Angular confinement and concentration in photovoltaic converters
Introduction
In a photovoltaic conversion process the irreversible generation of entropy leads to a reduction of the theoretically possible efficiency. One of these losses is due to the étendue expansion between absorbed and emitted radiation [1]. The étendue expansion of a PV converter is defined by two angles. The first angle θinc describes the cone of radiation incident on the PV converter. The second angle θext describes the cone into which the PV converter emits radiation. This cone is called the angular acceptance range [2]. The initial situation is that the PV converter receives radiation from the narrow cone in which solar radiation impinges on the earth. This cone is defined by a polar angle of θinc=4.7 mrad for direct sunlight only (including circumsolar radiation it is about θinc=44 mrad). The cone into which the PV converter emits radiation, however, is the complete hemisphere to which a polar angle of θext=π corresponds.
In this work we want to investigate how the entropy generation induced by the two angles θinc and θext influences the performance of a solar cell. This entropy shall be called optical entropy within the scheme of this work and it shall be the only source of entropy investigated. In the solar cell models used, other sources of entropy, like e.g. thermalization, also occur but they will not be further referred to in the text, though it is important to bear in mind that they exist. Furthermore, a system in which no entropy is generated because of a parity of θinc and θext shall be called an optically conservative system. A system in which entropy is generated because of a disparity of θinc and θext shall be called an optically non-conservative system.
To reduce the amount of entropy created it is possible to change the cones of absorption and emission by means of optical elements. In this process, changing the cone of incidence θinc is nothing more than concentration. The optical elements used here are e.g. lenses or mirrors. The potential of concentrating systems is well known and described, e.g. by Miñano [10].
Changing the cone of emission θext is what we call “angular confinement” (see Fig. 1). The optical elements we want to use for this process are angularly selective optical elements. With both techniques the generation of optical entropy is reduced, and at the limit of maximum concentration or maximum angular confinement an optically conservative system is created.
First formulations of concepts using angular confinement have been made by Araujo and Marti [3]. Badescu [4] investigated angularly and spectrally selective emitting surfaces for PV systems under one sun illumination. The thermodynamic limit for the maximum voltage was discovered by Markvart [5]. Further thermodynamic considerations are given by Badescu and Landsberg [6], [7]. Recently, the utilization of angularly selective filters for angular confinement was proposed. Suitable filters were investigated by Fahr et al. [8] and Ulbrich et al. [9].
In this work we compare the theoretical efficiency of a PV converter that integrates concentration to one that integrates angular confinement. We will find that both effects reduce the generation of optical entropy in a similar way and result in the same efficiency limit. This investigation is presented on a very general level but shall always be given regarding angularly selective filters. Here we will find that the utilization of angularly selective filters and concentration concern complementary angular ranges and may be combined. Finally, we will find that a combination of both techniques will result in the same efficiency limit as concentration or angular confinement alone.
To investigate the efficiency limit, we will start with a short introduction into the concept of étendue. In the corresponding section we will calculate the étendue balance for the initial situation of a PV converter without any optical element and the balance for a concentrating and angularly confined system. We will also calculate the amount of entropy generated by a PV converter. Subsequently we will analyze the effect of angular selectivity and concentration represented by the two angles θinc and θext on the voltage and the current density, and on the efficiency of a PV converter. Following that we will investigate how the Shockley–Queisser limit [11], as derived by Würfel [12], is affected by angular confinement and concentration.
Section snippets
étendue, optical entropy and optically conservative systems
The entropy σ per photon generated by the disparity of the absorbed and the emitted light cone is defined over the étendue of the incident εinc and the emitted εext radiation. It is given by [1]
Already from this equation it is clear that the optical entropy generation is influenced pari passu by incident and emitted radiation as it only depends on the ratio. Furthermore no optical entropy is generated if the étendues for incident and emitted radiation are equal. The étendue, being
The Shockley–Queisser limit
As was shown in the last section, a difference between the solid angles of absorption and emission results in a generation of entropy. Following the theorem of Gouy and Stodola, a linear correlation exists between entropy generation and exergy loss [13]. Therefore the solid angles must also influence the maximum efficiency of a solar cell. This effect is investigated in the following section, where the influence of the solid angles on the Shockley–Queisser limit is investigated.
The assumption
Discussion
The consideration given until now shows that a close relationship exists between the two solid angles that define the angular ranges from which a solar cell receives radiation and into which a solar cell emits radiation. However, the techniques to influence these angles differ. Concentration is obtained with lenses and mirrors, while a confinement of the angular emission range requires angularly selective elements. For this reason a discussion shall be given here concerning differences and
Summary
The optical entropy generation in a solar cell is defined by two angles, θinc and θext, that describe the angular spreading of incident and emitted radiation. Initially these two angles are different, which causes a generation of entropy and consequently a reduction in solar cell efficiency. The angle θinc is changed by concentration, while the angle θext is changed by angular confinement. In this work we have compared the impact of these techniques on the efficiency of a solar cell in the
Acknowledgments
We thank the reviewers for suggestions and advice, the DFG for their financial support in the project Nanosun (Pak88) and the BMBF for their financial support in the project Nanovolt. Jan Christoph Goldschmidt gratefully acknowledges the scholarship support from the Deutsche Bundesstiftung Umwelt (DBU) and the ideational support from the Heinrich Böll Stiftung and the Studienstiftung des deutschen Volkes.
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