Download

Click to download the Prometheus System Calculator v1.0.4
For maximum access, the program is written to work in Open Office Spread Sheet (Calc) (though many other spread sheet programs can read the format as well).

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Read Me

Please note this Program and Read Me are in beta version. Please write eerik(at)solarfire.org for any confusion or thoughts you may have.

What does it do?

The Prometheus System Calculator v1.0.3 takes inputs representing the basic dimensions of a Prometheus concentrator and calculates stuff, difficult or tedious to calculate on paper.

Presets:
When you open the program the inputs are pre-set as an example. The pre-set values represent a Prometheus-325 (325 individual reflectors) at 10 degree inclination with 30 x 30 cm individual reflectors.

These values can of course be changed to represent any size machine you imagine.

Inputs

To change an input simply click on the value and type in the new desired number, then press enter.

Many of the inputs and outputs are self explanatory and you’re encouraged to simply start typing in new numbers to see what happens, but for a better understanding of the program read on.

Outputs

The machine is left-right symmetric, and so all the outputs only represent the left hand side of the machine. However, since the machine is tilted toward the sun it is not top-bottom symmetric: the top of the machine will be higher off the ground and thus closer the focal point than the bottom of the machine. Footnote 1.

The outputs represent a machine of 18 cross-rows with 18 reflectors on each row (325 reflectors total). This is because 18x18 just large enough or larger than required (for a smaller machine just ignore the unneeded outputs), can be be based on 3 m steal lengths (half of the standard 6 m steal is usually bought) and the outputs fit on a single page.

Hence there are 9 positive rows (found above the centre of the machine) and 9 negative (found below). However, there are only 9 positive columns representing the reflectors for the left hand side, as the values will be the same for the right hand side.

All measurements starts from the centre of the machine, which is NOT the geometric or gravity centre, but IS where the main beam meets the vertical post (the centre of the reflector plane).

Smaller machines can be represented by simply plugging in the desired inputs and ignoring (colouring or deleting) the superfluous outputs.

Larger machines can be represented in 2 steps: 1. calculating the 18x18 centre of the machine and then 2. increasing the "1st row and 1st reflector space" to the 19th row and reflector respectively (which will be the distance to the outer row and reflector plus the row and reflector space respectively). Thus, the second round of calculation will begin with the 19th row and reflector and the outputs will represent the outside of the machine; the steps can be repeated for still larger machines. Or, if a larger output table is desired: the output blocks can be increased if you are very familiar with spreadsheet, or you can commission a larger output space from the author.

Focal Distance

The first section calculates the focal distance from every individual reflector. Under INPUTS there is the basic dimensions that will determine the position of each reflector and the height of the focal point.

Because it can, around the bottom and right hand edge of the output block are the measurements of each row or reflector from the centre (to the right of "Reflector spacing:" and below Row-spacing"), just so you don’t have to do the arithmetic when building the machine.

Focal height
The height of the focal point (as far as the program is concerned) is measured from the main beam (in a real machine there is a further space between the main beam and the ground, but this is irrelevant for the calculations and will depend on any additional clearance).

Cross-row space
The cross row space is the space between each cross row. Usually this space is about the same for every cross row since it is enough for the reflectors on rows simply not to hit eat other.

The two rows adjacent to the centre may be a common exception, see below.

However, you are not a prisoner of equidistant rows by any means, by adjusting the tweak values (replacing the 0 by something) of a row it will adjust the distance (positive or negative) of that row. However, the program will not automatically displace all the rows after a tweaked row, so if you want to tweak to do so you must tweak all the outer rows accordingly.

Reflector space
Like the rows, the reflectors must be spaced from each other so that they do not interfere with each other, but this time it’s light. If a reflector reflects to the back of the one next to it, some mirror surface is wasted. The inside mirrors reflect almost perfectly up so they must only be spaced slightly more (1 cm) than their width, but the outer reflectors must incline more to hit the focal point and so must be spaced a bit further (4-6 cm). (A later version of the program will hopefully calculate this distance automatically, but for now you may simply adopt an average of 33 cm (for 30 cm wide reflectors), or work it out with accurate drawings, math or empirically.

Again there is a usual exception for the first reflector space, and there is a tweak value for the precise distances.

1st row space
The 1st row space is with respect to the centre of the machine, defined above. If you plan to put nothing in the centre requiring more space, than the 1st row space will simply be half of the normal row space (one half from the centre to row 1 and one half from the centre to row -1). However, if, for instance, you wish to place a beam going from the centre to the outside (left and right) of the machine to support the outer structure, then the 1st row space may need to be increased so the rows do not obstruct.

1st reflector space
The 1st reflector space is usually not simply half the average reflector space, as there is the main beam and the handle (L piece) that attached the row to the main beam. So, normally, the 1st reflector space will be the 1/2 the width of the main beam + width of the handle + 1 to accommodate slight inclination +1-4 cm safety margin in case your reflectors attachment pieces (hands) are not perfectly centred.

Beam inclination
The Beam inclination is the inclinanation toward the sun. If the inclination is 0, then in the morning and afternoon each row will cast a shadow on the row behind it, which is resolved with even a minor inclination of 10 degrees. The stronger the inclination the the earlier in the morning and later in the afternoon the machine becomes fully functional, but the less efficient the machine will be approaching noon (as the reflectors must incline more relative the sun beams and thus reflect less light), and the higher off the ground the machine must be (so the lower end is off the ground) and thus taller, more expensive and less convenient. For a large machine 10-15 degrees is recommended; the early morning and late afternoon is not so important since the sun is not so strong at those times anyway and a large machine is usually meant to simply capture as much energy as possible so optimization for strong sun hours is best. Whereas a smaller machine, meant say for cooking, can approach 30 degrees as early morning or late afternoon cooking is usually desired more than maximum noon energy.

Focal distance outputs

The focal distance outputs gives all the distances from each reflector to the focal point. This information is used to decide which reflectors will be formed with which mould. A primary reflector made for a close focal point will be very ineficient placed at a focal point farther away. Generally, a primary reflector is only good for positions within 10-20 cm focal distance. Though other factors also play minor rolls.

An basic test the accuracy of the focal distances can be done by setting the beam inclination to 0. When this is so, the focal distances can be calculated by the Pythagorean theorem:

1. square-root(row-distance^2 + reflector-distance^2) = distance-from-centre.
2. sqaure-root(distance-from-centre^2 + focal-height) = distance-from-reflector-to-focal-point.

If the answer is different between your answer and the corresponding output than either you or I made a mistake.

However, the above formula will not work if the beam is inclined as then the reflector plain is no longer at a right angle to the vertical axis where rests our focal point, and so the Pythagorean theorem does not apply. With inclination of the main beam, the top reflectors will be closer to the focal point and the bottom further away, than would be with no inclination.

So the test is only for a limited case, but confidence should be relatively high if progressively more inclined beams render outputs that make sense.

For a more complete test, vector geometry is the fastest and what is used in the program. Simply add up row-vector and reflector-vector, which will require some cos and sin, than subtract the resulting vector with the focal-height-vector, which will give the difference between the focal point and reflector position. The focal distance is then simply the magnitude of this reflector-to-focal-point-vector.

Control Cable

If you wish to have one control cable which rotates all the rows simultaneously, then you will run into the problem that not all the rows have the same relative inclination. The outer rows are inclined more sharply in order to hit the focal point, and so if the cable attachment was placed at the same distance on every handle, the cable itself would not be straight but in an arch. The outer rows would have their attachments closer to the beam. So, the outer rows must have the cable attachments further away from their pivot so that all the attachments lie on a straight line, and there be not cable guidance wonkyness.

To calculate this we must calculate the inclination of each row for our optimum sun angle of the sun (there will be still some wonkyness as the sun is away from this angle, but the effect is insignificant for most purposes if the cable is straight for sun optimum). Then we decide what height we want the cable to be from the main beam, and finally solve for distance between the beam and each intersection of each row inclination with the cable height. Fortunately, the program does all this for you given a control-cable height and sun optimum (pre-set at 15cm and 45 degrees respectively).

Reflector angles

The reflectors can be oriented on the machine empirically, by simply rotating the machine to face the sun (at the time of day you which to optimize the machine for), placing a reflector on it’s position pointing it at the focal point and fixing the bolts, and repeating this until all the reflectors point at the focal point. Afterwhich, only the whole machine must rotate and the rows incline. However, though this only need be performed once, it is remains a long and arduous task.

If we can calculate all the angles of all the reflectors before hand we can pre-set their arms indoors, and then only slide in the reflector hand (into some sort of slot or boost), or make a guide that allows us to attach the reflector straight on the arms (that align the arms and the hands). For, all the reflector should be 90 degrees to the arms they are attached on. The ability to adjust the hand (which is attached to the reflector) perpendicularly to the arm (which is attached to the row) is only necessary if accurate construction is not possible.

But even without accurate construction, by presetting the arms, only minor adjustments must be made to the reflectors, and 1 round of adjustment may be enough (and render better results) than 3 rounds of adjusting under the sweltering sun.

With accurate construction (hands 90 degrees to the reflectors, positioned straight on the arms, each set at the appropriate angle on their rows), no adjustment on installation maybe necessary at all. (Though the ability to adjust may still be desired in case something goes wrong.)

One further advantage of being able to calculate the angles of every reflector for some given sun angle, is that it allows us to master a complicated aspect of the machine, explained under sun limit.

Quick angles

The angles of each reflector is unique, but each columns of reflectors is close. So, if final adjustment is required anyway, then all the rows can be pre-set the same. The quick angles simply copy the angles of row -2, from the optimum angles below, for easy reference.

Precise angles

For the sake of simplicity, we are only talking about one side of the machine, however, keep in mind all the angles are the same for the other side.

For each each angle of each reflector to strike the focal point for some angle of the sun, is unique. This is difficult to visualize, but I assure you it’s true; if you imagine the sun vectors raining down all parallel to each other then must strike each reflector that must have a normal vector precisely half way between the sun-vector and the focal distance vector; since all the focal distance vectors are unique all the reflection normal vectors must also be unique. When the sun changes angle, reflectors fixed for a different angle of the sun, will no longer be perfect, but not by much. The difference is very slight, slight enough for the system to work well many hours around the optimum sun angle the machine is callibrated for. What’s more each sun angle (except for solar noon) is reached twice during a day, once in the morning and once in the afternoon, so if we optimize for slightly more than the middle of the morning (say 45 degrees) then we are not efficient at very low sun angles (but the sun is not very strong and may be obstructed by building or trees anyway) but move closer and closer to the optimum then arrive at optimum, then move away as we approach noon (but the sun is the strongest, and our reflectors can reflect more sunlight as they face more directly the sun), then back towards optimum in the afternoon, and finally efficiency starts to reduce significantly in the hours that interest us the least.

Sun limit

Sun limit recalculates the angle of each reflector for a different angle of the sun, so that we can compare them to sun optimum.

For, the disadvantage of the system is that if all the reflectors focus perfectly on the focal point for some sun angle (sun optimum), they will not and cannot focus perfectly for some other sun angle, the focal point will thus be larger for all other sun angles. For many applications the many advantages of a fixed focal point outweigh the disadvantage of a changing focal point, but it remains important to know how much in order to determine the size of the capture area, the expected performance, and other things. So, if we know the reflector angles for two sun angles, sun optimum and sun limit (beyond which reduction in power is not important) then we can calculate the degree to which sun limit will miss the focal point if the machine is calibrated for sun optimum.

Miss calculator

By deciding our sun limit (the angle of the sun where we still want a maximum, or close to max, amount of sunlight to strike the collector surface), then we can determine the amount each reflector will miss the focal point at that angle and determine our collector surface area accordingly.

You will notice that the misses are most extreme in the upper right hand corner, this is because the lower part of the machine already has a more oblique angle in order to hit the focal point (the focal point is behind), and so changes in the sun’s angle is less significant. Also, you may notice that the lower the focal point the greater the miss. So, for machines with very low focal points intended for use at any time, a solution to the miss problem is simply building the machine without the outer top corners.

By playing around with the sun optimum and sun limit angles you may obtain negative results for misses, this means your reflection misses on the other side of the focal point.

The Prometheus System Calculator is developed by Eerik Wissenz, licensed under GPLv3+: GNU GPL version 3 or later; http://www.gnu.org/licenses/gpl.html. Free software: you are free to change and redistribute it. There is NO WARRANTY, to the extent permitted by law.