An analytical method for the optical analysis of Linear Fresnel Reflectors with a flat receiver

Highlights

• An analytical optical method for Linear Fresnel Reflectors with a flat receiver is presented.

• Analytical results agree very well with ray tracing for both factorized and biaxial models.

• Higher errors are seen for high incidence angles, especially for the longitudinal incidence.

• Intercept factor errors can be high up to 132%, although energetic differences can be high up to 3%.

Abstract

The optical analysis of Linear Fresnel Reflectors solar concentrators deals with the calculation of the absorbed flux at the receiver. Due to concentrator discrete and complex geometry ray tracing numerical simulations is the main used method, in parallel with some analytical approaches that have been developed for specific cases. This paper presents an analytical method for Linear Fresnel Reflectors with a flat receiver. It is based on Zhu’s vector-based method and Rabl’s concepts of acceptance and effective source, although new shading (including receiver shading), blocking, and cosine losses analyzes are presented, with the inclusion of end-losses effect. As shown, the problem is better described by the concept of intercept factor, the effective aperture area used to collect the incident sunlight. Comparison tests with ray tracing simulations performed in SolTrace for three different effective sources were carried out to validate the analytical model for both factorized and biaxial models of the intercept factor, including energetic evaluations for Évora, Portugal. In general, analytical results do agree very well with ray tracing, better for the factorized model than the biaxial. Errors in the analytical estimative of intercept factor can be high up to 132% for the biaxial model at high longitudinal incidence angles; on the other hand, errors in the amount of annual absorbed energy were high up to only 3%.

Introduction

The Linear Fresnel Reflectors (LFR) is a solar concentrator technology used for thermal applications (Collares-Pereira et al., 2017, Zhu et al., 2014). It is composed of a discrete number of mirrors – the primary field, or heliostats –, slightly elevated from the ground, which purpose is to reflect (and concentrate) the incident sunlight on a fixed linear receiver, located above the primaries (Mills, 2012, Rabl, 1985). Each heliostat has a single-axis tracking mechanism to follow the sun’s daily movement, and the receiver has an absorber element to convert the reflected radiation to thermal energy, transferring it to a heat transfer fluid.

One main problem in LFR studies is the optical analysis: the calculation of the absorbed flux at the receiver. Usually, such flux is defined by a relative measure with respect to the incident flux at the concentrator aperture.

Rabl (1985) presents two definitions for the so-called optical efficiency, ${\eta }_{\text{opt}}$. Eq. (1) shows the first one: the ratio between absorbed flux (power), $\mathit{Q}$, and the incident flux, $\mathit{A}·\mathit{I}$,

$${\eta }_{\mathrm{o}\mathrm{p}\mathrm{t}}=\frac{Q}{A·I}$$

where A is the concentrator aperture area, in m²; and $\mathit{I}$ is the direct normal solar irradiance (DNI), in W/m². While studying the performance of LFR concentrators one must take care with the definition of the incident flux $\mathit{A}\mathit{·}\mathit{I}$. Occupied and reflective surfaces are different, and this can lead to misinterpretations. Here, all definitions are taken to the reflective area, unless specified.

The other definition for ${\eta }_{\text{opt}}$, shown in Eq. (2), is based on the split of losses due to optical properties and geometrical characteristics (Rabl, 1985): $\mathrm{\Pi }$ represents the concatenation of surfaces optical properties (reflectivity, transmissivity, and absorptivity), and $\gamma$ is the intercept factor, which accounts for losses due to concentrator geometry.

$${\eta }_{\text{opt}}=\mathrm{\Pi }\gamma$$

In realistic modeling, optical properties are functions of incidence angles (Howell et al., 2016), and we should write $\mathrm{\Pi }=\mathrm{\Pi }\left({\theta }_{\text{T}},{\theta }_{\text{L}}\right)$, where ${\theta }_{\text{T}}$ and ${\theta }_{\text{L}}$ are the transversal and longitudinal incidence angles, respectively, as defined in Section 2.1. The geometric relations that dictate how much radiant flux hits the receiver are also functions of incidence angles, and, thereby, $\gamma =\gamma ({\theta }_{\text{T}},{\theta }_{\text{L}})$. In conclusion, ${\eta }_{\mathrm{o}\mathrm{p}\mathrm{t}}={\eta }_{\mathrm{o}\mathrm{p}\mathrm{t}}({\theta }_{\mathrm{T}},{\theta }_{\mathrm{L}})$.

Usually, the specific literature uses the sun at normal incidence (${\theta }_{\text{T}}$ = ${\theta }_{\text{L}}$ = 0) as a standard position, and optical efficiencies from other directions are treated as relative losses. This defines the Incidence Angle Modifier (IAM) (Carvalho et al., 2008, Rabl, 1985) as shown in Eq. (3).

$$\mathrm{I}\mathrm{A}\mathrm{M}({\theta }_{\mathrm{T}},{\theta }_{\mathrm{L}})=\frac{{\eta }_{\mathit{opt}}({\theta }_{\mathrm{T}},{\theta }_{\mathrm{L}})}{{\eta }_{\mathrm{o}\mathrm{p}\mathrm{t}}(0,0)}=\frac{{\eta }_{\mathrm{o}\mathrm{p}\mathrm{t}}({\theta }_{\mathrm{T}},{\theta }_{\mathrm{L}})}{{\eta }_{\mathrm{o}\mathrm{p}\mathrm{t}0}}$$

Nevertheless, it is usual to consider optical properties as constants (Bellos et al., 2018a, Bellos et al., 2018b) – hemispherical average values (Howell et al., 2016) –, and $\mathrm{\Pi }\left({\theta }_{\text{T}},{\theta }_{\text{L}}\right)=\mathrm{\Pi }$. With this simplification, simple relations between $\gamma$ and IAM can be derived, as shown in Eq (4).

$$\mathrm{I}\mathrm{A}\mathrm{M}({\theta }_{\mathrm{T}},{\theta }_{\mathrm{L}})=\frac{{\eta }_{\mathrm{o}\mathrm{p}\mathrm{t}}({\theta }_{\mathrm{T}},{\theta }_L)}{{\eta }_{\mathrm{o}\mathrm{p}\mathrm{t}0}}=\frac{\mathrm{\Pi }\gamma ({\theta }_{\mathrm{T}},{\theta }_{\mathrm{L}})}{\mathrm{\Pi }\gamma (0,0)}=\frac{\gamma ({\theta }_{\mathrm{T}},{\theta }_{\mathrm{L}})}{{\gamma }_0}$$

In this sense, one can correctly write:

$$Q({\theta }_{\mathrm{T}},{\theta }_{\mathrm{L}})={\eta }_{\mathrm{o}\mathrm{p}\mathrm{t}}({\theta }_{\mathrm{T}},{\theta }_{\mathrm{L}})A·I=\mathrm{\Pi }\gamma ({\theta }_{\mathrm{T}},{\theta }_{\mathrm{L}})A·I$$

which shows that the optical analysis problem is basically the calculation of ${\eta }_{\text{opt}}$ or $\gamma$.

The most used method for optical analysis is ray tracing numerical simulations – an approach closely related to the definition presented in Eq. (1) since $\mathit{Q}$ is one of the simulation outputs. This method is based on following the path of light rays throughout the optical system, simulating their interaction with surfaces and materials, calculating the reflected, transmitted, and absorbed fractions of the incident rays (Osório et al., 2016, Wendelin and Dobos, 2013). The number of rays must be large enough to produce accurate results. It has the advantage to deal with any kind of geometry and mapping the flux distribution on the absorber. The computational cost can be high, which makes it difficult to use for optimization studies (Ajdad et al., 2019, Moghimi et al., 2017).

On the other hand, LFR main optical losses are well-known: shading, blocking, cosine effect, heliostats defocusing, finite acceptance (sun shape and optical errors), and end-losses. In this sense, analytical approaches have been used to compute the optical analysis problem. They are usually related to the definition presented in Eq. (2), and have the advantage of being faster than ray tracing simulations, even though are not “general-purpose” and their accuracy depends on the models that are used to describe the optical losses.

To analytically compute the absorbed flux is not a trivial task. On one hand, heliostats and absorber surfaces interact with sunlight just once, and primaries reflectivity and absorber absorptivity (hemispherical averages values) composition in $\mathrm{\Pi }$ is just a simple product; also from Eq. (5), the term $\gamma ({\theta }_{\text{T}},{\theta }_{\text{L}})·\mathit{A}$ can be interpreted as the effective aperture area used to collect the incident radiation $\mathit{I}$ – a purely geometric factor –, and, thus, shading, blocking, and cosine effect losses can be easily considered. On the other hand, defocusing and finite acceptance losses (sun shape and optical errors) are harder to calculate; and when the analyzed LFR includes a glazed receiver with a secondary optic it gets even more complicated: for instance, secondary’s acceptance function, multiples reflections and transmissions, and total internal reflection are effects that must be properly handled to precisely compute the absorbed flux at the receiver.

In this context of analytical methods, the concept of an effective aperture area, $\gamma ({\theta }_{\text{T}},{\theta }_{\text{L}})·\mathit{A}$, has been used to address the LFR optical analysis problem: Heimsath et al., 2014, Hongn et al., 2015 presented analytical models for end-losses; Sharma et al. (2015) presented equations to model shading, blocking, cosine effect, and end-losses in LFRs; Bellos and Tzivanidis (2018) developed simple equations for both transversal and longitudinal components of the IAM, through shading, cosine effect and end-losses modeling; and Yang et al. (2018) presented a vector analysis to compute end-losses and cosine effect, and also simple equations for shading and blocking.

The analytical methods referred above do not include curved-shape heliostats (parabolic or cylindrical) neither compute losses due to finite acceptance (sun shape and optical errors). Abbas et al. (2012) presented an analytical study about the curved-mirrors influence on the optical performance of LFRs, although no formulation for defocusing losses was presented. Rabl (1985) proposed the idea to compute losses due to finite acceptance by the concepts of acceptance function and effective source (the convolution of a sun shape and optical errors). Finally, Zhu’s vector-based method (Zhu, 2013) includes it all, although for the particular case of a tubular receiver: it considers primary mirrors as a set of discrete points, and uses a reflection plane analysis to account for shading, blocking, and defocusing by conditional equations; then, uses Rabl’s concepts to compute losses due to finite acceptance; cosine effect is accounted by a factorized model, and end-losses are not included.

Recent optimization studies (Ajdad et al., 2019, Moghimi et al., 2017) have faced a problem associated with the high computational cost of using ray tracing tools to compute the optical analysis of LFRs. An analytical approach would solve that, although it is not easy and gets more complicated if the receiver contains a secondary optic. On the other hand, the aperture of a secondary optic (CPCs, trapezoidal, SMS-like) can be modeled as a flat receiver and Zhu’s method presents a good start point on primary mirrors optical losses modeling. This is one of the major reasons why this work is presented as the authors came across such difficulties while developing methodologies for advanced optimization of LFRs systems.

This work develops an analytical method to compute the optical analysis problem of LFRs with a flat receiver. It considers Zhu’s vector-based approach (primaries discretization and conditional equations) and Rabl’s concepts (acceptance and effective source), however, through new shading (including receiver shading), blocking, and cosine effect analyzes. Furthermore, it includes end-losses and definitions about how to quantify each optical loss. In this sense, the present method is not only a review and improvement of the current state of the art. It is also a tool that shall be used in future works involving optimization processes.

The paper is organized as follows: Section 2 presents the method and the developed equations. Section 3 shows optical analysis results of the proposed method, comparing it against ray tracing simulations. Finally, in Section 4, some conclusions and perspectives for future research activities are presented.