INTRODUCTION

The test case under consideration, shown in Figure 1, consists of a Parabolic Trough having an aperture of 10m, total length of 8 m, the focal length is located at 6.0454798 m at the Y-axis with maximum rim angle at φmax =1/4*(π-2θs)=0.78307rad [1]. The sun source is modeled in Tonatiuh as a ‘’pill-box’’ shape with the sun half angle set to 4.65 mrad, while the irradiance is set to 1000W/m2. The sun position is set at an azimuth of a 0.0o and elevation of 90.0o, i.e. the sun rays are cast normal to the reflective surface of the trough. The virtual surface, is placed parallel to the ground and centered in the vertical axis, 8 meters above the focal line.

Figure 1. Overview of the parabolic trough as simulated in Tonatiuh, Aperture 10m, focal length at 6.0454798 m, Virtual Roof at 8 m  

Note: Please request all the relevant files mentioned in this tutorial from [email protected]

FLUXTRACER SETUP

For this particular test case, we will use the voxel traversal and line analysis functionalities. As per Figure 2, set-up the input ray path, the output results path and select the voxel traversal and point analysis functionalities. This can be done by copying the relevant command lines from the list of available functionalities (lines 15-33). The corresponding command lines have to be pasted after the Rays folder line (line 4) as shown below. For this example, the voxel traversal functionality is pasted in line 6 while the line analysis functionality is pasted in line 7.

Figure 2: Setting up FluxTracer

VOXEL TRAVERSAL

For the voxel traversal functionality, the user needs to add the following information:

<VoxelTraversal cornerMin="-0.055, -0.597, -4" cornerMax="0.055, 0.612, 4" dimensions="90, 90, 430" output="VoxelTraversal.vtk"/>

This means that a bounding box of (X,Y,Z) = (0.11x0.15x8.0) m will be used along the focal line of the trough, fixed at equally spaced distances along Y and X axis around the focal line. The bounding box is spaced using 90 voxels in X and Y direction, while in the Z direction the box is spaced with 430 voxels. The output file will be called VoxelTraversal and the form of the output for this functionality is a vtk form. Figure 3 shows the results of applying the voxel traversal functionality on the parabolic trough. The contours shown are colored in terms of the number of rays that traverse each voxel within the region of the bounding box.

Figure 3: Voxel traversal results taken at X-Y plane at Z=0.0m showing contours the distribution of the radiant energy along the focal line of the trough

LINE ANALYSIS

The point analysis functionality in FluxTracer is called TubularReceiverVoxelized as shown below. For the line analysis functionality, the user needs to add the following information:

<TubularReceiverVoxelized center="0., 6.0454798, 0." direction="0., 0., 1." cornerMin="-0.055, -0.597, -4" cornerMax="0.055, 0.612, 4" dimensions="90, 90, 430" output="TubularReceiver.vtk"/>

This means that a bounding box of (X,Y,Z) = (0.11x0.15x8.0) m will be used along the focal line (0,0,6.0454798), fixed at equally spaced distances along Y and X axis around the focal line. The bounding box is spaced using 90 voxels in X and Y direction, while in the Z direction the box is spaced with 430 voxels. The direction parameter controls the direction vector of the line along with the analysis will be performed. For this case is along Z-axis, i.e. (0. 0. 1.). The output file will be called TubularReceiver and the form of the output for this functionality is a vtk form. 

Since the line analysis provides information regarding the minimum distances of the rays to a user-defined line of interest, this type of analysis can be used to get an idea of the minimum region that should be analyzed around the line of interest to get insight on the distribution of radiant energy around that line. If the line of interest defined by the user contains the focal point of the solar concentrator being analyzed, this type of analysis could provide insight on what should be the dimensions and cross section of a cylindrical receiver to capture a given fraction of all the solar energy sent by the solar concentrator to the receiver. As in the case of point analysis, this type of analysis could also facilitate the resolution of specific receiver related optimization problems without the need to run again the Monte Carlo ray tracer or of generating new sets of rays.

Figure 4 shows the results obtained by carrying out a line analysis with FluxTracer for the parabolic trough presented in the radiant density analysis subsection. It provides the geometry of the smaller convex receiver that will intersect all the rays reflected by the trough, assuming a pillbox sunshape distribution of 4.65 mrad half-angle.

Figure 4. Schematic showing a portion of the smaller convex receiver shape for a parabolic through of 8 m cross-section and a focal length of 6.045 m to intercept 100% of reflected rays assuming a pillbox sunshape of 4.65 mrad half-angle.

A preliminary validation of the results above was performed by comparing the result against the exact analytical solution. Thirty four years ago, Prof. Carlos Gomez-Camacho derived the equations of the cross-sections of the minimum convex receiver tube that will intersect all sun rays reflected by an ideal parabolic trough if the sunshape were a pillbox of a given half-angle θs [2]. He identified four different equations, which apply to four different cases determined by the relative position of the focal length of the parabolic trough with respect to the plane of the input aperture of the parabolic trough reflector. For purpose of this comparison, only one parabolic trough configuration has been analyzed. This configuration correspond to a parabolic trough whose focal line is above the the level of input aperture of the parabolic trough reflector, which is the case for the trough we are using in this example. According to the analysis of Prof. Gomez-Camacho, for this configuration, the optimal receiver shape in polar coordinates is given by:

where r is the distance from the focal point, ϕ is the deviation angle from the horizontal line passing from the focal point, defined in the range [ϕl,2π−ϕl], where ϕl is the root of the following equation

The above equation can be particularized for a given length of the cross-section of the parabolic trough reflector and a given focal length. To compare with FluxTracer, the parabolic trough geometry was particularized for a length of the cross-section of the parabolic trough reflector of 8 m and a focal length of 6.045 m. Figure 5 shows the comparison between the results obtained by FluxTracer and the analytical solution. As Figure 5 shows this approximation is indeed very accurate.

Figure 5: Preliminary validation of FluxTracer. (a) analytically calculated boundaries of the cross-section of the minimum convex receiver geometry to intercept all rays reflected from a parabolic trough of 8 meters input aperture cross-section and a focal length of 6.045 m assuming a pillbox sunshape of 4.65 mrad half-angle. (b) Cross-section of the density distribution in three-dimensional space of the point cloud resulting from FluxTracer’s line analysis of the same parabolic trough.

To estimate the accuracy of the FluxTracer predictions against the analytical solution, we consider the mean percentage error (MPE),

are employed to quantify their differences, where refers to the analytical predictions, refers to the estimated ones, while N is the sample size, set to be equal to the total number of the outermost voxels, as shown in Figure 5. As expected, the errors are marginal, yielding a value for the mean percentage error less than 1%.

REFERENCES

[1] Blanco M. J., The size of the focal spot of an ideal parabolic dish solar concentrator, CENER 

[2] C. Gomez Camacho, Receptor de concentrators parabolicos para mayor relation de concentration, II Congreso Iberico de ISES. Lisboa, October 1984, 5.51-55, Organizado por las secciones espognola y portuguesa de la International Solar Energy Society (ISES).