Evaluation of the Spot Shape on the Target for Flat Heliostats

energies

Article Evaluation of the Spot Shape on the Target for Flat Heliostats

David Jafrancesco, Daniela Fontani ID , Franco Francini and Paola Sansoni * CNR-INO National Institute of Optics, Largo E. Fermi, 6-50125-Firenze, Italy; [email protected] (D.J.); [email protected] (D.F.); [email protected] (F.F.) * Correspondence: [email protected]; Tel.: +39-055-23081

Received: 23 May 2018; Accepted: 18 June 2018; Published: 21 June 2018

Abstract: The aim of this study is to evaluate the changes of the spot shape on the target in dependence of the variations of size and faceting of a flat heliostat or an array of heliostats. The flat heliostat, or a flat heliostat array, is a layout common for Concentation Solar Power (CSP) plants. The spot shape is evaluated by means of a numerical integration of an appropriate function; in order to confirm the results, both an analysis based on the Lagrange invariance and some simulations are performed. The first one validates the power density value in the central part of the spot, while the simulations assess the spot shape, which in its central part differs less than 3% from the calculated result. The utilized numerical method does not require specialized software or complex calculation models; it determines an accurate spot shape but cannot take into account shading and blocking phenomena.

Keywords: optical design; heliostat; solar; concentration

1. Introduction During the last years, many theoretical and experimental researches were carried out in order to increase the efficiency and the ratio benefits/costs of solar plants. Many efforts were made in order to improve optical layouts, thermal exchange systems and materials (a not very recent but comprehensive analysis is given in [1]), but researchers turned their attention also to increasing the density of the heliostat field [2,3] or to studying the dynamic of the receiver [4,5]. The correct sizing of the receiver was investigated in many scientific articles (e.g., [5]), as well as the flux distribution on the receiver [6,7]. A key point of these studies was the correct sizing of mirrors and target, also referring to multi-faceted plane mirrors that represent a useful alternative to spherical or parabolic mirrors, which are more expensive and difficult to manufacture. The shape of the spot on the target is a fundamental parameter because it strongly influences the size of the target, which has to be a trade-off between the requirements for radiation collection and energy losses [8]. The aim of the present work is to determine the spot shape on the target in a heliostat field by means of an easy but efficient mathematical analysis. The examined system is a central tower with flat or faceted heliostats, which is a common layout of commercial Concentration Solar Power (CSP) plants. The proposed methodology permits the correct sizing of the mirroring elements also taking into account the concentration limit achievable in the actual layout. In this framework, a key element of the design phase is represented by the determination of the spot shape on the target that can be assessed in two different ways: by a raytracing (generally using a Monte-Carlo method) or by the development of a suitable analytic function [9]. The first way led to the creation of some dedicated software tools such as STRAL [10] or SolTrace [11]. In order to obtain useful results they have to trace a large number of rays, so a large processing power is requested; despite the increased performance of the computers, the raytracing requires relatively long time.

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The problem of the determination of an analytical function that describes the spot shape on the receiver was studied by many authors, which very often proposed complex methods of convolution of solar divergence and errors. Among them, Elsayed and Fathalah in [12] tried to evaluate the flux density function on the receiver, but they utilized the average radiation concentration as input parameter and the angle between the adjacent sides of the principal image of the heliostat (that is the image generated only by collimated sun rays) as control parameter, which are very difficult data to evaluate. Lipps and Wanzel in [13] performed a very careful analysis for a flat polygonal heliostat, taking into account also shading and blocking phenomena. However, the beam enlargement is expressed by a polynomial approximation instead of a standard deviation value, and in general, the expression of the data is quite convoluted, requiring the calculation of the image of each single heliostat based on mirror boundaries definition. Collado et al. in [14] supplied an analytic description of the beam, but only for spherical-shaped heliostats (the convolution function in this case convolves three functions: sun shape, errors and concentration) with the receiver at suitable distance to produce the minimum circle of confusion; in this paper there is also a table with a succinct description of the characteristics of some previous works. In this field, large efforts were devoted to develop efficient software for the evaluation of the heliostats fields. An early example was DELSOL, developed from 1979 at Sandia Laboratoires. The upgraded version, DELSOL3, is described in [15], but now it seems no longer updated [16]; this software utilizes an analytical Hermite polynomial expansion/convolution-of-moments and includes an economic analysis too. HELIOS has the same scope of DELSOL, but is more accurate and slow; in [17] is stated: “HELIOS is used where a detailed description of the heliostat is available and an extremely accurate evaluation of flux density is desired”. Both these software tools are built to evaluate the performance of a whole heliostats field, while the aim of the proposed work is to choose the correct size of heliostats in an early stage of design. Others software tools for the study of the flux distribution generated by a spherical heliostat, assuming a circular symmetry, are UNIZAR and HFLCAL, analyzed by Collado in 2010 [18]. In particular, an interesting application is the program HFLCAL, presented by P. Schwarbözl et al. at the SolarPaces 2009 conference [19], where the authors utilize a mathematical method of analysis very similar to the one shown in the present work. The difference is that HFLCAL, like UNIZAR, considers spherical faceted mirrors (with spherical curvature both on the facets and on the envelope). These assumptions lead to formulas that are to all appearances similar to the ones in the following sections of this paper, but actually they are quite different. In fact, considering the spot, in HFLCAL it always has a shape that is basically Gaussian and modified by the astigmatism. The method utilized in this work concerns flat heliostats, which are very often employed in CSP plants with a central tower to decrease the costs [20]. The suggested approach is much simpler than the previous ones (it requires only a standard worksheet like Excel and few VBA (Visual Basic for Applications) program lines, but no dedicated software codes) and permits one to perform a useful analysis on the characteristics of the spot and on how the heliostat parameters influence them. The key elements are the approximation of the angular shape of the beam reflected by a point on the heliostat surface to a bi-dimensional Gaussian distribution (centered on the direction of the specular reflection) and its equivalence with the irradiance space distribution generated by the same beam on the target plane. Hence, the irradiance on a point of the target is calculated as the sum of the contributions from all the points of the heliostat (the atmospheric attenuation is neglected). The final results include the maximum power concentration on the spot center achievable with a faceted heliostat, the variation of the spot shape depending on the heliostat distance and the minimum size of the facets (under which no advantages arise). In order to confirm the results achieved by the mathematical analysis, a set of simulated models have been developed using a non-sequential optical computer-aided design (CAD, by Lambda Research TracePro). The use of a non-sequential optical CAD permits to evaluate the irradiance distribution on the target potentially taking into account all the physical phenomena involved (reflection, refraction, scattering ... ). In fact, while a sequential optical CAD, where the order in which the rays hit the surfaces is defined by the user, is dedicated to the evaluation of aberrations, Energies 2018, 11, 1621 3 of 18 in a non-sequential optical CAD the rays hit the surfaces depending on their optical path, so it is focused on the calculation of radiometric and photometric quantities as irradiance maps on the target. The non-sequential optical CADs were developed originally for illumination engineering; in [21] a short description of their characteristics is provided. Moreover, the application of Lagrange invariance confirms the results obtained from the calculation and the simulations.

2. Mathematical Analysis This section provides a concise and precise description of the mathematical analysis. Firstly, the study considers a rectangular-shaped flat heliostat with the target on the axis. The beam angular distribution essentially depends on solar divergence, heliostat pointing errors and mirrors surface features (slope errors, gloss). The solar divergence has its peculiar shape, that can be roughly assimilated to an uniform distribution with semi-aperture 4.7 mrad [22], as well as the heliostat pointing errors ([23], mean value = 3 mrad). Regarding the mirrors surface features, the angular shape of the reflected beam is Gaussian, with standard deviation (std. dev.) σ = 5 mrad [24]. In effect, Rabl in [25] demonstrated that the error cone due to the reflection on a mirror surface is elliptical, but Guo and Wang in [26] realized an equivalence between circular and elliptical Gaussian distribution. Moreover, it has to be considered that the sun position varies during the day and the seasons, then the axes directions of the elliptical Gaussian distribution on the receiver plane greatly vary. Consequently, the assumption of a Gaussian circular distribution is more adequate to the aim of this study, which is the correct sizing of heliostats or heliostat facets in a solar plant that works on a large range of sun positions. Researchers demonstrated that in many cases the convolution between the enlargements, both intrinsic of solar light and due to the errors, can be considered as a Gaussian distribution when the first cause is less important than the second one [27]. Then, in this case it is worth to approximate √ the previous uniform distributions to Gaussian ones with std. dev. σ = a/ 3, where a is the half width of the uniform distributions [28]. Thus, from a practical point of view the beam enlargement of the beam reflected by the heliostat field can be treated as Gaussian with std. dev. σ = 5.9 mrad [29]. If I(θ) is the intensity distribution with respect to the reflection direction and σθ is its standard deviation, for a single point on the heliostat surface this relation is valid [30]:

θ2 ( ) = − I θ I0exp 2 (1) 2σθ where:

- I0 is the peak intensity value; - θ is the angle between the normal to the heliostat surface and the direction of the specular reﬂection;

- σθ is the standard deviation of the angular distribution taking into account all the causes of beam enlargement.

It is preferable to rewrite that formula in terms of irradiance E(x, y) on the generic target point P = (x, y). Utilizing the expression E = I/d2 (where d is the heliostat-target distance [31]) and expressing the angle θ of Equation (1) as a function of (x, y):

 √ 2  x2 + y2  x2 + y2  arctg d 2 2   2 x + y E(x, y) = E exp−  =∼ E exp − d = E exp − (2) 0  2  0  2σ 2  0 2  σx,y  x,y 2σx,y 2 arctg d d2 Energies 2018, 11, 1621 4 of 18 Energies 2018, 11, x FOR PEER REVIEW 4 of 18

DueDue to to the the fact that a single pointpoint isis considered,considered, obviouslyobviouslyσ σxx == σσyy = σx,,yy == ( (σσθθ·d·d),), where where dd isis the the distancedistance between between the the heliostat heliostat and and the the target, target, indicate indicatedd in in Figure Figure 11.. Equation (2) is equivalent to the EquationEquation (7) (7) in in [26] [26] when when the the center center of of the the refe referencerence system system coincides coincides with with the the point point of of maximum maximum irradianceirradiance EE00. .

Target

x xF

d Heliostat

2LF

Figure 1. Scheme of the projection and enlargement from the heliostat to the target. Figure 1. Scheme of the projection and enlargement from the heliostat to the target.

TheThe next next step step is is to to consider consider an an extended extended squared squared heliostat; heliostat; the the situation situation is is more more complicated, complicated, becausebecause the the convolution between aa squaresquare (the(the heliostatheliostat surface) surface) and and the the Gaussian Gaussian distribution distribution has has to tobe be taken taken into into account. account. In In this this case, case due, due to theto the integration integration on on the the surface, surface,ESUN ESUN(the (the solar solar irradiance irradiance on onthe the heliostat) heliostat) must must substitute substitute the Ethe0 constant E0 constant (irradiance (irradiance on the on center the center of the target),of the target), with the with suitable the suitablenormalization; normalization; the actual the normalization actual normalization corresponds tocorresponds a bi-dimensional to a Gaussianbi-dimensional distribution Gaussian with distributionzero correlation with between zero correlation the two variablesbetween thexF andtwo yvariablesF [9]. For x aF squareand yF [9]. heliostat For a withsquare side heliostat 2LF: with side 2LF: Z L Z L 2 ! 2 ! ESUN F F (x − xF) (y − yF) ( ) = − − E x, y 2 exp 2 exp 2 dxFdyF (3) , 2πσ= xy −L−L − 2σx,y − 2σx,y (3) F F , ,

EquationEquation (3) (3) is is similar similar to to the the Equation Equation for for a Gau a Gaussianssian enlargement enlargement on the on thetarget target utilized utilized by the by softwarethe software HFCAL HFCAL (see (seethe theEquation Equation (7) (7)in [18]), in [18 ]),but but the the total total flux ﬂux from from the the focusing focusing heliostat heliostat in in EquationEquation (3) (3) is replaced by the sun irradiance ESUN withwith an an integration integration of of the the exponential exponential terms terms on on thethe heliostat heliostat surface. surface. From From an an optical optical point point of of view, view, it it means means that that the the irradiance irradiance on on the the target target point point ((xx,, yy)) is is obtained obtained as as the the sum sum of of the the contributions contributions from from all all the the points of the flat ﬂat heliostat. Substituting:Substituting: x − xF tx = −√ = σ 2 x,y √2, y − yF ty = √ 2σx,y − = √2,

The final Equation is:

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x−L y−L √ F√ F √Z 2σ √Z2σ ESUN − 2 − 2 tx ty E(x, y) = e e dtxdty (4) , = π (4) x+L y+L √ F√ F 2σ 2σ √ √ It is thethe bi-dimensionalbi-dimensional Error Error Function Function (erf) (erf) Equation Equation (integral (integral of Gaussianof Gaussian function) function) that that does does not notadmit admit analytical analytical solution. solution. However, However, it is it very is very easy easy to perform to perform a numerical a numerical integration. integration.

3. Numerical Integration and Simulation Results The irradiance EE((xx,, yy)) on on the the target target can can be be obtained obtained by by numerical numerical integration integration of Equation of Equation (4). (4). In FigureIn Figure 2 the2 the behavior behavior of ofthis this function function along along the the diagonal diagonal (for (for x =x y=) isy) shown is shown for for different different values values of of L . The irradiance values are calculated for a square heliostat with side 2L , considering four LF. TheF irradiance values are calculated for a square heliostat with side 2LF, consideringF four different different sizes (L ) for the heliostat. The heliostat-target distance is 50 m, σ = 0.264 (total angular sizes (LF) for the heliostat.F The heliostat-target distance is 50 m, σx,y = 0.264 (totalx,y angular divergence divergence = 5.9 mrad) with the E constant set to 1. Obviously in Figure2 the curves for different = 5.9 mrad) with the ESUN constantSUN set to 1. Obviously in Figure 2 the curves for different values of LF values of L refer to different ordinate values in the center of the target, because for example the mirror refer to differentF ordinate values in the center of the target, because for example the mirror with LF = L L with0.25 mF has= 0.25 area m 16 has times area smaller 16 times than smaller the mirror than the with mirror LF = 1 with m. F = 1 m.

Figure 2. Irradiance E(x, y) on the target for a square heliostat with side 2LF. Figure 2. Irradiance E(x, y) on the target for a square heliostat with side 2LF. Some simulations were performed by means of TracePro (TracePro 2018 Expert Release 18.2, LambdaSome Research simulations Corporation, were performed Littleton, byMA, means USA), of a TraceProsoftware (TraceProdedicated 2018to lighting Expert simulation. Release 18.2, In theLambda following Research text, Corporation,“calculated results” Littleton, or “calcula MA, USA),ted profiles” a software refer dedicated to the values to lighting obtained simulation. by the numericalIn the following integration text, “calculated of Equation results” (4), “simulated or “calculated results” profiles” or “simulated refer to theshape” values refer obtained to the values by the obtainednumerical by integration TracePro raytracing. of Equation The (4), irradiance “simulated ma results”ps and orprofiles “simulated are presented shape” referin Figures to the 3 values and 4 forobtained two of by the TracePro examined raytracing. cases. These The irradiancetwo simulation mapss andwere profiles carried are out presented keeping the in Figures other parameters3 and4 for astwo in of the the previous examined numerical cases. These calculation. two simulations were carried out keeping the other parameters as in the previousThe target numerical irradiance calculation. map a and the corresponding vertical/horizontal profiles b are plotted in FigureThe 3 for target mirror irradiance side LF map= 0.5 am and and the in Figure corresponding 4 for LF = vertical/horizontal 2 m. profiles b are plotted in Figure3 for mirror side LF = 0.5 m and in Figure4 for LF = 2 m.

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(a)

(b)

Figure 3. Target irradiance map (a) and corresponding central profiles (b) with mirror side LF = 0.5 m. Figure 3. Target irradiance map (a) and corresponding central proﬁles (b) with mirror side LF = 0.5 m.

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(a)

(b)

Figure 4. Target irradiance map (a) and corresponding central profiles (b) with mirror side LF = 2 m. Figure 4. Target irradiance map (a) and corresponding central proﬁles (b) with mirror side LF = 2 m.

Energies 2018, 11, 1621 8 of 18 Energies 2018, 11, x FOR PEER REVIEW 8 of 18 Energies 2018, 11, x FOR PEER REVIEW 8 of 18 There is good agreement between the theoreticaltheoretical calculation results and the irradiance map proﬁles obtained from the TracePro simulations. Some examples of comparison are presented in profilesThere obtained is good from agreement the TraceP betweenro simulations. the theoretica Somel calculationexamples ofresults comparison and the are irradiance presented map in Figures5 and6, for LF = 0.5 m (Figure5) and for LF = 2 m (Figure6). Figuresprofiles 5obtained and 6, for from LF = the0.5 mTraceP (Figurero 5)simulations. and for LF =Some 2 m (Figureexamples 6). of comparison are presented in Figures 5 and 6, for LF = 0.5 m (Figure 5) and for LF = 2 m (Figure 6).

Figure 5. Comparison between simulated and calculated irradiance profiles for LF 0.5 m. Figure 5. Comparison between simulated and calculated irradiance proﬁles for LF 0.5 m. Figure 5. Comparison between simulated and calculated irradiance profiles for LF 0.5 m.

Figure 6. Comparison between simulated and calculated irradiance profiles for LF 2 m.

Figure 6. Comparison between simulated and calculated irradiance profiles for LF 2 m. Please Figurenote the 6. Comparisondifferent shape between of the simulated spot on and the calculated receiver, irradiancedue to the proﬁles effect forof theLF 2 divergence. m. WhatPlease happens note the with different faceted shape mirrors of the is spotshown on in th Fie receiver,gure 7: the due heliostat to the effect(formed of theby manydivergence. mirrors) has stillPleaseWhat the happens notesame the size with different (side faceted 2L shapeF = mirrors 2 m), of the but is spot shown the onfacets in the Fi receiver,numbergure 7: the changes due heliostat to the from effect (formed 1 to of 256. the by divergence.manyIt is supposed mirrors) thathas stillallWhat the the happensfacets same lay size with of (sidea faceted spherical 2LF mirrors= 2 envelope m), isbut shown theand facets inare Figure oriented number7: the in heliostatchanges order to (formedfrom reflect 1 toits by 256.central many It mirrors)is ray supposed toward has thestillthat center theall samethe offacets sizethe (sidelayreceiver of 2 LaF spherical =(the 2 m), little but envelope angle the facets differen and number arece oriented is changes neglected). in from order 1The toto 256. reflectirradiance It is its supposed central along ray thatthe toward target all the diagonalfacetsthe center lay is of ofplotted a sphericalthe receiver in Figure envelope (the 7 for little and increasing angle are oriented differen number ince order ofis facetsneglected). to reflect per side, its The central from irradiance ray1 up toward to along16 (256the the centerfacets target inof total).thediagonal receiver is plotted (the little in angle Figure difference 7 for increasing is neglected). number The of irradiance facets per along side, the from target 1 up diagonal to 16 (256 is plotted facets in Figuretotal). 7 for increasing number of facets per side, from 1 up to 16 (256 facets in total).

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Figure 7. Irradiance E(x, y) on the target for faceted mirrors. Figure 7. Irradiance E(x, y) on the target for faceted mirrors. The case of the single-facet heliostat seems different from the other curves; actually, it is in Theaccordance case of with the them. single-facet FigureIn fact 7.the Irradiance heliostat 1 × 1 curve E seems(x, yis) onequal differentthe targetto the for irradiance from faceted the mirrors. variation other curves; along the actually, diagonal it is in accordanceof the heliostat with them. modified In fact by thethe 1flux× 1enla curvergement. is equal Correctly, to the irradianceits maximum variation irradiance along is ESUN the because diagonal of a flat heliostat does not focus the beam. The increase of the facets number (canted towards the target) the heliostatThe modified case of the by single-facet the flux enlargement. heliostat seems Correctly, different its maximumfrom the other irradiance curves; is actually,E because it is in a flat makes the faceted heliostat more and more similar to a spherical mirror focused on SUNthe target (it heliostataccordance does not with focus them. the In beam. fact the The 1 × increase 1 curve ofis equal the facets to the number irradiance (canted variation towards along thethe target)diagonal makes representsof the heliostat a limit modified case). by the flux enlargement. Correctly, its maximum irradiance is ESUN because the faceted heliostat more and more similar to a spherical mirror focused on the target (it represents a a flatPlease heliostat note does that not the focus areas the under beam. the The curves increase of Figure of the 7 facets are not number equal because(canted towardseach plotted the target) curve limit case). ismakes the shape the faceted of the function heliostat along more a andline (themore diagon similaral), to and a sphericalit does not mirror represent focused the energy on the collection target (it Pleaseonrepresents the plane. note a thatlimit thecase). areas under the curves of Figure7 are not equal because each plotted curve is the shapeItPlease of is the interesting note function that theto along noteareas athat lineunder the (the the concentration diagonal),curves of Figure and in x it 7= doesare 0 seemsnot not equal represent to havebecause a the maximum.each energy plotted collection This curve is on the plane.confirmedis the shape by of Figure the function 8, which along shows a line the (the irradiance diagonal), in and the ittarget does notcenter, represent at x = the0, depending energy collection on the Itvalueson is the interesting plane.of the heliostat-target to note that thedistance. concentration In order intox see= 0 how seems convenient to have a could maximum. be to Thiscompose is confirmed the by FigureheliostatIt8, is which surface interesting shows with tonumerous the note irradiance that facets, the inconcentrationthe the irradiance target center,isin plotted x = at0 seemsversusx = 0, tothe depending have number a maximum. of on facets the valuesper This side. is of the heliostat-targetconfirmed by distance. Figure 8,In which order shows to see the how irradiance convenient in the couldtarget center, be to compose at x = 0, depending the heliostat on the surface with numerousvalues of the facets, heliostat-target the irradiance distance. is plotted In order versus to see the how number convenient of facets could per be side. to compose the heliostat surface with numerous facets, the irradiance is plotted versus the number of facets per side.

Figure 8. Irradiance in the target center for different heliostat-target distances d.

The result is quite intuitive: as far is the target from the heliostat, as useless is to divide the heliostat in manyFigure facets. 8. Irradiance in the target center for different heliostat-target distances d. Figure 8. Irradiance in the target center for different heliostat-target distances d.

The result is quite intuitive: as far is the target from the heliostat, as useless is to divide the heliostat in many facets.

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The result is quite intuitive: as far is the target from the heliostat, as useless is to divide the Energies 2018, 11, x FOR PEER REVIEW 10 of 18 heliostat in many facets. The map of the target is shown in Figure9 for heliostat size 2 × 2 m (2L = 2 m) with 8 × 8 facets, The map of the target is shown in Figure 9 for heliostat size 2 × 2 m (2LFF = 2 m) with 8 × 8 facets, × target size 2 × 22 m, m, and and heliostat-target distance distance dd = 50 50 m. The The result is expressed as concentration (with respect to the solar irradiance on the heliostatheliostat = 1). For For comparison, comparison, in in Figure Figure 1010 therethere is the result obtained from the simulation for the same configuration.configuration. All the results obtained from the TracePro simulations are in accordance with the theoretical calculation. Figure Figure 1111 reportsreports the the comparison between between the the calculated calculated profile profile and and the the two profiles profiles of the simulated irradiance map. The The calculated calculated profil profilee is diagonal on the target and the two simulated profilesprofiles (horizontal and vertical on the target) areare diagonalsdiagonals withwith respectrespect toto thethe heliostat.heliostat. Considering thethe datadata ofof Figure Figure8 and8 and the the related related calculation calculation methods, methods, it is it possible is possible to estimate to estimate that thatfor the for heliostat-target the heliostat-target distance distance of 50 of m 50 the m limit the li concentrationmit concentration on the on target the target center, center, when whenthe facets the number → ∞ (i.e., L → 0), is 7.313. This result coincides with the maximum concentration value facets number  ∞ (i.e.,F LF  0), is 7.313. This result coincides with the maximum concentration value obtained by means of a simulation with a parabolic mirror (the limit for a faceted mirror when L → 0): obtained by means of a simulation with a parabolic mirror (the limit for a faceted mirror whenF LF  × 0):the the averaged averaged concentration concentration on on a targeta target of of 20 20 ×20 20 mm mm is is 7.316. 7.316. TheThe softwaresoftware utilizesutilizes a Monte Carlo method to generate the ray tracing, then the accordanceaccordance with the previous resultresult isis veryvery good.good.

FigureFigure 9. The 9. Themap map of the of the target, target, expressed expressed as calculated concentration, concentration, for 2forLF 2=L 2F m= 2 with m with 8 × 8 8 facets. × 8 facets.

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Figure 10. Simulated irradiance map on the target for 2LF = 2 m with 8 × 8 facets. Figure 10. Simulated irradiance map on the target for 2LF = 2 m with 8 × 8 facets. Figure 10. Simulated irradiance map on the target for 2LF = 2 m with 8 × 8 facets.

Figure 11. Comparison between simulated irradiance profiles and calculated profile for an 8 × 8 heliostat field. Figure 11. Comparison between simulated irradiance profiles and calculated profile for an 8 × 8 × Figureheliostat 11. field.Comparison between simulated irradiance profiles and calculated profile for an 8 8 heliostatAnother field. additional confirmation of this result derives from the Lagrange invariant (for paraxial configuration,Another additional see [32] (pp.confirmation 18–19)). Since of this only result the derivesrays with from direction the Lagrange near the invariant central axis (for of paraxial the solar configuration,beam can be seeconcentrated [32] (pp. 18–19)). in the centerSince only of the the target rays ,with the sizedirection of this near zone the can central be calculated axis of the starting solar beamfrom can the be divergence concentrated with in respect the center to the of centralthe target axis, the of thesize solar of this beam. zone Considering can be calculated a divergence starting of −8 from0.01913 the divergence× σ (the Gaussian with respect std. dev.), to the the central related ax solidis of anglethe solar is 4.0 beam. × 10 Considering sr. The product a divergence of this value of

0.01913 × σ (the Gaussian std. dev.), the related solid angle is 4.0 × 10−8 sr. The product of this value

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Another additional confirmation of this result derives from the Lagrange invariant (for paraxial configuration, see [32] (pp. 18–19)). Since only the rays with direction near the central axis of the solar beam can be concentrated in the center of the target, the size of this zone can be calculated starting from the divergence with respect to the central axis of the solar beam. Considering a divergence of 0.01913 × σ (the Gaussian std. dev.), the related solid angle is 4.0 × 10−8 sr. The product of this value by the area of the mirror (4 m2) must be equal to the product of the angle subtended by the target toward the mirror area (0.001599 sr) multiplied by the area of the target central zone. Consequently the central zone of the target has an area of 1.0 × 10−4 m2. Then, the part of the solar beam that has divergence less than or equal to 0.01 × σ is evaluated and the irradiation on the central zone of the target is calculated: the final result is a concentration value of 7.312, very near to the previous value. The difference between the value calculated by means of the integral on the target (7.313) and the theoretical value evaluated by the Lagrange invariance (7.312) depends on some very little approximations in the integral calculation, e.g., the modifications of the irradiance values due to the facet angles are not considered. Considering Equation (4), it is easy to realize that in the point (x, y) = (0, 0) the irradiance value depends only on the ratio d/(2LF) (because σXY ~σθ·d, where σθ is constant). Thus, it is useful to evaluate the irradiance on the center of the target E(0, 0) in dependence of the facets number and of the ratio d/(2LF), where d is the heliostat-target distance and 2LF the side length of the squared heliostat. The results are shown in Table1; the last row is the limit value of concentration, evaluated by means of the Lagrange invariance.

Table 1. Irradiance on the center of the target (ESUN = 1), where d is the heliostat-target distance and 2LF is the length of the heliostat side; the limit is calculated by means of the Lagrange invariance.

Ratio d/2LF Facets No. Per Side 5 10 15 20 25 30 40 50 75 100 1 1.00 1.00 1.00 1.00 0.999 0.991 0.933 0.828 0.553 0.364 2 4.00 4.00 3.96 3.73 3.31 2.84 2.02 1.46 0.739 0.431 3 9.00 8.92 7.96 6.38 4.95 3.86 2.43 1.65 0.783 0.445 4 16.0 14.9 11.4 8.08 5.82 4.34 2.61 1.72 0.799 0.45 6 35.7 25.5 15.4 9.73 6.59 4.74 2.74 1.78 0.811 0.455 8 59.7 32.3 17.3 10.4 6.89 4.89 2.79 1.80 0.815 0.46 12 102 38.9 18.9 11.0 7.12 5.01 2.83 1.82 0.818 0.456 16 129 41.7 19.5 11.2 7.21 5.05 2.84 1.82 0.82 0.457 20 146 43.1 19.8 11.3 7.25 5.07 2.85 1.82 0.82 0.457 24 156 43.9 20.0 11.3 7.27 5.08 2.85 1.83 0.82 0.457 limit 181 45.6 20.3 11.4 7.31 5.10 2.86 1.83 0.821 0.457

4. Mirror Field As example a field of mirrors with five rows and five columns (with row distance = 4 m and column distance variable) and a receiver at 20 m height can be considered. The horizontal distance between the central heliostat (3rd row/3rd column) and the target projection on the ground is 20 m, the std. dev. = 5.9 mrad; moreover, the target is inclined toward the central heliostat. For the sake of simplicity, the sun elevation is 45 deg and the azimuth is 0 (it corresponds to noon, because the field is located north of the tower). The calculation takes into account the cosine factor due to the inclination of the beams with respect to the heliostats other that the central one. The data were obtained by means of a simple VBA procedure in an Excel datasheet, considering the x-direction and y-direction behaviors of the target irradiance for a set of single heliostats with side 2 m (LF = 1 m). The calculation was repeated for another set of 25-faceted mirror heliostats with the same size but with different distance between the columns. Figure 12 shows the behavior for column distance 3 m, Figure 13 for column distance 10 m. Energies 2018, 11, 1621 13 of 18

Energies 2018, 11, x FOR PEER REVIEW 13 of 18

Figure 12. Target irradiance for a 5 × 5 heliostat field with row distance 4 m and column distance 3 m. Figure 12. Target irradiance for a 5 × 5 heliostat ﬁeld with row distance 4 m and column distance 3 m. Figure 12. Target irradiance for a 5 × 5 heliostat field with row distance 4 m and column distance 3 m.

Figure 13. Target irradiance for a 5 × 5 heliostat field with row distance 4 m and column distance 10 m. Figure 13. Target irradiance for a 5 × 5 heliostat field with row distance 4 m and column distance 10 m. FigureFor the 13. configurationTarget irradiance of forFigure a 5 × 125 heliostatthe spot field is a with circle row (the distance behavior 4 m and in columnx and distancey direction 10 m. is the same,For then the only configuration the y curve of is Figure represented 12 the on spot the is gr aaph), circle while (the inbehavior the case in of x Figureand y 13direction the different is the same,behaviorFor then the in only the configuration x the direction y curve of with is Figure represented respect 12 the to spot theon ythe is direction a gr circleaph), (the becomeswhile behavior in theclear in case xand and of the Figure y directionspot 13shape the is thedifferentbecomes same, thenbehavioran oval. only in the the y curvex direction is represented with respect on the to graph),the y direction while in becomes the case ofclear Figure and 13the the spot different shape behaviorbecomes inan theoval.The x direction same behavior with respect is obtained to the yfor direction the simulated becomes field: clear for and comparison, the spot shape Figure becomes 14 displays an oval. the resultsThe obtained same behavior behavior from TracePro is is obtained obtained with forthe for theparameters the simulated simulated of field:Figure field: for 13. for comparison, The comparison, values areFigure Figureaveraged 14 displays14 overdisplays four the theraytracings,results results obtainedobtained using from 2.5 from TracePro× 10 TracePro8 rays with per with raytracing;the the parameters parameters each of raytracing, of Figure Figure 13. 13 based.The The values valueson the are are MonteCarlo averaged averaged overmethod, four 8 raytracings,starts with a using different 2.5 × 10random108 raysrays per perseed. raytracing; raytracing; The centra each eachl profiles raytracing, raytracing, of the based based simulated on on the MonteCarloirradiance mapmethod, are startscompared withwith with a a different different the calculated random random seed.profiles seed. The in centralThe Figure centra profiles 15.l profiles of the simulatedof the simulated irradiance irradiance map are comparedmap are withcompared the calculated with the profilescalculated in Figureprofiles 15 in. Figure 15.

Energies 2018, 11, 1621 14 of 18

Energies 2018, 11, x FOR PEER REVIEW 14 of 18

(a)

(b)

Figure 14. Simulated target irradiance for a 5 × 5 heliostat field with row distance 4 m and column × Figuredistance 14. Simulated 10 m, irradiance target irradiance map (a); corresponding for a 5 5 heliostatcentral profiles ﬁeld with(b). row distance 4 m and column distance 10 m, irradiance map (a); corresponding central proﬁles (b).

Energies 2018, 11, 1621 15 of 18 Energies 2018, 11, x FOR PEER REVIEW 15 of 18

Figure 15. Comparison between simulated irradiance profiles and calculated shapes for a 5 × 5 Figure 15. Comparison between simulated irradiance profiles and calculated shapes for a 5 × 5 heliostat heliostat field with row dist. 4 m and column dist. 10 m. field with row dist. 4 m and column dist. 10 m. As shown, the accordance between TracePro simulation and calculated spot shape (in x and y directions)As shown, is very the good, accordance the difference between between TracePro calc simulationulated and simulated and calculated profile spot is less shape than (in 2.5% x and up yto directions 0.25 m from) is verythe target good, center the difference and less betweenthan 5% calculatedup to 0.5 m and from simulated the target profile center. is less The than reported 2.5% updifference to 0.25 mis similar from the to target the variation center and among less thanthe resu 5%lts up of to the 0.5 four m from raytracings the target with center. different Thereported random differenceseeds. Therefore, is similar the to proposed the variation method among is applicable the results for of the fourevaluation raytracings of the with entire different heliostat random fields seeds.too. Therefore, the proposed method is applicable for the evaluation of the entire heliostat fields too.

5. Discussion 5. Discussion The analysis, founded on the TracePro simulation, presents some results that have to be pointed out. Concerning the method, basically it is a simplifiedsimplified convolution analysis that does not take into account blocking and shadowing effects, but allows to obtain results in good accordance with the simulations. In particular, it should be noted that there are not many studies on flatflat heliostats, due to the factfact that that the the most most important important analysis analysis software software (as HFLCAL (as HFLCAL for example) for example) manages manages focusing focusing mirrors. Anmirrors. example An ofexample study aboutof study the flatabout heliostats the flat is helio [12],stats where is Elsayed[12], where et al. Elsayed presented et experimentalal. presented resultsexperimental (for a laboratory results (for set-up); a laboratory in the conclusion, set-up); in they the noticed conclusion, that the they range noticed over whichthat the the range irradiance over iswhich constant the irradiance is reduced is whenconstant the is mirror reduced is movedwhen the further mirror away is moved from further the receiver away or from the the mirror receiver size decreasesor the mirror (point size 4, pagedecreases 409). In(point the present4, page work, 409). theIn the same present conclusion work, derives the same from conclusion a confrontation derives of Figuresfrom a confrontation3 and4 and from of Figures the following 3 and 4 Figureand from 16 ,the which following reports Figure the irradiance 16, which variation reports the on irradiance the spot forvariation two different on the spot distances for two from different the target. distances Figure from 16 plots the simulatedtarget. Figure irradiance 16 plots profiles simulated for single-facet irradiance heliostats with side 2LF = 1 m at distance 10 m and 30 m. profiles for single-facet heliostats with side 2LF = 1 m at distance 10 m and 30 m. Clearly, in Figure 16 the maximum irradiance value on the target is ESUN (the heliostat is not Clearly, in Figure 16 the maximum irradiance value on the target is ESUN (the heliostat is not focusing); this value can be increased if the heliostat is faceted, but the concentration has a maximum that depends on the heliostat-target distance distance (Fig (Figureure 88).). The issue was examined by LandmanLandman et et al. al. in [33]: [33]: for for canted canted flat flat facets, facets, they they define define the the image image dimension dimension as asthe the sum sum of the of theprojected projected size size of the of thesingle single facet facet and a and term a that term takes that into takes account into account the solar the enlargement solar enlargement (Equation (Equation (3) page 129); (3) page correctly, 129); correctly,the cited Equation the cited shows Equation thatshows the image that dimension the image has dimension a minimum has awhen minimum the facet when size thetends facet to zero size tends(it is the to solar zero (itenlargement is the solar by enlargement the distance). by However, the distance). due However,to the fact that due there to the is fact the that contribution there is the of contributiona single facet of only, a single it is facetnot immediately only, it is not apparent immediately that apparentsuperimposing that superimposing the spots from the many spots facets from manythere is facets a limit there to isthe a limitcenter to irradiance the center irradianceon the target. on theIn target.the present In the work present it is work shown it isthat shown the thatconcentration the concentration maximum maximum is related, is in related, addition in to addition the presence to the of presence aberrations of aberrations (for faceted (for heliostats faceted heliostatsthat corresponds that corresponds to an approximate to an approximate canting), to canting),a theoretical to a limit, theoretical as demonstrated limit, as demonstrated by means of the by meansLagrange of theinvariant Lagrange in invariantSection 3. in SectionThe fact3. that The factwith that a withfaceted a faceted heliostat heliostat it is possible it is possible to reach to reach an irradiance value on the center of the receiver similar to the irradiance generated by a parabolic

Energies 2018, 11, 1621 16 of 18

Energies 2018, 11, x FOR PEER REVIEW 16 of 18 an irradiance value on the center of the receiver similar to the irradiance generated by a parabolic heliostat is highlighted also by Ri Riveros-Rosasveros-Rosas et al. in [34], [34], where a specific speciﬁc layout of a solar furnace is studied.

Figure 16. Comparison between simulated irradiance profiles at two distances from the target. Figure 16. Comparison between simulated irradiance profiles at two distances from the target. Surely, the method and results presented in this work constitute a rough evaluation of the irradianceSurely, on the the methodtarget, because and results neither presented the blocking in thisand workshadowing constitute effects, a nor rough the asymmetry evaluation of the irradiancereflection onbeam the from target, the because heliostat neither are thetaken blocking into consideration. and shadowing Moreover, effects, nor the the canting asymmetry was consideredof the reflection always beam perfect, from that the heliostatis economically are taken plausible into consideration. only for special Moreover, applications the canting (as some was furnacesconsidered or alwaysscientific perfect, devices). that Theoretically, is economically it plausibleis possible only also for to special modify applications the software (as in some order furnaces to take intoor scientific account devices).imperfect Theoretically, canting, atmospheric it is possible absorption also to and modify blocking the software or shadowing, in order but to takeit would into constituteaccount imperfect a significant canting, complication atmospheric (in absorptionparticular concerning and blocking the or last shadowing, two phenomena), but it would undermining constitute thea significant aim of this complication study. Nevertheless, (in particular in the concerning early design the phase last two it is phenomena),important to have undermining a fast evaluation the aim of the this best study. heliostat Nevertheless, facet size in both the earlyfor structural design phaseand economic it is important considerations; to have athis fast is evaluation the scenario of wherethe best theheliostat present facet work size is applicable. both for structural and economic considerations; this is the scenario where the present work is applicable. 6. Conclusions 6. Conclusions The aim of the present study is to supply methods, important parameters and results concerning the powerThe aim density of the distribution present study of is the to supplyspot generate methods,d on important the receiver parameters from a andflat resultsheliostat concerning or a flat heliostatsthe power array. density These distribution methods and of the data spot are generated particularly on useful the receiver in the design from aphase flat heliostat of a solar or field a flat to guideheliostats the array.optical These designer methods in the and devel dataopment are particularly of the field useful configuration. in the design phase of a solar field to guideThe the layout optical designerwith a flat in theheliostat, development or a flat of the heliostats field configuration. array, is very often utilized in CSP (ConcentrationThe layout withSolar a Power) flat heliostat, plants or with a flat central heliostats tower. array, This is verymirror often configuration utilized in CSP can(Concentration be associated, asSolar a first Power) approximation, plants with to centrala Gaussian tower. beam This enlargement, mirror configuration which emerges can as be the associated, convolution as aof first the solarapproximation, beam and tothe a Gaussianpointing errors beam enlargement,due to components which emergesand assembly as the tolerances. convolution It ofpermits the solar to elaboratebeam and a the suitable pointing function errors that due describes to components the variation and assembly of the irradiance tolerances. on Itthe permits target. to elaborate a suitableIn this function work, thatthe variation describes of the the variation spot depending of the irradiance on the heliostat on the size target. or the facets size of a faceted heliostatIn this is work,evaluated the variation by means of theof spotboth depending the numerical on the integration heliostat size of or that the facetsfunction size and of a faceteda non- sequentialheliostat is evaluatedraytracing by software, means of Lambda both the Research numerical TracePro. integration The of thateffect function of various and aparameters non-sequential are analyzed:raytracing side software, of the Lambda facets (2 ResearchLF from 0.5 TracePro. m to 4 Them), effectheliostat-target of various distance parameters (d from are analyzed: 25 to 100 sidem), facetsof the number facets (2 (fromLF from 1 × 0.5 1 to m 16 to × 4 16). m), A heliostat-target test of the method distance on an (d arrayfrom of 25 heliostats to 100 m), demonstrates facets number its (fromapplicability 1 × 1 to for 16 a× rough16). A evaluation test of the of method the performance on an array of ofa solar heliostats plant demonstrates too. its applicability for aThe rough results evaluation obtained of the from performance the numerical of a solar integration plant too. are in good agreement with the corresponding TracePro simulations. The validity of the method is confirmed by an application of the Lagrange invariant to the Gaussian beam.

Energies 2018, 11, 1621 17 of 18

The results obtained from the numerical integration are in good agreement with the corresponding TracePro simulations. The validity of the method is confirmed by an application of the Lagrange invariant to the Gaussian beam. An example of the utility of this analysis is represented by Table1, which synthesizes the concentration obtainable in a heliostat field changing the number of facets and varying the ratio between the heliostat-target distance and the facets side. The opportunity of having such an overview of the performance (as irradiance on the target center) of numerous possible configurations for the faceted heliostat field can address the designer in the selection of the most suitable field for a specific application. In particular, these results allow estimating the utilization of flat mirrors instead of concentrating mirrors or the minimum useful faceted size for a faceted heliostat, depending on the heliostat-target distance. Moreover, it is important to remark that there is a lower limit of the mirror size depending on the distance from the receiver: under this value, there is no increase of the concentration on the target. These results are readily available for the designer of a solar field when a fast analysis of the project is requested. The proposed method is applicable to flat heliostats, to faceted plane heliostats, and also to arrays of flat heliostats. The method can be easily modified to take into account the different positions of the sun and to more accurately describe the behavior of a faceted heliostat.

Author Contributions: D.J. conceived and developed the work; F.F. collaborated in the development of the simulations; D.F. analyzed the data and realized all the figures; P.S. controlled the results, wrote the paper and refined the presentation of the work. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

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