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(2)TRough Parabola Model, Spot Size and power distribution attempt

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Posted by Erdemal on October 12, 2007, 1:40 pm

Continues Topic "Tough Parabola Model"

---- first post : Erdy > Erdemal wrote: ---------------
Is there somewhere a model for tough parabola solar
reflector ?

I'd like to design and optimized one for the hotest
focus point given a known surface imperfections.

I could write it and probably reinvent the wheel.

Cant find any searching with 'tough parabola model'

Morris Dovey wrote:
 > Erdy > Erdemal wrote:
 > | Not being a great 'handyman/DIY', 'imperfections' are expected
 > | on that parabola :). The purpose is too to evaluate the quality
 > | of the sunlight focus given :
 > |
 > |     - apparent sun diameter
 > |     - quality of the 'real' parabola surface
 > |       which will not be perfect.
 > |     - parabola parameter (p   in   y^2 = 2px)
 > |
 > | In example, divide the 'trough parabola' surface in
 > | squares of side of one centimeter (or more or less),
 > | suppose these surfaces plane, set a random tilt error
 > | (1, 5, ... degree) for each of these surfaces,
 > | apply reflection law to them then compute the
 > | total 'ligth' received by a surface positionned at
 > | focus or elsewhere.
 > Ok. The "easy" way to approach this one miight be to take a
 > representative crosswise "slice" that's one square wide, then pick a
 > target pipe diameter. Now pick numbers of squares in that
 > representative slice that have 0-degree, 1-degree, 2-degree, etc.
 > errors and calculate how much of the light reflected from that square
 > will hit (or miss) the target.
 > Note that focal length and target sizes are significant. Since for a
 > given target size a small angular error might give a partial hit if
 > the focal length were 'small', but might be a complete miss if the
 > focal length were 'large'...
 > | To an old man like me, it would take few+ hours to
 > | write it. A model or charts doing that must exist
 > | somewhere. Such plots for UHF and SHF can be found
 > | (gain versus surface average 'bump', size, frequency, ...).
 > Yeah right. An experienced old duffer like you could probably knock
 > off a better fast approximation than an inexperienced young guy like
 > me could compute. :-)

Not that old. Maybe just very lazy and a little bit old :)
I am a french speaking Belgian, I dont get much of your
explanation. Let's hope the following will be understood,
my 'englishus kitchenus' wont probably help.


Modeling parabola surface imperfection is not as complex
as I wrote in a previous post :). Sitting and drawing a
schematic of it shows in one minute that it's much more

Quality of light spot at focus:
- the average apparent - seen from earth -, diameter of
   the sun is 0.53°
- each point of the parabola reflects the image of the
   sun in a cône of 0.53° of aperture : the farer from
   the miror the larger the spot.
- surface imperfections will deviate the cône, this will
   result in larger spots

1 - Perfect Parabola
'The farer from the mirror the larger the spot'
So the 'total' spot will be envolved of two concentric circles :
   - the smaller issued from the vertex of the parabola
         D_min = 2 * d_min * tg (0.53/2)
         D_min ~= d_min * tg (0.53) = d_min * 0.00925
   - the larger issued from the edge(s) from the parabola
         D_max = 2 * d_max * tg (0.53/2) = d_max * 0.00925
         D_max ~= d_max * tg (0.53) = d_max * 0.00925
   (*) D for diameter, d for distance (just to confuse you :)

   - in the smaller circle, the power will be maximum all accros
     the circle.
   - in the annular region, the power will decrease from maximum
     to zero at the external edge. ??? The decrease will be
     inversly linked to parabola area ??? ... I guess ???
     Simple math will say.

2 - Imperfect Parabola
Let's make it simple :), imprefections will just spread a
certain amount of the power on a third concentric circle,
larger than the two previous. The size of that circle will
depend on the maximum deformation of the imperfections,
you can evaluate it in degree and simply add it to the 0.53°
of the sun apparent diameter (it will have the same effect
as a blury sun :). The amount of the power deviated by
impeffections has to be evaluated and will depend of the
quality of the construction.

??? For the power distribution, a gaussian distribution
??? with a known/evaluated maximum at center and a know
??? evaluted value at ??? will probably be the best
??? approximation.

I see many other possibilities ... :)

3 - Conclusion
The hottest spot is obtained with the shorter focal distance
and the larger parabola. For a same parabola, as the physical
parabola gets larger, the annular region increases of the focus
spot increases and the added power for size increase becomes
marginal. An perfect parabola of infinite size will not give
an infinite temperature at spot.

Let's do the math ... later !


All this must have been already computed and published ! Maybe
Archimedes already did it 2200 years ago in Syracusa (Sicily).


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