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Concentrators - Sine Law

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Posted by altsolarthermal on October 9, 2006, 9:09 pm
 
Hello all

Just a question about the Sine Law and its application.  It is my
understanding that the sine law states that the maximum concentration
of
100% of radiation that can be achieved from a symmetrical trough
concentrator is 1/sine(half-angle).

This then implies that the best theoretical concentration achievable on
an
East-West trough that tracks North-South daily is 229, calculated with
a
half-angle of half the solar radius (0.25).

With a non-tracking East-West trough the best theoretical concentration
achievable is 2.488 which is calculated from a half-angle of 23.7, or
the
change in solar height between equinox and solstice (23.45) plus half
the
solar radius (0.25).

In research publications this concentration is actually the size of an
exit
aperture at the base of the trough, not the size of a collector pipe
within
the trough.  It generally seems to be transposed to be applied to pipe
collectors, although there are substantial differences.

One of these differences is that the amount collected through an exit
aperture drops to zero nearly instantaneously when the incident angle
moves
outside the accepted range whereas with the pipe this collection
generally
drops 1 (direct non-reflected light only) for some angle range until
the
shadow of the trough wall falls upon the pipe.

A second difference is that the incident radiation does not drop off so
instantaneously with the pipe as it does with the exit aperture.  This
could
have an effect.  Concentration is defined as (proportion of incident
light
collected/collector size), so if the proportion collected drops slower
than
the required increase in the collector size the concentration will
increase.

So my question is: Is it generally considered true that the sine law
fully
applies to symmetrical trough concentrators with pipe collectors?  Is
2.488
considered to be the best concentration achievable from symmetrical
non-tracking trough concentrators?

Thanks for your time

Glenn Thorpe


Posted by Duane C. Johnson on October 10, 2006, 3:55 am
 
Hi Glenn;


 > Just a question about the Sine Law and it's
 > application. It is my understanding that the sine
 > law states that the maximum concentration of 100%
 > of radiation that can be achieved from a symmetrical
 > trough concentrator is 1/sine(half-angle).

 > This then implies that the best theoretical
 > concentration achievable on an East-West trough that
 > tracks North-South daily is 229, calculated with a
 > half-angle of half the solar radius (0.25).

We usually refer to the North-South tracking of an
East-West trough as "seasonal" rather than daily as
the movement each day is small.

I assume you are describing a perfect parabolic trough
with a depth equal to the focal length and a circular
receiver pipe. And the Sun's diameter is .5 degree
in diameter.

I don't think you have the calculation correct.
Concentration ratio is the ratio of the frontal width
of the parabola divide by the circumference of the
receiver pipe.

FL = Focal Length
X  = Concentration Factor
FL * 4 = 4 = Width
FL * pi * (2 * tan(.5 deg / 2)) = .0274 = Circumference
Width / Circumference = Concentration Factor
4 / .0274 = 145.9X

However, other concentration factors can be obtained
with parabolas of greater or lessor depths.

 > With a non-tracking East-West trough the best
 > theoretical concentration achievable is 2.488 which
 > is calculated from a half-angle of 23.7, or the
 > change in solar height between equinox and solstice
 > (23.45) plus half the solar radius (0.25).

 > In research publications this concentration is
 > actually the size of an exit aperture at the base of
 > the trough, not the size of a collector pipe within
 > the trough. It generally seems to be transposed to
 > be applied to pipe collectors, although there are
 > substantial differences.

You are not describing a conventional parabolic trough
I think you are describing a CPC, Compound Parabolic
Concentrator. Basically these non imaging designs are
from Roland Winston and others. These are generally
made with 2 parabolas on each side and are quite
deep.

Your 2.488X is close to the value in the figure in
the excellent book:
"Solar Engineering of Thermal Processes"
Duffie & Beckman
ISBN 0-471-51056-4
1991

 > One of these differences is that the amount
 > collected through an exit aperture drops to zero
 > nearly instantaneously when the incident angle moves
 > outside the accepted range whereas with the pipe
 > this collection generally drops 1 (direct
 > non-reflected light only) for some angle range until
 > the shadow of the trough wall falls upon the pipe.

True.

 > A second difference is that the incident radiation
 > does not drop off so instantaneously with the pipe
 > as it does with the exit aperture. This could have
 > an effect. Concentration is defined as (proportion
 > of incident light collected/collector size), so if
 > the proportion collected drops slower than the
 > required increase in the collector size the
 > concentration will increase.

 > So my question is: Is it generally considered true
 > that the sine law fully applies to symmetrical
 > trough concentrators with pipe collectors?

No. It is about 64% of the value and not sine.
I use tangent instead of sine but for these small
angles the difference is insignificant.

The major error is in not using the circumference
and the width of the parabola is 4 times the focal
length.

 > Is 2.488 considered to be the best concentration
 > achievable from symmetrical non-tracking trough
 > concentrators?

About right. Although the CPC system size is
considerably larger than a convention parabolic
trough with an over sized receiver.


Lastly there is another problem with East-West
designs in general. There is a cosine loss. You
will get 100% of the incident power only at solar
noon. As the sun position deviates from noon the
collected power decreases by the cosine of the suns
angle from solar noon, decreasing to 0% at the
horizon. (OK, we are ignoring atmospheric effects.)

When the trough is alined North-South along the
polar axis the collected power remains at 100% all
day. In addition the maximum collected power varies
seasonally from 100% at the equinoxes to 92% at the
solstices. (Again, ignoring atmospheric effects.)

Of course the North-South trough requires an active
solar tracker throughout the day. But these are
quite low in cost.

 > Thanks for your time
 > Glenn Thorpe

Duane

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Posted by SJC on October 10, 2006, 4:53 am
 I was just waiting for Duane to log on :)



Posted by nicksanspam on October 10, 2006, 5:48 am
 

Why the circumference, vs the diameter?

Nick


Posted by Anthony Matonak on October 10, 2006, 5:54 am
 nicksanspam@ece.villanova.edu wrote:

Presumably because that is what is receiving the light,
the surface of the receiver pipe.

Anthony

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