A mathematical procedure to predict optical efficiency of CPCs with tubular absorbers
Introduction
Solar radiation incidents on the earth at a rate of about 81016 W, which is more than 10000 times the present world energy consumption, and it is considered as one of the cleanest sources of energy [1]. Nowadays, applications of solar energy is increasing due to rapidly rising fossil fuel prices and increasing energy consumption. The end use of global energy consumption can be broken down into three categories which are power generation, transportation and thermal energy processes. These three categories roughly constitute around 20, 30, and 50% of the total energy demand respectively [2]. The thirty percent of heat requirements for industrial processes in the world are in the temperature range of 100–300 °C. Solar thermal collectors have an inherent advantage over solar PV in that they convert solar irradiance into heat at efficiencies 3–4 times those achieved by PV technologies (15–20%). Solar collectors are broadly classified by their operating temperature as low (<100 °C), medium (100–300 °C), or high temperature (>300 °C) collectors. In recent years, solar collectors operating at low temperatures have been widely used for water and building heating, as well as the drying of agricultural products [[3], [4], [5]]. Collectors operating at high temperatures are widely used for electricity generation [6,7], but the applications of solar collectors operating at medium temperatures are limited. Potential applications in the medium temperature range of 100–300 °C include space heating, air conditioning, solar cooking and industrial processes. A compound parabolic concentrator (CPC) with a tubular absorber is a typical ideal concentrator with inherent advantages of low cost, a simple structure and no need of expensive sun-tracking devices. Such CPCs are potential solar collectors which can be commercialized for solar thermal processes operating at temperatures in the range of 100–250 °C.
In 1974, Rabl and Winston first developed and tested non-evacuated CPC collectors, and found that the heat losses through reflectors were critical [8]. To reduce heat losses from tubular absorbers to the ambient air, various CPC solar collectors with an evacuated solar tube as the receivers were developed and tested [[9], [10], [11], [12]].
A compound parabolic concentrator designed based on the cover tube of an evacuated metal tube with a geometric concentration of 1.18 was developed for medium temperature applications by Winston et al. [13]. An array of CPC collectors with area 53 m2 was used to drive a 23 kW double effect (LiBr) absorption chiller. Measurements showed that the collector system stably operated at 160–200 °C with an average daily efficiency of 36.7% and instantaneous efficiencies of up to 40% were achieved. The experimental tests by Jiang et al. showed that an evacuated tube CPC solar array could achieve an efficiency of up to 50% when operation at 200 °C [14]. Recently, Widyolar et al. tested a new external compound parabolic concentrator (XCPC) with an evacuated pentagon-shaped absorber, and experimental results confirmed an optical efficiency of 62% and a thermal efficiency of nearly 50% at 200 °C [2]. These recent works demonstrated that CPC solar collectors are technically viable to provide solar process heat up to 250 °C.
To theoretically investigate the performance of CPCs with a tubular absorber, it is essential to find the optical efficiency of CPCs for the radiation incident at any incident angle, and ray-tracing analysis is commonly used in the past due to sophisticated multiple reflections of solar rays on the way to absorber resulting from the cusped reflectors near the bottom of tubular absorbers [[15], [16], [17]]. The ray-tracing analysis is simple but time-consuming, especially as one investigates effects of the geometry of CPCs on the optical and thermal performance of CPC based solar heating systems. In this work, an attempt is first made to develop a mathematical procedure to determine the optical efficiency of the CPC for radiation incident at any projected incident angle. This because of the fact that the optical performance of linear CPCs is uniquely determined by projected angle of solar rays on the cross-section of CPC troughs [18,19]. To evaluate the reliability of the suggested mathematical models for predicting the optical performance of CPCs, comparisons of the annual collectible radiation of east-west oriented CPC collectors calculated based on the optical efficiency expected from different models and ray-tracing analysis are compared.
Section snippets
Equation of reflectors
As well known, the reflectors’ profile of CPCs is uniquely determined by the geometry of absorbers. As shown in Fig. 1, for the CPC with a tubular absorber, the reflectors consist of two parts, and any point on the upper reflector in the suggested coordinate system is described by Ref. [17]:where is the angle, measured from the
Annual collectable radiation of east-west oriented CPCs
It is assumed that the CPC in question is oriented in the east-west direction, the length is infinite as compared to the width, and the radiation reflecting from the ground is not considered, thus, the radiation received by unit area of the tubular absorber at any time of a day is given by:where Ib is the intensity of beam radiation, is the incident angle of solar rays on the aperture and given by Refs. [1,19]:where is the
Angular ranges of incident solar ray for multiple reflections
Given the geometry of CPCs, the angular ranges of solar rays incident on the upper reflector that arrives on the tube after more than n-reflections (n = 2,3,4) can be found by numerical calculations. It is seen from Table 1 that, for more than n-reflections ranges from negative to positive value except the truncated CPC with = 20° and Ct = 2.5. As aforementioned, the optical performance of symmetric CPCs for radiation incident at is identical. Hence, (n = 2,3,4), the minimum =
Conclusions
Modelling analysis and ray-tracing analysis show that multiple-reflections commonly take place for radiation incident on the reflectors near the aperture at small angles, and multiple-reflections after the third reflection usually occur on the cusped reflectors near the bottom of tubular absorbers, hence truncating reflectors near the aperture can improve the optical efficiency of CPCs with a poor reflectivity.
Analyses presented in this work show that, for CPCs with a tubular absorber, a
Acknowledgements
This work is in partial fulfillment of the funded research program 51466016, financially supported by National Natural Science Foundation of China.
References (27)
- et al.
Non-tracking East-West XCPC solar thermal collector for 200 celsius applications
Appl Energy
(2018) - et al.
Experimental investigation on solar drying of salted greengages
Renew Energy
(2006) - et al.
Nocturnal reverse flow in water-in-glass evacuated tube solar water heaters
Energy Convers Manag
(2014) - et al.
Design and test of non-evacuated solar collectors with compound parabolic concentrators
Sol Energy
(1980) - et al.
Relative cost-effectiveness of CPC reflector designs suitable for evacuated absorber tube solar collectors
Sol Energy
(1986) - et al.
Development and performance analysis of compound parabolic solar concentrators with reduced gap losses—“V” groove reflector
Renew Energy
(2002) - et al.
Effect of orientation of a CPC with concentric tube on efficiency
Appl Therm Eng
(2018) - et al.
Energy matrices of U-shaped evacuated tubular collector (ETC) integrated with compound parabolic concentrator (CPC)
Sol Energy
(2017) - et al.
Performance of a 23KW solar thermal cooling system employing a double effect absorption chiller and thermodynamically efficient non-tracking concentrators
Energy Procedia
(2014) - et al.
Characterization of novel mid-temperature CPC solar thermal collectors
Energy Procedia
(2015)
Optimized reflectors for non-tracking solar collectors with tubular absorbers
Sol Energy
Comparison of fixed asymmetrical and symmetrical reflectors for evacuated tube solar receivers
Sol Energy
Comparison of solar concentrators
Sol Energy
Cited by (13)
Assessment of the optical efficiency in solar collectors: Experimental method for a concentrating solar power
2023, Thermal Science and Engineering ProgressOptical efficiency and performance optimization of a two-stage secondary reflection hyperbolic solar concentrator using machine learning
2022, Renewable EnergyCitation Excerpt :The optical efficiency, defined as the ratio of the energy absorbed by the receiver to the energy incident on the concentrator aperture, is a vital parameter for optimizing solar parabolic collectors [2–4]. The power from solar radiation incidents on the earth is approximately 8 × 1016 W, which is more than 104 times the current world energy consumption and represents one of the cleanest energy sources [5,6]. Traditionally, optical analysis of a radiation concentrator is performed using a computer ray-tracing program.
Model construction and optical properties investigation for multi-sectioned compound parabolic concentrator with particle swarm optimization
2021, Renewable EnergyCitation Excerpt :According to Fig. 9(a), the optical efficiency of S-CPC at 0.75 reflectivity is slightly smaller than that of some M-CPCs in the range of 0°–10°, which should put it down to the more reflections than M-CPCs while solar ray passes through reflector to the absorber. The conclusion is identical with Xu et al. [30]. In the meantime, the lowest optical efficiency in Fig. 9(a) for S-CPC within 0°–10° and 10°–30° respectively are 85% and 75%, and drop by 15% and 25% compared with the S-CPC of Fig. 9(c).