How To Build a Solar Generator

To go one further, for a solar trough collector it's also convenient to place your collection pipe level with the endpoints of your parabola.  This is good for providing mechanical support and easily finding your focal point.  This requires an extension to the basic parabola formula.  Unfortunately I did that derivation 30 years ago when I was far more mathematically astute and don't have it handy any more.

If the ellipse is eccentric enough, it will be close enough to a parabola to do the job.

Ways to draw an ellipse:

http://www.uwgb.edu/dutchs/MATHALGO/Ellipses.HTM

How to draw parabola with it's exact focal point, kindly confirm.

An ellipse is not a parabola.  However, a pipe is also neither a point nor a line. 

You only need a 'perfect' parabola if you need to focus to a perfect point/line.  Your target is a pipe, something of sufficiently larger diameter, and it blocks a fairly wide angle of a poorly focused reflected beam. 

Depending on the pipe size, the reflector can be somewhere between a half circle, a half elipse and a parabola and still work.  If the pipe surface is dark enough (dark forest green absorbs IR very well, black is OK but also radiates a lot of heat) it will still capture most of the reflected beam.

You can approximate the parabola by holding/pinning the ends of a string as long as the width of your reflector sheet against the edge of a piece of cardboard, and just let it hang.  The uniform weight-per-unit-length of the string causes it to sag in the shape of a parabola. Tilt the cardboard so the string is just touching to the piece of cardboard and  mark the line. 

The focal point for the parabola is on a vertical line half way between the two pins. (Draw the vertical centerline on the cardboard.)

Problem:  You can make as many parabolic shapes as you wish with one string, by moving each end toward or away from the centerline (equidistant).  However the focus of the parabola moves up/down the centerline with each change in width of the pins. 

Now the amount of sunlight captured depends on the distance between the pins... and the maximum energy reflected is, of course a flat panel, but the focus is at infinity...which is not good. If you bring both ends of the string together the focal point is at the very bottom of the 'loop' of string. Maximum collection by reflection, but zero collection surface... also not good.

The "right answer" is somewhere between. 

If you have a flexible reflector sheet, use the string to draw a 'good' parabola, say about half as deep as it is wide.  Then hold the panel against the cardboard roughly pointed toward the sun and see where the 'focus' hits the cardboard (WEAR one or two pair of SUNGLASSES!!!!).  Adjust the width/curve of the panel (moving both edges closer to or away from the center line) until you get the focused beam near the edge of the cardboard, centered between the pins. 

That is close to optimum.  Remember you will have a pipe there to collect the beam, not a point on the cardboard.

Practical Approach: To really play with it, draw  a black disk or dark circle of paper the diameter of the heater pipe, centered on the line at the edge of the cardboard, and see how the beam collects as you change width of the parabola.

Theoretical Approach: Do a lot of math.

Pragmatic Approach: Look at someone else's design and take a few measurements.

Now it becomes a design/fabrication issue. Is your pipe going to be 'inside' the curve of the parabolic sheet so that the pipe must pass through the end support pieces?  Then move the pipe 'down' closer to the curved sheet, move the
'ends' of the parabola closer and lose a little area of sunlight capture.  Or do you want to keep the pipe independently supported so it is not locked to the parabolic dish?  Then move the pipe 'Up' away from the reflector and widen the parabola. This increases reflective area, but exposes the pipe more to the air/winds.  These are design/fabrication considerations.

Then when you build (at least your first unit), you should have some nominal adjustment of the pipe position, an inch or so up or down, so you can tweak its position on the focus line for optimum operation.

In the 1960's I built a solar hot dog cooker using a photographer's tintype and plywood ends.  The wire for the hot dogs rested in notches in the plywood,  The unit did not focus 'perfectly on the wire, but it did capture all the energy in the wider 'dogs'... it worked very well. 

The largest variable in heating rate for it and for solar heaters is how well your parabola is pointed at the sun, so solar tracking is a must for optimum use.

Many years ago I saw an ingenious, home made, passive hydraulic solar tracking system.  It used a set of four black hydraulic cylinders, positioned on a frame, one on each side of and behind the solar panel.  When pointed directly at the sun each of the cylinders was halfway shadowed from the sun by the solar panel... they were all balanced at equal 'temperatures'.

As the 'sun' moved it would expose one or two of the cylinders more than the others, heating them up more and moving the frame until it was balanced again... and pointed directly at the sun.  The unit had to be reset to point eastward each morning, but even that could be 'automated' with a few weights and an internal hydraulic 'leak'.

Has that been commercialized yet?