1. Introduction
When generating electricity using alternative energy sources, it is appealing to incorporate technologies that concentrate solar thermal energy, such as heliostats, parabolic troughs and parabolic dishes connected to a Stirling engine [1]. The latest solar thermal technologies have high costs of installation, operation and maintenance and a decreased solar energy-to-electricity conversion efficiency [2]. Having a Stirling engine at the focal point of a parabolic dish decreases the effects of this efficiency problem by incorporating a solar concentrator with a fixed focal point. Scheffler developed a solar concentrator for solar cookers that are fixed inside the house [3]. This concentrator also has other applications, such as oil extraction [4], distillation [5] and sterilisation applications [6]. To incorporate this technology to generate electricity using Stirling engines, it is necessary to make some adjustments, such as incorporating an azimuth axis that tracks the solar height and form continuously and incorporates the reflector surface with the least amount of imperfections, which is motivated by a study of ray tracing [7].
Different mathematical models have been developed using various mathematical tools and software to optimise the design of assembly systems. The model proposed by Harris and Lenz [8] analysed the thermal behaviour of a parabolic dish concentrator that used the view factor to estimate the amount of radiation reaching the receiver cavity in cylindrical, conical, elliptical and spherical receivers. Additional results for various concentrator geometrical shapes have been provided in Badescu [9]. Shuai et al. [10] used the Monte Carlo method to determine the performance of the radiation concentrated by a parabolic dish receiver for various geometries. Kumar and Reddy [11] presented a software implementation of a CFD (computational fluid dynamics) study to determine the optimal size of the aperture opening area of a spherical cavity coupled to a parabolic dish, which depends on the item’s minimum convection. Chin-Hsiang and Hang-Suin [12] presented a study to optimise geometrical parameters for Stirling engines based on a theoretical analysis. Nepveu et al. [13] presented a model for a parabolic dish concentrator, known as a Eurodish, coupled to a 10 kW Stirling engine.
Halit et al. [14] presented an experimental study on the development of a beta-type Stirling engine for low and moderate temperature heat. Fraser [15] presented a model to estimate the performance of an Alpha-type Stirling engine, and the energy that is transmitted to the receiver is calculated by the equation proposed by Duffie and Beckman [16].
This equation states that the energy concentrated in the receiver of a solar concentrator is directly proportional to the direct radiation, the aperture opening area of the reflector, reflectivity and intercept factor. The latter concept is particularly useful in establishing the dimensions of the concentrator because reducing the receiver aperture opening diameter as a percentage of the solar image and incorporating the thermal model of the receptor defines the dimensions of the receiver for minimum heat loss.
There are different models available to estimate the intercept factor; Jaffe [17] and Badescu [18] presented a model that involves optical, thermal and conversion aspects of energy to optimise the dimensions of the parabolic dish concentrator. Romero [19] developed software for Sandia Laboratories that determined the energy that is intercepted by the circular or rectangular segments of a parabolic dish, which involves the optical aspects of the concentrator. Fraser [15] and Badescu [20] provided the results that are the most appropriate for parabolic dish concentrators that are coupled to a Stirling engine for different capacities. The models proposed by Badescu [21], Stine and Harrigan [22] considered both the influence of varying the receiver aperture opening diameter, which depends on the rim angle and height of the focal point, as well as the standard deviation of the errors caused by the geometrical factors and the optical concentration system.
After reviewing the literature, it was determined that no other previous model can be applied directly to evaluate the performance of an STSC because these models do not incorporate the geometric model of the STSC; therefore, in this study, a new mathematical model for an STSC coupled to a cavity receiver of a Stirling engine is presented. An analysis of the performance is carried out to compare an STSC and a parabolic dish to determine the technical viability of using this technology.