2.1. The Energy Density Ratios of the Reflected Ray
Solar rays from various positions will reach different positions on the receiver surface after being reflected by a certain point of the heliostat. Optical errors at different points of the heliostat will cause the reflected ray to deviate from the ideal direction, as shown in
Figure 1. We calculated the flux density contribution of the reflected solar ray to a certain point on the receiver surface by integrating all of the reflection points on the heliostat [
9]. Therefore, the distribution of the flux density on the whole receiver surface could be computed.
The rays reflected by the heliostat were divided into three main parts [
11]:
,
, and
, as shown in
Figure 2. The incident rays first reach the upper surface of the glass, and part
is directly reflected, accounting for approximately 5%. The rest of the rays are transmitted through glass, reflected by the metal reflecting surface, and transmitted out of the upper surface. These are the main reflected rays, namely part
, accounting for approximately 91.5% when the incidence angle is small. The remaining part is reflected again from the upper surface, and again from the reflective metal surface. When it reaches the upper surface, it transmits out part
, accounting for approximately 3%. The rest of the rays still travel inside the glass, but the fraction is negligible.
The focuses of the three parts of the beams are different, but the part mainly determines the distribution of the flux density on the receiver plane, so, in practice, the rays of part are aligned with the center of the receiver plane. When the distance between the receiver plane and the heliostat is large enough, the difference in the focal positions of the three parts can be ignored. Meanwhile, the contribution of part and part to the flux density is not tiny. When a more accurate flux density distribution is required, the optical errors of these two parts should be taken into account, and the flux density distribution is different from that of part . In particular, part and are not taken into account when calculating the intercept factor, and the curvature of the intercept curve will differ significantly from the actual situation, especially when the intercept factor reaches 90%.
The proportion of , , and is related to the incident angle. There are also polarization changes in reflection and refraction on the glass surface. We first decomposed the amplitude vector of the ray into two directions. The one perpendicular to the vibration of the incident plane is called the S component, and the one parallel to the vibration of the incident plane is called the P component. They have identical parts in solar rays.
The incident angle is and the ratios of the refractive index of the two sides of the reflected glass interface are and , respectively. According to the refraction law, the refraction angle inside the glass can be calculated as .
According to Fresnel’s equations [
12], the reflectivity of the reflected ray in the
P and
S components can be calculated:
Similarly, the transmissivity of the refracted ray in the
P and
S components can be calculated:
As light passes through the metal reflecting surface, namely the silver plating layer, it is refracted and reflected as it reacts with the silver. Part of the refraction into the silver plating layer is equivalent to loss:
and
are the reflectivity of light on the silver surface in the
P and
S components.
is the incident angle reaching the silver surface.
is the ratio of the refractive index of the silver plating layer relative to the glass, and, here, it was equal to 0.051585.
k is the dielectric constant, and, here, it was equal to 3.9046. The wavelength of the incident ray was 587.6 nm [
13].
As light passes through the glass, some of it is absorbed by the glass. Heliostats usually use ultra-clear glass. According to the experimental results, approximately 1% [
14] of light is absorbed when it passes through the glass perpendicularly, mainly absorbed by impurities in the mirror.
The incident ray is natural light, and the light intensity is , which has the same amplitude in the P and S components. Thus, the light intensity in the P and S components is equal.
By solving the energy density of the reflected and refracted ray at the interface boundary each time, the energy density of the three parts of rays reflected off the glass surface can be obtained. Finally, the proportion of three parts of the reflected ray can be calculated.
where
and
are the reflectivity of the first reflections in the
P and
S components, respectively.
and
are the reflectivity of the second reflections in the
P and
S components, respectively.
and
are the transmissivities of the transmission into the glass in the
P and
S components, respectively.
and
are the transmissivities of the transmission out of the glass in the
P and
S components, respectively.
is the absorption ratio of glass to light, which is calculated by:
where
d is the thickness of the glass, and
is the absorption coefficient, which should be determined from the test data [
14] for the specific mirror.
Finally, the energy density ratios of the three parts can be calculated:
2.2. Solution of Flux Density on Receiver Plane
Although the energy density distribution formed by the three parts of the reflected ray is different, the calculation process of the flux density of each part is similar. It only needs to calculate the flux density on the receiver plane of a single part and then multiply it by the energy density ratios to obtain the flux density of the three parts. Finally, we can calculate the flux density distribution on the whole receiver plane.
where
is the flux density of a single part.
For a solar tower system, it is assumed that a receiver plane is a square plane and the heliostat is a rectangular spherical mirror, as shown in
Figure 3. Suppose that the heliostat reflector is
, the receiver plane is
,
and
are normals of the cell surfaces on the heliostat and the receiver,
and
are the points on the cell surfaces on the heliostat and the receiver, and
and
are the included angles between the line
and normals of each cell surface. For a single part, the analytic formula of the flux density at point
in the receiver surface is [
15]:
where
r = |
| is the distance between the reflection point and the receiver point,
I is the total intensity of the reflected ray of a single part, and
B is the intensity distribution of reflected rays, obtained from the convolution calculation of the sun shape and optical error.
To calculate the intensity distribution of the reflected ray, we need to determine the solar intensity distribution firstly, which is described by a circular Gaussian function. An approximate description of the intensity of sunlight is the only assumption in this paper. Due to many factors, such as sun position and atmospheric absorption, the solar intensity changes from time to time. However, the standard deviation of the solar intensity distribution
can be determined by direct solar irradiation
[
8]:
The optical error of the reflected ray must then be determined. This paper adopted the elliptic Gaussian function to describe the slope error on a heliostat. It is necessary to consider that rays will be reflected by the upper and lower surfaces of the mirror. The equations for calculating optical errors are also different for each part of the reflected ray.
The optical error of each point on the heliostat is different [
9,
16]. To improve the calculation speed, we can integrate the whole heliostat and replace the optical error of each point with the average optical error. To compare the two calculation results, we calculated the difference with ten heliostats in
Section 3.2. From the results, the average difference between the two is less than 0.05%. Therefore, in the subsequent calculation of the optical error, the optical error of each point was no longer considered separately but was only replaced by the average slope error of the surface.
For the first part of the reflected ray
, the upper surface of the glass directly reflects it, so the average optical errors were calculated as follows:
where
and
are slope errors in the X direction and Y direction, respectively.
For the second part of the reflected ray , it will be refracted by the lower surface and reflected by the upper surface. Given mirrors’ relatively mature manufacturing technology, the slope errors on the upper and lower surfaces can be considered equal.
Therefore, the standard deviation of the average error in two directions was calculated as follows:
where
is the refractive index ratio of glass to air, constant at 1.51 in general, and
is the incident angle of the heliostat reflection point, respectively.
For the third part of the ray
, the ray passes through the glass again, and its optical error is two more reflection errors than that of the ray
:
The total error of the reflected ray at each point of the heliostat was calculated by convolution of three Gaussian function errors as follows:
where
and
are the tracking errors in the X direction and Y direction. The tracking errors of all models used in this paper were 0.
In the elliptic Gaussian model, the distribution of reflected light intensity can be calculated as:
where
is the angle between
in the X direction and the reflected ray from the center of each cell on the heliostat, and
is the angle between
in the Y direction and the reflected ray from the center of each cell on the heliostat.
and
are the total errors in the X direction and Y direction, respectively.
In this paper, based on the elliptic Gaussian model, we simplified the calculation process of an optical error and built a circular Gaussian model, which was applied to verify the model’s accuracy by comparing it with SolTrace [
17]. It can also be applied to quickly compute a flux density distribution when the accuracy requirement is not high. In the circular Gaussian model, the reflected ray was no longer divided into three parts and was only calculated as the same reflected ray. The slope error
was simplified to radial, and the tracking error
was taken into account, so the total error was calculated as [
18]:
In the later model verification, the optical error in SolTrace will be consistent with that in the circular Gaussian model and calculated by the least square method.
Therefore, the intensity distribution function of the reflected ray in the circular Gaussian model can be expressed as:
is the angle between and the reflected ray from the center of each cell on the heliostat, and is the total error.
Therefore, for a single part of a reflected ray, the flux density distribution function that the heliostat devotes to a specific point on the receiver plane can be transformed into: