Ray Tracing For Concentrator Design

The light funnel is a simple solar concentrator that is used widely in solar cooking and other low temperature solar thermal processes. Despite its wide use, there is a general lack of understanding of the properties of this deceivingly simple concentrator. The common belief is that reflectors inclined at 45 to 60 degrees is ideal, with 60 degrees requiring more tracking but providing greater heat. The development of an open source ray tracing package in the python programming language has enabled a review of this under scrutinized concept. Through ray-tracing and geometric arguments inspired by ray-tracing results, a significantly steeper design is here provided. A solar oven of significantly higher temperature may be achieved, with a concentration ratio of 13x, compared to the limit of 4x in a 60 degree funnel.

Please see the bitbucket repository for source code and details about the program used here.

The data in the following graphs was generated using around 2500 rays to trace a short trough type light funnel. The rays were positioned from an evenly spaced grid and then each ray was randomly offset in two dimensions. Every grid square has contains exactly one randomly placed ray origin.

The model had a target width of 5 and so an example with a mirror length of 10 has reflectors twice as long as the target area. The simulation is necessarily three dimensional, so a short trough light funnel was modeled, though rays had no length in the depth dimension of the funnel.

Concentration ratios and optical efficiencies here given are the properties of a long trough type funnel. A Cone focusing in two dimensions wold have a concentration ratio and optical efficiency that is the square of the trough version. An example given with a concentration ratio of 2x and optical efficiency of 80% would translate to a cone of concentration 4x at 64% efficiency. It is possible that the cone would under-perform this extrapolation, as this is the case with a well studied concentrator using similar principals: the compound parabolic trough and its two dimensional concentrating version.

The reasoning that follows is for a funnel that is moved to track the sun every hour, give or take half an hour. On the equinox, the sun travels 180 degrees in 12 hours in its fastest changing direction, or 15 degrees in one hour. The sun varies little over the course of one day in its other principal axis. A trough with an acceptance angle of +/- 10 degrees would be tracked seasonally, and would not need to track the sun more than a six or seven times a year. Thus a funnel with an acceptance angle of +/- 10 degrees would not need to be moved more than once an hour. The best way to concentrate light with only one tracking adjustment a day (at noon) is a vertical mirror moved from one side of the target to the other as the sun passes directly overhead. This would work better than a cone type light funnel if one were not going to track the sun.

Principals of a Light Funnel

A light funnel is built cheaply to heat food or for other low temperature processes. It will necessarily reject light and have less than ideal optical efficiency.

The principals of the light funnel can be understood through its edge-rays. edge-rays are the most extreme rays that enter the collector. In proper design, edge-rays hit the appropriate edge of the target, and geometry guarantees that less extreme rays will make it to less extreme areas of the target. Edge rays define the limits of the funnel, and describe its performance collecting light over a range of angles .

Consider a light funnel with sides inclined at 60 degrees, and confined to light rays coming straight down (an unrealistic approach because the sun itself has some angular width and any collector will be imperfectly tracking the sun). If the funnel is made too large, then the outermost rays will not hit the target but instead hit the mirror on the opposite side. At these angles, that ray of light would bounce back exactly along the path it came in on.

Rays outside of these edge rays will all be rejected, and thus the funnel is restricted to a certain useable size. For 60 degrees, the maximum concentration ratio is equal to 2 in a line focusing "trough" setup, and 4 in a two dimensional "point" focusing setup.

In the 60 degree example, the usable funnel size is limited by the shallowness of the reflected light. If the light were reflected at a steeper angle, then a deeper funnel could be used. Steeper sides would cause steeper reflected light, and although steepening the sides would apparently make the entry aperture smaller for a given depth of funnel, steeper reflected rays means that the funnel can be deeper. Being able to deepen the useful funnel actually allows for an increased entry aperture and thus a higher concentration ratio.

Steeper sides allows for higher concentration ratios

Assuming only one reflection, an equation for the concentration ratio can be algebraically determined from this edge ray. This equation is R = 1 – 2*Cos(2*theta). For 60 degrees this equation gives 2. For 75 it gives 2.73. The minimum concentration is 1 at 45 degrees (ie: none of the light that hits the reflectors hits the target), and the equation losses significance at 90 degrees. At 89.999 degrees and an almost infinitely long funnel, 3 times concentration is predicted.

At 75 degrees also, an extreme ray that hits the opposite mirror instead of the target is not immediately rejected and could still hit the target. So for any inclination greater than 60, the actual concentration ratio is greater than this. Given infinite reflections, simulation shows that the concentration ratio is almost exactly 1 / Sin(90 – theta). For 60 degrees, these two equations both predict 2x concentration. For 75 degrees, perfect reflectors would be capable of 3.86. Since each reflection loses energy, it is unreasonable to expect third and fourth reflections in even steeper funnels to be practically worthwhile.

Behavior with rays at some acceptance angle

Light will not fall perfectly parallel and straight along the central axis of the concentrator. Due to the angular width of the sun and much more than that to inaccurate tracking of the sun, a range of angles must to be considered. A properly designed funnel will behave favorably with off axis light within some acceptance angle.

Given a light over a range of angles, some light will very likely be rejected. Analysis of a light funnel should report its optical efficiency (how much light that enters through its aperture hits the target) as well as its effective flux concentration ratio. The effective concentration is the ratio (entry area/target area) times the optical efficiency. *(Remember that the concentration ratios and optical efficiencies given here are for a one dimensional concentrator. An ideal cone type two dimensional concentrator would have concentrations and efficiencies that are the square of these.)

A light funnel with sides of 60 degrees built just deep enough to achieve 2x concentration for light coming straight in will perform less well with off axis light. Edge rays that deviate from 0 degrees straight in will be rejected from one side or the other at an even lower point on the funnel. Thus we expect a light funnel to have an optical efficiency vs incoming angle curve that is %100 at zero degrees and slowly declines as angle increases. Simulation verifies this:

Though an extreme ray is rejected from one side, its parallel ray at the other side still bounces in. By making the funnel deeper then, light from various angles can still be collected.

A light funnel with sides extended to allow some light to be rejected

A funnel that has an entry aperture 3 times the width of its target, but which spills 2/3 of its light, effectively delivers light at 2x the intensity as incoming light. In this example, the optical efficiency and thus the effective concentration of the system remains level for a good +/- 10 degrees. At every angle, some light is rejected. But over a range of angles the effective concentration remains 2x. An infinitely long funnel reaches the limit of 2x concentration over the range of +/- 30 degrees (90-60).

A 75 degree funnel (infinitely long) would be capable of collecting light at +/- 15 degrees. Because rays at 10 degrees are much closer to its limit, the steeper collector must be made relatively much deeper to perform ideally over this range.

a steep funnel with reflectors 7 times longer than the aperture width

This collector with steeper 75 degree sides has a concentration ratio of about 3.85, almost twice the limits of the 60 degree cone. (The slight increase in collection efficiency (and thus concentration) with increasing angle is an artifact of the model and not a real phenomena. Rays were generated just above the entry aperture and so angled rays enter inward from the extreme edge of the model. These are rays that would be rejected, and since they are not counted, the efficiency seems to increase.)

This model that allows up to 10 reflections shows the 75 degree funnel performing well above the single reflection projection. In fact, if one allows infinite reflections a funnel would concentrate at 1/sin(90 – theta), which is the thermodynamic limit given the acceptance angle. Even considering this limit with an actual collector would lead to excessively large collectors and low optical efficiencies. Because of this, light funnels should be designed to operate at well below their theoretical acceptance angle.

Some of the calculated superiority of the 75 degree funnel over the 60 degree is due to secondary reflections that are not rejected. In reality, with aluminum foil as the reflector, these secondary reflections are less beneficial than the simulation would predict. Not all of the increased performance however is due to second reflections.

The graphs below represent the perfectly reflecting funnel and the same funnel but with only single reflections counted:

For the 60 degree funnel, the two cases (a single or many reflections) are indistinguishable. For the 75 degree funnel, the constrained model under-performs the perfectly reflecting funnel. The 75 degree funnel however does not rely upon excessive internal reflections, and with one secondary reflection allowed the 75 degree funnel continues to concentrate at the theoretical limit.

Still, for +/- 5 degrees, the singly reflecting 75 degree funnel has an effective concentration ratio of about 2.7. The 60 degree funnel reaches its maximum at 2. The 75 degree funnel maintains above a 2x concentration beyond +/- 10 degrees. Though it falls off sooner, it maintains superiority over the 60 degree funnel.

For comparison, an unrealistically steep and deep funnel was modeled, and the singly reflecting funnel gave 2.9 times concentration:

In general, steeper sides cause higher peak concentration ratios but also a smaller range of angles over which that ratio is achieved. The 75 degree funnel, though falling in performance at a smaller angle than the 60 degree, still maintains equal to or above the performance of its shallower sibling over the range of angles simulated. The singly reflecting models underestimate the effectiveness of of an actual funnel, as light bouncing twice off even aluminum foil would still have some intensity, even if only %60 that of single reflections.

The 60 degree funnel requires reflectors twice as wide as the target in order to concentrate 2x over +/- 10 degrees. This is probably the lower limit to the usefulness of a light funnel. This performs well over the range of angles a poorly tracking solar concentrator would experience. Since the concentration ratio falls off slowly after 10 degrees, the collector could be adjusted after the sun has moved even as much as 30 degrees. As the reflector steepness is increased, higher concentration ratios are achievable.

One more design consideration

Not all areas of the reflectors are utilized equally. In the example of the 60 degree funnel fully accepting at +/- 10 degrees, the outer areas of the mirror are active only for extreme angles, while the bottom portion is actively collecting light from all angles.

The 75 degree funnel can utilize secondary reflections to achieve better performance. Because secondary reflections give very shallow rays, only the bottom most section of the reflector is actively used in useful secondary reflections.

For these reasons, if one had mixed materials to work with, some excellent reflectors and some less reflective materials also, then using the best reflectors at the bottom is the most obvious choice, as the bottom 10 or 20 percent of the funnel is most responsible for the performance of the concentrator.

A concentrator with its reflectors more completely used would offer greater capacity per bulk, awkwardness and reflector cost. Although the 60 degree example achieves 2x concentration over +/- 10 degrees, the outer portion of the reflectors are under utilized. A 75 degree funnel with full concentration over this range of angles would be very long. But if one were to strip it down to the bottom most section, to the reflectors that are most actively used, we should have a concentrator with greater capacity for the same amount of reflector as the 60 degree.

Ray-tracing has shown this to be the case. Compare this graph to the 60 degree model discussed earlier:

Using the same reflecting panels as the 60 degree model (reflectors twice as wide as the target), the 75 degree funnel gives equal concentration over a greater range of angles. Although it’s aperture is larger, the 60 degree model is spilling energy while the 75 degree model is under utilized. One could actually still add reflector length to the 75 degree model and increase its concentration and energy intercept if desired, while adding reflector length to the 60 degree model would not increase concentration and only very minimally increase energy intercepted.

Below is a graph of the minimal length of funnels of different tilt and with rays coming in at 10 degrees. Reflectors of this minimal length are active over all angles of incoming rays. Beyond this length, the reflector area becomes increasingly specialized. Data below 50 degrees is erroneous, where the reflected extreme ray is horizontal and no concentration takes place. This graph with rays coming in at 0 degrees would begin to be erroneous under 45 degrees.

It happens that the 75 degree funnel with a mirror to target width ratio of 2 is about the maximum of this plot. The maximum shifts only a degree or so if incoming rays are changed from 10 to 0. Below about 75 degrees there is less mirror area to actively collect light, and above 75 degrees the angle is too steep to get a satisfactorily large opening.

The Ideal Concentrator

The mathematical ideal concentrator has been achieved over 30 years ago. The compound parabolic concentrator was the design that achieved this. A CPC trough achieves 3.82x concentration over the range of +/- 15 degrees. A similar light funnel (tilted to 75 degrees) could reasonably achieve 3.6x concentration over +/- 5 degrees or over 1.8 times concentration over +/- 15 degrees.

Using flat reflectors of any length, a tilt of 75 degrees is desireable. Beyond this, there is no generalized design best for all situations. The important factor is the temperature at which work is being done by the solar oven. Given reflectors of a fixed size, the reflectors could be moved as far apart as giving a reflector length to target width ratio of 2. This would result in the greatest ammount of energy being collected bt only at 2x concentration. Moving these reflectors closer together and decreasing the target size increases the concentration ratio up to about 3.62 (reflector to target ratio of 6), but decreases the ammount of energy collected.

The stagnation temperature (temperature an empty oven reaches) is determined by the degree of insulation and the concentration ratio. The ammount of useable energy collected is realted to the ratio of the absolute working temperature to the absolute stagnation temperature. Bringing the reflectors closer together would increase the temperature of the oven but decrease the ammount of total energy.

Future sudies should be aware of this interplay, and the thermodynamics
of baking bread or cooking rice may be studied.

Currently, the recomendation for solar oven design brought forward by this study is to use reflectors tilted at 75 degrees, brought as close as possible to the object being heated (reducing the target size to the minimum), and to use reflectors as long as possible, from twice the target width up to 6 times the width of the target.

-elliot