωopt(θc) at ζ = ζopt but, other than that, they have to be used in their respective ranges of validity above. Further, having calculated ω, the bandgap width θcut-off is readily calculated as: θcut~-off=asin[sin(θc)sin(ω)](8) Clearly, the angles θcut-off and ω define the z-components (r’ 1)z = –sin(θcut-off) and (r’ 2)z = –sin(ω). The choice of θcut-off as the bandgap width (and not the angle of rotation ω) was intentional, since this choice guarantees that all rays entering the edge ζ, irrespective of their horizontal components and having a magnitude of the z-component smaller than sin(θcut-off), are refracted. As an illustration, assuming n1 = 1.5 and n2 = 1, i.e. θc = 41.81o, Equations (5), (6) and (8) yield ζopt = 70.53o, ωopt = 37.76o and θcut-off = 24.09o, respectively. It is also noted that ω(θc,π/2 – θc) ≡ω(θc,2θc) ≡ 0, i.e. the bandgap width is zero at these points. We reiterate the requirement that the area of the cross-section Γ of the optical elements decreases along the negative z-axis (cone-like shape of the optical elements), which condition guarantees that TIR rays continually lose forward (vertical) component until their magnitudes fall within the bandgap of an edge and are consequently refracted. Hence, in analogy with Equation (2), one upper limit for the divergence of intrinsic rays in the 3D case is given by: Ψintr = π/2−γs−α/2 = π/2−asin[sin(θcut~-off−ε).n1/n2]−α/2(9) since the minimum incidence angle at the point of refraction is γt = (θcut-off – ε), where ε(α) defines the maximum angular decrement in the magnitude of the ray’s vertical component between two successive collisions. Clearly, in 3D, ε < α, where α/2 defines the slope of the sidewalls through cot(α/2). Here, again, ψintr < π/2. At this point, a small but important clarification is in order. In the above reasoning, we implicitly assumed that the vertical component of any given TIR ray (which has not refracted stochastically prior to that) will eventually fall within the bandgap. This is only true if the bandgap width θcut-off is larger than ε. Reiterating that ψintr is an increasing function of α and a decreasing function of θcut-off, it is obvious that ψintr is minimal when the difference (θcut-off – ε) is as large as possible. Hence, we arrive at the following relations: 0 < α ≪ θcut~-off(10) In other words, the bandgap width is to be much larger than twice the slope of the sidewall. This is called the apex criterion and its fulfilment guarantees that the vertical component of all unrefracted TIR rays does eventually enter the bandgap and they are all subsequently refracted out of the cone at minimal angles ψintr with respect to the cone axis. Closer inspection of inequalities (10) reveals that they implicitly contain inequalities (4), since, according to inequalities (10), θcut-off must be greater than zero, which is fulfilled if and only if inequalities (4) are satisfied. For practical purposes, the apex criterion for a given combination (n1, n2) can be approximated by the ratio between the height and the average lateral dimension of the front surface of the optical element, which is to be >10 for the cases considered below. Thus, it follows from the above discussion that the apex criterion suffices to guarantee that all TIR rays are eventually refracted out of the optical element in an orderly and optimal manner, and hence the primary escape-cone effect with regard to TIR rays is totally suppressed. In this way, we proved in 3D too that both extrinsic and intrinsic rays can be projected onto a finite area in the focal plane, irrespective of their initial angle of incidence onto the front surface. Fig. 4 represents an experimental confirmation of the above analysis. Specifically, it shows a triangular pyramid made of glass with a front surface 1 × 1 × 1 cm and a height of 15 cm. Under these conditions, all rays entering the pyramid through the front surface are intrinsic (see Equation (3) for very small values of α). The front surface is illuminated by a rastered laser beam under an angle of ~45o. Comparison between Fig. 2 and Fig. 4 identifies the ‘diffuse image’ in the focal plane formed by intrinsic rays. Fig. 4: Open in new tabDownload slide Experimental verification of ‘diffuse image’ 2 Principle of operation For reasons apparent from the above discussion, we consider the case involving intrinsic rays only. This provides the first focusing mechanism since all rays initially travel downwards through TIR through an OE of a decreasing cross-section. The latter, in addition, provides one further focusing mechanism in a diffuse-light concentrator by allowing the OE to be brought in close proximity to each other but still separated by narrow gaps of a lower-refractive-index optical material such that their respective ‘diffuse images’ overlap to a great extent, as illustrated in Fig. 5a. Due to this overlap, the area of the ‘Exit aperture’ is smaller than that of the ‘Input aperture’, which manifests the second confinement mechanism in the concentrator. At the same time, the plurality of the optical elements thus formed gives rise to collective effects, as follows. We present here a qualitative description only. Considering, say, ray r5 in Fig. 1, it is clear that it is ‘discharged’ in the midst of a laterally reflective grating formed by the multiple gap/sidewall interfaces between neighbouring OEs, as illustrated in Fig. 5b. The point at which the ray exits the OE is called a ‘burst point’, since, at this point, the ray bifurcates through a concurrent reflection/refraction event into two fractal rays. Thus, ray r4 splits into one refracted ray r5 and one reflected ray r6. Further, ray r5 collides with the sidewall of a neighbouring optical element and bifurcates into fractal rays r7 and r8, respectively. Analogously, ray r7 bifurcates into rays r9 and r10. The above process takes place at each and every interface, generating in this way an avalanching cascade of fractal rays. Clearly, the lateral components of both rays r8 and r10 are opposite to that of ray r5. Hence, each optical element returns (reflects back) a fraction of the energy emanating from a neighbouring optical element. At the same time, the intensity of ray r9 is smaller than that of r5. Thus, this mechanism eventually results in a skew of the light intensity over the ‘diffuse image’ of a given optical element towards its centre. Figuratively said, the gap/sidewall interfaces act as generators of fractal rays and, in doing so, they define at the same time a lateral fractal reflector that restricts the lateral flow of energy. Consequently, the fractal cascade of rays drifts downwards towards the exit aperture while being at the same time laterally confined by the lateral fractal reflector. Since the final stage of ray propagation is effected through a fractal cascade, such concentrators are called fractal. On the other hand, it can be readily shown that, in the limit of zero gap width, each reflection at any face is equivalent to an escape-cone collision event in terms of loss of the forward (axial) component. In other words, the forward propagation of all fractal rays within the cascade is equivalent to that of a ray propagating through an OE (conical waveguide) with ideally reflecting walls. Thus, all rays eventually lose the forward component and are rejected. Therefore, the focal plane is positioned at a distance from the front surface that is covered by the majority of rays incident onto the front surface within a predefined angular range which in fact defines the acceptance angle. Consequently, the OEs are truncated at the focal plane. It should be pointed out that the gap width is non-zero, which results in a downward drift of refracted rays through the gaps. Thus, assuming that the neighbouring faces are parallel to each other, then ray r4 is parallel to r7 but the latter is somewhat displaced downwards, which is also called gap displacement or gap drift. Hence, the finite gap width results in an additional downward drift of traversing rays. Fig. 5: Open in new tabDownload slide (a) Primary focusing mechanism. (b) Illustration of ray bifurcation. (c) Illustration of fractal cascade. (d) Illustration of TIR trapping. (e) Additional focusing by peripheral mirrors. Fig. 5c illustrates the above description through standard ray tracing. Specifically, it shows the propagation and subsequent fractal cascade formation of a ray incident at an angle of 60° onto the front surface of an optical element in a concentrator consisting of 150 pyramidal OEs having a triangular front surface of 12 × 12 × 12 mm and a height of 250 mm where the focal plane is positioned at a distance of 105 mm from the front surface. To avoid clutter, the vertical edges of the optical elements are not drawn. Their silhouettes, however, are visually outlined by deleting the ray trajectories within the gaps and aligning the viewing angle with the gap planes. In addition, only rays with intensity above a certain limit are drawn for clarity. Thus, the ray enters the top surface and propagates downwards through the optical element through TIR while continually losing the vertical component until, stochastically, the condition for TIR is disobeyed at a particular point (denoted as ‘stochastic burst’ in Fig. 5c), after which the fractal cascade develops according to the description above. In other words, from that point onwards, the fractal rays typically propagate through constant bifurcation within the lateral reflector. One important detail, though, requires a closer look. As noted earlier, the burst point occurs through a probabilistic process and hence its occurrence is not guaranteed. Further, its occurrence does not in any way guarantee that the fractal cascade develops fully. Thus, one or both fractal rays after the point of a stochastic burst may become trapped in neighbouring optical elements and continue their propagation through TIR, and so on. Whatever the sequence of events, however, any ray propagating within the concentrator continually loses vertical direction until the latter falls within the bandgap of an edge, which in turn is guaranteed for all rays by the design chosen, i.e. an adequate apex criterion. Rays may still get TIR trapped by the cooperative action of neighbouring OEs due to the symmetry of the structure. Nevertheless, this type of TIR trapping always involves interface crossing and hence fractal rays always experience the gap drift. Notably, such TIR trapping, if not dominating, is a very common occurrence, as illustrated in Fig. 5d, which shows the propagation of the same ray as in Fig. 5c but with a slightly rotated azimuth. One final flux confinement, however trivial, is readily achieved by the addition of peripheral mirrors, as illustrated in Fig. 5e. These mirrors play a 2-fold role. First, they define the spatial dimensions of the concentrator and thus restrict the sideways drift of the fractal rays by reflecting them back into the core of the concentrator. Second, the lower aperture of the mirrors defines strictly the exit aperture. Thus, the ratio between the input and exit apertures of the mirror structure defines the geometrical concentration coefficient Cg. 3 Major loss mechanisms We now consider typical loss mechanisms in the light concentrator excluding the primary escape-cone effect discussed above, since it has been totally suppressed in the design through inequalities (4). These losses are divided into two categories: propagation and rejection losses, respectively. The propagation losses result in energy deposition in the concentrator, whereas rejection losses represent the light intensity returned back to the ambient. The first category includes mainly absorption losses along the propagation path of the rays and absorption losses in the peripheral mirrors. Noteworthy, the propagation losses are dispersive, i.e. a function of the wavelength, as are the indices of refraction, for that matter. All these are materials- and technology-related issues common to all optical systems and are outside the scope of the current presentation. As for the rejection losses, these include reflection losses from the top surface, light scattering in the bulk and from interfaces due to surface roughness, impurities, particle inclusions, contaminations, etc. One type of rejection loss, i.e. the so-called secondary escape-cone effect, is always present for fractal rays propagating within the concentrator through bifurcation. Referring to Fig. 5b, we consider the fractal ray r6 and its subsequent reflections off the sidewall (not drawn) that remain within the optical element. Clearly, the intensity of these rays decreases exponentially, since it decreases proportionally to the reflection coefficient at each and every collision with the sidewalls. More important for the argument is the fact that the lateral component of successive reflected rays gradually increases due to the geometrical escape-cone effect until it is reversed and the residual intensity at this stage, however small, is rejected. This is a manifestation of the so-called secondary escape-cone effect and refers to a fractal ray propagating exclusively within any given conical structure. Virtual cones can be defined by any arbitrary combination of (partially) reflecting surfaces in the concentrator, including the peripheral mirrors, thus increasing the possible rejection paths. In other words, this rejection mechanism exists within any virtual cone and not within an optical element only. Fortunately, as noted above, its intensity diminishes exponentially and, with careful design, the net effect can be made sufficiently low through strengthening the apex criterion, i.e. by increasing the ratio between the OE’s height and its lateral dimension. Optimal performance is achieved for a ratio for which the secondary escape-cone effect is practically suppressed within the transmission range of the concentrator and is typically of the order of 10. Much higher ratios result in increased absorption losses with little additional benefit. The secondary escape-cone effect, however, becomes dominating around the acceptance angle. In other words, this is the point at which the radial velocity of the fractal rays becomes very significant. Light entering the concentrator through the gaps between the optical elements is obviously not concentrated and thus the net surface area of the gaps needs to be taken into account when calculating or measuring the optical efficiency. 4 Concentrator modelling As an illustration of the performance of the concentrator, we now present selected numerical modelling employing a ray-tracing algorithm based on strict Fresnel reflection/refraction at each and every interface. Reflection off the top surface is initially disregarded but added in the final calculations. We limit the discussion to low concentrations in which the acceptance angle is very large and its decline around the acceptance angle is not very steep, for which reason we refer to the angular dependence of the transmission coefficient as peripheral vision. The concentrator simulated consists of 150 optical elements representing triangular pyramids with a front surface of 12 × 12 × 12 mm, a height of 250 mm and a refractive index of 1.583. Truncation is applied at a height corresponding to the desired geometrical concentration. Thus, their height at Cg = 3 is 105 mm and the gap width is 0.2 mm. The optical-absorption coefficient of the air gaps is assumed to be zero, while that of the optical elements is equal to 0.004 cm–1. The peripheral mirrors are assumed to be made of aluminium. The reflection coefficient for Al is taken from Hébert [10] assuming a refractive index of 1.3 and an extinction coefficient of 7.46 for a wavelength of 619 nm. All calculations have been done with a precision guaranteeing that ≥99% of the energy in a cascade is traced to the end. Fig. 6 shows the calculated performance at a geometrical concentration Cg = 3 for light incident at the centre of the concentrator. Specifically, Fig. 6a shows the transmission, absorption and rejection coefficients as a function of the angle of incidence θ. The calculations are aggregated for 100 rays incident at random points of incidence with a full azimuthal scan. As seen, the transmission coefficient τ(θ) remains well above 90% up to a 65o angle of incidence. Fig. 6b shows the angular dependence of the transmittance η(θ) = τ(θ)[1 – R(θ)]—that is, the corrected transmission by subtracting the reflected fractional intensity R(θ) at the front surface. Assuming a uniform angular distribution of the incident flux, we define the total power density Φo of diffuse light incident onto a unit area parallel to the surface of Earth as: Fig. 6: Open in new tabDownload slide (a) Angular dependence of transmission and loss coefficients at Cg = 3. (b) Transmittance at Cg = 3. Φo=∫π/20A.cos(ϑ)dϑ=A(11a) where A is an arbitrary irradiance constant. Analogously we define the energy (exergy) efficiency Eeff of the concentrator for diffuse light as: Eeff=∫π/20[A.cos(ϑ)η(ϑ) Φ o]dϑ=0,8456(11b) Fig. 6b shows also the transmittance of an OE with ideally reflecting walls. The ideal OE has otherwise the same geometrical dimensions and absorption as the OE in the concentrator at the same Cg. It is seen that the peripheral vision of the ideal OE is seemingly somewhat lower than that of the concentrator. The explanation is that the concentrator has a slightly lower optical concentration coefficient due to the finite gap widths, which effectively reduces the input aperture. On the other hand, the effect of gap drift discussed above to which all fractal rays are subjected through the bandgap ejection results in a wider peripheral vision. Thus, the bandgap ejection makes use of the gap widths to extend the peripheral vision to the limit. Fig. 7a and b show the same calculations for a concentration Cg = 4, where the energy efficiency Eeff = 74%, which reflects the obvious fact that the peripheral vision is a decreasing function of Cg. An interesting case is presented in Fig. 8a and b, which illustrate the performance of the same concentrator with OEs of the refractive index of water, 1.33, at a concentration Cg = 2.5. The energy efficiency in this case Eeff = 82%. Thus, this suggests that OEs made of water in combination with anti-freezing agents may potentially be used in low-end applications. The obvious conclusion, however, is that the higher the refractive index, the higher the performance following standard etendué conservation. More importantly, the above findings demonstrate that the performance of fractal concentrators is practically identical to a conical waveguide with ideally reflecting walls and the same absorption with the same absorption. Fig. 7: Open in new tabDownload slide (a) Angular dependence of transmission and loss coefficients at Cg = 4. (b) Transmittance at Cg = 4. Fig. 8: Open in new tabDownload slide (a) Angular dependence of transmission and loss coefficients at Cg = 2.5 and n = 1.33. (b) Transmittance at Cg = 2.5 and n = 1.33. It should be noted that the overall peripheral vision of the concentrator is notably broader than calculated above, since the concentrator itself occupies a sizeable solid angle, typically 20–30° from normal incidence. In other words, there are always OEs that have a smaller incidence angle than that of the central OE for non-normal incidence up to the angle at which mutual shadowing between neighbouring concentrators occurs. Finally, the gap widths take up a few percent of the input aperture, which needs to be included in the overall performance of the concentrator. The mirror losses have been found to be negligible for all of the above cases. The average mirror losses of rays entering the concentrator through OEs at its very periphery are calculated to be <2%. Thus, the average mirror losses for the whole concentrator are well below 1%, for which reason they are not included in the above calculations. 5 Fabrication and applications The optical elements and other elements may be fabricated of low-cost and recyclable polymer materials of high optical quality such as PMMA, PC, etc, say, by injection moulding and other similar methods. The gaps between the optical elements are defined by appropriate spacers, as illustrated in Fig. 9. The exit aperture, along with the top protective screen, encapsulate the concentrator and provide additional thermal insulation at the same time. The absorber/solar cell is in intimate contact with the exit aperture and is optically matched to the latter. The whole concentrator is assembled in a dust-free environment. Further, since the absorption losses are sufficiently low, fractal solar concentrators may advantageously be thermally insulated from the ambient, as illustrated in Fig. 9. Thus, the concentrated light flux is fed onto a light absorber, which may be a solar cell designed for the intended light concentration or just a light-to-heat converter. In either case, the absorber is thermally insulated from the ambient meaning that most of the heat generated will be transferred to the heat-carrier fluid. In other words, deducting the obvious losses, the remainder of the energy is transformed into usable forms of energy. These types of systems are called concentrating photovoltaic–thermal (CPVT) systems. In practice, arrays of solar concentrators such as that shown in Fig. 9 are to be assembled into panels with a common heat-carrier system thermally insulated from the ambient. Generally, both consumer and industrial applications in the form of CPVT and CSP power-generation systems are feasible. The heat generated in CPVT systems is generally rated as low-grade, since the temperature in the carrier fluid is limited by the maximum operating temperature of the solar cells, typically <90oC, and is suitable for household use, greenhouse heating, etc. Where suitable, a heat reservoir may be used in conjunction for short-term heat storage. Not least, a combination of CPVT and a Sterling engine would increase the electrical efficiency in the warmer months. A heat exchanger is needed in the daytime to dump excess/waste heat to the ambient as well as during the night-time to cool the heat reservoir for daytime cooling purposes. A further possibility is to use the excess heat in the warmer months for absorption cooling. Medium-grade-heat CSP systems may be possible if the thermal insulation is improved. Water desalination is a likely application too. Building integrated solar is another possible application where panels of solar concentrators are to be integrated with the building facades making them an integral part of the building insulation and energy system. Fig. 9: Open in new tabDownload slide Thermally insulated concentrator with active cooling 6 Discussion Certain justification for the applications proposed above is in order. To this end, we consider the example in Fig. 6a. Assuming thermal conductivity of 0.2 W/K/m (typical for plastics) and a height of the OE of 10 cm as well as a temperature difference between the heat-carrier fluid and the ambient of 60oC, we arrive at a thermal loss of 120 W/m2. This loss is then divided by Cg, since the emitting surface now is Cg times smaller. Hence, we have an effective thermal loss of 40 W/m2. Assuming further spring irradiance of 250 W/m2 as well as an energy efficiency Eeff of the concentrator of 80%, we arrive at an operating energy efficiency of 64% under the above conditions. This figure becomes 76% at solar irradiance of 1000 W/m2. In reality, one would vary the temperature difference within certain limits to maintain optimal operating efficiency. The example in Fig. 8 would certainly have an inferior performance due in part to the lower Cg and in part to the higher thermal conductivity of water compounded with convection losses. Nevertheless, this case, perhaps, is not to be dismissed, particularly in smaller geographical latitudes. Further, calculating the CAP FOM for the concentrators in Figs 6 and 7, we arrive at CAP >1.5. One might question the use of CAP FOM in this case but nevertheless it provides a qualitative comparison with the current state of the art. As stated above, the fractal concentrator performs at the limits of what is possible with ideally reflecting concentrating waveguides. This performance is achieved partly by the fractal cascade and to some extent by the bandgap, both of which extend the peripheral vision to the physical limit. Further examples of bandgap use for optimizing performance while minimizing cost will be presented in coming communications. Next, it is thought that spectral uniformity is not an issue with fractal concentrators despite the fact that the refractive index is dispersive. If the latter makes any difference, it would be at the edge of the peripheral vision, which constitutes a negligible fraction of the total energy harvested. The spatial uniformity of the concentrated flux has not been studied, although it is also not thought to be an issue too due to the stochastic nature of concentration and the lack of preferred focusing mechanisms. Concentrator transmission, though, may be spectrally sensitive since both the refractive index and absorption are dispersive. The absorption dependence, however, is readily calculated, since it is additive in nature. Finally, light concentration means that one uses correspondingly smaller (in area) solar cells. In view of the low concentration coefficients, however, it would hardly make any economic sense to use fractal concentrators in PV applications alone. The situation changes radically when bi- and tri-generation systems are considered, since now one can harvest the dominant fraction of the available solar energy and convert it into usable forms of energy. 7 Conclusions A novel approach to concentrating both direct and indirect light based on the individual and collective confinement properties of an array of refractive optical elements has been presented. A theoretical framework relating the geometry of the optical elements to their confinement properties has been developed. Consequently, the principle of operation and major design rules of a light concentrator have been established. The main mechanisms of light propagation through the concentrator as well as associated losses have been described in detail. The concentrators are shown numerically to exhibit acceptance angles for CAP FOMs of the order of 1.5. The main advantage of the proposed concentrator is when it is used in bi- and tri-generation solar systems. Thus, it is numerically demonstrated that such concentrators may be used in a range of applications such as CPVT and CSP with operating energy efficiencies >70%. It is also concluded that materials with higher refractive indices yield higher performance. Current polymer materials in this context have moderate refractive indices and low-cost optical materials with higher refractive indices are needed to increase concentration and, hence, performance as well as to reduce material use. Acknowledgments The publication of this article has been sponsored by Vinnova through contract 2019–04927. The work is done under contracts VK-1931:02 and RK-1932:08. Tribute: In memory of George Carter and Mike J. Nobes. Conflict of Interest None declared. References [1] Fraunhofer Institute for Solar Energy Systems . Photovoltaics Report . https://www.ise.fraunhofer.de/content/dam/ise/de/documents/publications/studies/Photovoltaics-Report.pdf ( 12 August 2020 , date last accessed). [2] Boretti A , Castelletto S, Al-Zubaidy S. Concentrating solar power tower technology: present status and outlook . Nonlinear Engineering 2019 , 8 : 10 - 31 . Google Scholar Crossref Search ADS WorldCat [3] Dixit S . Solar technologies and their implementations: a review . Materials Today: Proceedings 2020 , 28 : 2137 - 48 . Google Scholar Crossref Search ADS WorldCat [4] Gonzalo AP , Marugán AP, Márquez FP. A review of the application performances of concentrated solar power systems . Appl. Energy 2019 , 255 : 113893 . Google Scholar Crossref Search ADS WorldCat [5] Indira SS , Vaithilingam CA, Chong KK, et al. A review on various configurations of hybrid concentrator photovoltaic and thermoelectric generator system . Solar Energy 2020 , 1;201 : 122 - 48 . Google Scholar Crossref Search ADS WorldCat [6] Jia Y , Alva G, Fang G. Development and applications of photovoltaic–thermal systems: a review . Renew Sust Energ Rev 2019 , 102 : 249 - 65 . Google Scholar Crossref Search ADS WorldCat [7] Răboacă MS , Badea G, Enache A, et al. Concentrating solar power technologies . Energies 2019 , 2 : 1048 . Google Scholar Crossref Search ADS WorldCat [8] Buljan M , Mendes-Lopes J, Benítez P, et al. Recent trends in concentrated photovoltaics concentrators’ architecture . J Photonics Energy 2014 , 4 : 040995 . doi:10.1117/1.JPE.4.040995. Google Scholar Crossref Search ADS WorldCat [9] Katardjiev I . Optical system for manipulation and concentration of diffuse light . World Patent WO 2020/139186 A1. 7 July 2020 . Google Scholar OpenURL Placeholder Text WorldCat [10] Hébert M. Reflection and transmission of light by a flat interface, Fresnel’s formulae . Ph.D. Thesis. Paris, France : Institut d’Optique ParisTech , 2013 . http://paristech.institutoptique.fr/site.php?id=797&fileid=11468 ( 12 August 2020 , date last accessed). Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC © The Author(s) 2020. Published by Oxford University Press on behalf of National Institute of Clean-and-Low-Carbon Energy This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com © The Author(s) 2020. Published by Oxford University Press on behalf of National Institute of Clean-and-Low-Carbon Energy

Journal

Clean EnergyOxford University Press

Published: Dec 31, 2020

References