INTRODUCTION

By line analysis we mean the capability of FluxTracer, given a set of rays and a reference line, to compute the minimum distances from the rays to the line and to compute the points on the rays at which this minimum distance is achieved. In this analysis, FluxTracer computes the minimum distance from each ray, within the large set of rays generated by the Monte Carlo ray tracer, to a user-defined line within the region of interest. It provides information regarding the minimum distances of the rays to the line, and the position of the points on the rays corresponding to those minimum distances.

To carry out the line analysis, the user should specify to FluxTracer the following information:

• The size and location of the region of three-dimensional space of interest, as in the radiant density and point analysis.
• How the region of interest shall be voxelized, as in the radiant density and point analysis.
• The line of interest in regard to which the analysis should be carried out, specified by point of coordinates (x, y, z) and direction in terms of a unit vector (dx, dy, dz).
• The set of rays generated by the Monte Carlo ray tracer to be processed by FluxTracer.

As in the point analysis, because the number of rays usually processed by FluxTracer is relatively large, it is not usually practical to provide for each ray its closest distance to the line of interest and the coordinates of the points of closest distance. Here again, by default, FluxTracer provides instead the probability distribution of distances of the rays to the line of interest and the radiant density of this points in three-dimensional space using a similar voxelisation of the region of interest than in the radiant energy analysis.

Since the line analysis provide information regarding the minimum distances of the rays to a line of interest defined by the user, this type of analysis can be used to get an idea of the minimum region that should be analysed 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 analysed, 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 optimisation problems without the need to run again the Monte Carlo ray tracer or of generating new sets of rays.

COMPUTATION OF THE MINIMUM DISTANCE FROM A RAY TO A LINE

The line is described as

r=rL+λdL

The vectors d and dL define a set of planes with the normal vector

n=d×dL

The shortest distance between lines is

The corresponding points satisfy the system of equations

The solution is

and

If the lines are parallel n=0, the shortest distance is

POSSIBLE ANALYSIS OF INTEREST

Adding to FluxTracer the capability of computing the shortest distances from a set of rays to a user defined line and to compute the position or points on the rays at which this shortest distance is achieved will allow the following types of analysis:

Set of distances from the rays to the line:

• Compute the cumulative distribution function providing the number of rays (or the percentage of total rays) as a function of the distance from the line.
• From the cumulative function compute its derivative, i.e., the density function providing the probability that a ray be at a given distance from the reference line.

Cloud of 3D-points which are at the shortest distances from the rays to the line:

• Transform the cloud of 3D-points into a voxelised energy density in 3D-space.
• Compute the minimum enclosure containing the cloud of 3D-points.
• Compute the centre of gravity and the axis of inertia of the 3D-point cloud.

The analysis of the distances may be useful to determine the minimum radius of a cylinder that has as its axis the user defined ray that will intersect a given fraction of rays. The analysis of the cloud of 3D-points may be useful to determine the shape of the minimum size receiver extruded along the user defined line. To some extend the analysis of the voxelized energy density obtained from the cloud of 3D-points may also provide insight into the shapes of potential receiver geometries of interest.

OPTIMIZATION PROBLEMS

The study of the minimum distances from the rays to a given line, leads naturally to interesting optimisation problems, such as:

• What is the position and orientation of the reference line that will minimise the maximum "minimum distance to the line" of a given fraction of the total number of rays being analysed? Will this position and orientation change as a function of the fraction of rays?
• Once the line referred to in previous point has been obtained, what is the position and orientation of a line contained in a plane perpendicular to that line that will minimise the maximum ""minimum distance to the line" of a given fraction of the total number of rays being analysed?
• Will the lines obtained in the previous points provide insight on the location and shape of a minimum receiver that will intersect all the rays in the set? Will the point where they intersect coincide with the focal point of the concentrating system?