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BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to Fresnel reflectors and more specifically
to spiral Fresnel reflectors.
2. Description of the Prior Art
There is a great need, particularly in the poor third world countries, for
an inexpensive solar cooker which can cook food without the necessity of
using valuable fuel. Traditionally, solar cookers have been in the form of
paraboloids or hemispheres. However, such shapes are difficult to
manufacture and in order to keep their shape they must be formed from
metal, fiberglass or hard plastic. All of these materials are expensive
and the forming processes for these materials are expensive. As a result,
the finished cooker is also expensive. Further, such shapes are bulky and
require a disproportionate amount of space when shipped. Since shipping
volume is expensive, the cost is again increased.
A Fresnel Reflector is a reflector made from a flat sheet of material and
having concentric rings which have an identical focal point. The use of a
Fresnel reflector as a solar cooker would be advantageous since all of the
parts of the Fresnel reflector could be cut from one sheet of material and
there would be no complex three dimensional shapes to manufacture; the
Fresnel reflector could be made out of inexpensive materials such as
aluminized cardboard or aluminized plastic; and the assembled reflector
would have a very low profile and would be easy to transport and store. A
concentric ring Fresnel reflector is disclosed in "Compact Solar Energy
Concentrator" by Robert W. Hosken in Electro-Optical Systems Design,
January 1975, pages 32-35. However, the Fresnel reflector described in
this article provides rings which are machined into a blank of solid
material. Therefore, a high degree of precision is necessary is machining
the rings into the blanks with a resulting relatively high cost of
manufacture.
A Fresnel reflector using separate concentric rings has been proposed in
the past (for example, "EVALUATION OF SOLAR COOKERS" by Volunteers for
International Technical Assistance for the U.S. Department of Commerce,
Office of Technical Services). However, such a Fresnel reflector using
concentric rings has several disadvantages. Each ring of the Fresnel
reflector must be assembled and mounted separately, a time consuming task.
Further, each reflector is composed of many separate parts. Since each
ring is a separate part, there are many parts which can be misplaced or
damaged.
An inexpensive, easily transportable, Fresnel reflector would also be
advantageous in other areas of solar energy. For example, it could be used
for low to medium temperature steam generation for producing power. It
could also be used for the production of electric power by use of a
Brayton or Sterling cycle generator located at the focus of the reflector.
Further, direct electric power production by photovoltaic conversion would
also be possible by the placement of solar cells at the focal point of the
reflector. Accordingly, an inexpensive, easily transportable, Fresnel
reflector would be advantageous in all areas of solar energy production.
SUMMARY OF THE INVENTION
It is therefore an object of the present invention to provide a Fresnel
spiral reflector.
It is a further object of the present invention to provide a method for
making a Fresnel spiral reflector.
It is a further object of the present invention to provide a Fresnel spiral
reflector having a negative focal length.
It is yet a further object of the present invention to provide a Fresnel
spiral reflector which may be assembled in one simple and fast operation.
It is a final object of the present invention to provide a Fresnel spiral
reflector whose reflector is composed of a single piece of material.
If a mathematically defined spiral is developed on a piece of flat material
and the spiral is cut along its spiral line and "wound up" the arms of the
spiral will have an angle of inclination with respect to the plane of the
original sheet. The angle of inclination is proportional to the distance
of the arm from the center of the spiral. It has been found that the angle
of inclination of this spiral would be continuously changing so that the
point on the spiral would have the proper angle of inclination so as to
reflect sunlight through a focal point. The present invention therefore
involves the calculation of a spiral which can be formed on a sheet of
flat reflecting material and which, when "wound up" will have a
predetermined focal point. The spiral can be formed so that the resulting
reflector will have either a positive or a negative focal length.
Such a Fresnel spiral reflector will have all of the advantages of a
concentric circle Fresnel reflector and will have the additional
advantages of being quickly and simply assembled and of being composed of
a single piece.
The developed spiral can be formed by using a computer program to calculate
a spiral which, when wound up, will provide a predetermined maximum
diameter, focal length, concentration ratio, estimated blockage,
reflectivity and number of mounting rods. The computer program can then be
used to plot the developed spiral which can be transferred to a sheet of
reflecting material, cut out, and wound up to result in the desired
reflector. The wound up arms of the reflector can be held in place by
radial mounting rods whose positions can be determined by the computer
program.
Alternatively, a cam can be used to plot a developed spiral on a rotating
sheet, this developed spiral then being transferred to a sheet of
reflecting material as with the spiral developed by the computer program.
As a further alternative, the cam could be used to plot a developed spiral
on the sheet of reflecting material itself. In further alternatives, the
computer program could be used to generate a tape which could be used to
direct a numerically controlled machine to cut out the spiral, or a
stencil could be made from the spiral pattern and the stencil could be
used to mass produce identical spiral patterns.
BRIEF DESCRIPTION OF THE DRAWINGS
A more complete appreciation of the invention and many of the attendant
advantages thereof will be readily obtained as the same becomes better
understood by reference to the following detailed description when
considered in connection with the accompanying drawings, wherein:
FIG. 1 is a schematic plan view of the spiral Fresnel reflector;
FIG. 2 is a schematic cross-sectional view of the spiral reflector of FIG.
1;
FIG. 3 is a schematic plan view of a positive focal length developed
spiral;
FIG. 4 is a detail of a portion of the reflector of FIG. 2;
FIG. 5 is a schematic cross-sectional view of a negative focal length
spiral Fresnel reflector;
FIG. 6 is a schematic plan view of a negative focal length developed
spiral;
FIG. 7 is a schematic elevational view of a spiral Fresnel reflector
reflecting coverging light rays from a plurality of mirrors;
FIG. 8 is a schematic representation of an apparatus for plotting a spiral
on a sheet of material.
FIG. 9 is another embodiment of the apparatus of FIG. 8;
FIG. 10 is a plan view of a connecting rod arrangement;
FIG. 11 is a detail of the connection between one of the connecting rods
and the spiral;
FIG. 12 illustrates one embodiment of the centerpiece for the connecting
rods of FIG. 10;
FIG. 13 is a plot of a developed spiral produced by a computer;
FIG. 14 illustrates a frame defining an equilateral triangle grid;
FIG. 15 illustrates a frame defining a square grid;
FIG. 16 is a cross-sectional view of a type A-B conic reflector;
FIG. 17 is a cross-sectional view of a type A conic reflector;
FIG. 18 is an isometric view of the reflector of FIG. 17;
FIG. 19 is a cross-sectional view of a type P-B conic reflector; and
FIG. 20 is a cross-sectional view of a type P-A conic reflector.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
FIG. 1 shows a plan view of a wound up Fresnel reflector. The reflector has
a projected arm width of x and the spiral is at a distance d from the
center of the polar coordinate system, d varying with the angle .beta. of
the spiral. The same reflector is shown in cross-sectional elevation in
FIG. 2. As can be seen, the angle of inclination .phi. of each portion of
the spiral reflector arm varies with the distance d of that portion of the
arm from the center of the polar coordinate system. It can also be seen
that the focal length of a reflected light ray which is reflected from the
base of the arm portion, that base corresponding to the plotted spiral, is
equal to f.
FIG. 3 is a plan view of the spiral which is developed on a flat sheet of
reflecting material in such a manner that, when cut along the spiral line
and wound up, the Fresnel spiral reflector of FIGS. 1 and 2 results. The
developed spiral of FIG. 3 has a distance D from the center of the polar
coordinate system for any angle .PSI. of the spiral. In order to develop a
spiral on a flat sheet of reflecting material having a desired projected
arm width x and diameter d, it is therefore necessary to calculate what D
and .PSI. corresond to a wound Fresnel spiral reflector having a projected
arm width x, and a diameter d at a wound angle .beta..
##EQU1##
The above equation (1) relates the distance D of the developed spiral to
the desired focal length and the desired arm width at a wound up angle
.beta..
##EQU2##
In the above equations, (2) relates the angle .PSI. of the developed spiral
to the desired projected arm width and the desired focal length for any
wound up spiral angle .beta.. Therefore, a developed spiral can be plotted
for any desired resulting spiral Fresnel reflector having a predetermined
projected arm width, focal length and wound up maximum diameter.
The above equations, however, only relate to the inner most edge of a
portion of the spiral arm since it is that edge which is defined by the
spiral according to the above equations. However, the maximum
concentration ratio is not located at the focal point of light striking
the bottom of each portion of the arm but at the focal point ff of the
light striking the radial center of each portion of the spiral arm. As can
be seen in FIG. 2, it is at the focal length ff that the focal area is at
a minimum (mw) and it is at this minimum focal width that the
concentration of energy is greatest. Therefore, it is desirable to relate
a developed spiral to the concentration ratio within the minimum focal
width at the focal length ff. The relation between f and ff is found from
equation (3).
##EQU3##
The minimum width (mw) from equation 3 can be found according to equation
(4):
##EQU4##
G is percentage of blockage .rho. is reflectivity of the material
CR is concentration ratio
The percentage of blocking (G) from equation (4) is most easily understood
from FIG. 4. FIG. 4 shows a close up of the cross-sections of two portions
of a spiral arm. It can be seen from FIG. 4 that the light rays 1
reflected from each portion of the spiral arm 2 may be partially blocked
by an adjacent portion of the spiral arm. The percentage of the blocked
area 4 is represented by (G). The concentration ratio (CR) of equation 4
is simply the ratio, expressed in terms of "suns" by which the energy of
the sun is multiplied within the minimum focal width (mw).
Further, the projected arm width is not a parameter which one would
normally be initially aware of in order to produce a reflector having
certain characteristics. The projected arm width can be found from
equation (5) once (mw) and (ff) are found:
##EQU5##
Therefore, the above equations provide a complete description of a
developed spiral given an input of the maximum desired diameter, the
desired focal length, the desired concentration ratio, the desired
estimated blockage and the reflectivity of the reflector. An appropriate
computer program can then find the minimum focal width (mw) for the
outermost point of the spiral from equation (4), the projected arm width x
from equation (5) and the focal length of the inner edge of each point in
the spiral f from equation (3). From these parameters, the developed
spiral can be plotted from equations (1) and (2).
An additional equation which may be useful in plotting the developed spiral
is the equation relating the change of the diameter of the wound up spiral
to the angle .beta.. Such a relationship is expressed by equation (6):
##EQU6##
The above equations describe a spiral Fresnel reflector having a positive
focal length as seen in FIG. 2. FIG. 5 schematicaly illustrates a spiral
Fresnel reflector having a negative focal length while FIG. 6 illustrates
a developed negative focal length spiral which may be plotted on a flat
sheet of reflective material. As can be seen from FIG. 5 which is a
cross-sectional view through a wound up spiral, the focal point of the
reflected light in a negative focal length spiral Fresnel reflector is on
the opposite side of the reflector from the incoming light. Further, the
angles that the spiral arms 2 make with the horizontal plane are much
greater than those of a positive focal length reflector. Further, as can
be seen from FIG. 6, a negative focal length reflector must be wound in a
direction opposite to that of a positive focal length reflector so that
the outer coils of the developed spiral become the inner most coils of the
wound spiral reflector. The developed spiral for the negative focal length
Fresnel spiral reflector can be developed in a method similar to that for
the positive focal length reflector except that equations 1, 2 and 6 are
respectively replaced by equations 7, 8 and 9 below:
##EQU7##
Although the spiral Fresnel reflector will typically be used to concentrate
the direct rays of the sun, and the above equations provide a spiral
reflector for so concentrating parallel rays, the spiral Fresnel reflector
can also be used as a secondary reflector for concentrating converging
rays, as for example the rays reflected from a field of mirrors. FIG. 7 is
a schematic representation of a spiral Fresnel reflector 2 located between
a field of reflecting mirrors 6 and the apparent focal point F of the
field of mirrors. The Fresnel spiral reflector 2 is located at a height H
from the field of mirrors and the apparent focal point F of the field of
mirrors is at a height F from the field of mirrors 6. A spiral Fresnel
reflector which will concentrate the converging rays at desired focal
point DFP can be formed from a developed spiral plotted from the above
equation (2) where f is found from the following equations (10) and (11):
##EQU8##
Therefore, using equations (10) and (11), one need only preselect the
desired heights H and F as well as the other desired parameters such as
the maximum desired diameter, focal length, concentration ratio, estimated
blockage and the reflectivity of the reflector in order to plot a
developed spiral which may be wound up to form a spiral Fresnel reflector
usable with converging light rays.
An alternative method of plotting the developed spiral for the spiral
Fresnel reflector is by use of the apparatus shown in FIG. 8. FIG. 8
illustrates a schematic representation of a mechanical device for plotting
a developed spiral. A piece of flat reflective material 10, or a stencil,
is position for rotation about axis 12. The axis 12 is supported for
rotation on rigid guide 14 and includes a pinion 16. The pinion 16 meshes
with a rack 18 connected to cam 20 which guides pin 22 against the
reaction force of spring 24. Therefore, when material 10 is rotated in the
direction 26, or when cam 20 is moved in the direction 28, the rotation of
the material and the movement of the pin 22 due to the cam 20, cause the
pin to describe the spiral 30 on the material 10. The shape of the cam 20
can be predetermined based upon the equations. This method for developing
the spiral is useful when a large number of identical spirals are to be
produced.
Once the developed spiral is plotted, the developed spiral may be
transferred to a sheet of flat reflective material such as aluminized
flexible plastic or aluminized mylar bonded to low molecular weight
polyethylene or any other reflective sheet material. The developed spiral
can also be plotted directly onto the reflective material. Once the
developed spiral is plotted on the reflective material, the reflective
sheet is cut along the spiral line.
Other material which may be usable for the spiral Fresnel reflector are
masonite (such as 1/8" thick masonite) having aluminum foil glued to one
side, thin aluminum sheet and cardboard having an aluminum foil reflective
surface.
Once the developed spiral has been cut, it is necessary to wind up the
developed spiral in order to result in the spiral reflector. In the case
of a positive focal length Fresnel spiral reflector, the outermost portion
of the spiral arm of the developed spiral is placed at a fixed distance
from the center of the spiral and the center of the spiral is wound up. In
the case of a negative focal length spiral Fresnel reflector, the
innermost end of the developed spiral is placed at a fixed distance from
the center of the spiral reflector and the outermost end of the developed
spiral, which is the innermost end of the spiral Fresnel reflector, is
wound up. The degree of winding determines the angle of inclination of the
arms of the spiral, and therefore determines the focal length of the
resulting reflector. It is therefore possible to make minor changes in the
focal length by performing minor adjustments upon the degree of winding up
the arms of the spiral reflector.
Another version of the apparatus for plotting the developed spiral of FIG.
8 may be seen in FIG. 9. This device is useful for forming large
reflectors of varying diameters. The blank sheet 10 is fixed for rotation
on axis 32 which includes bevel gear 34. Bevel gear 34 meshes with bevel
gear 36 which is mounted on splined shaft 38 for axial movement only.
Bevel gear 40 is mounted on the other end of splined shaft 38 and meshes
with bevel gear 42 which is mounted on axis 44 together with pinion 46.
The axis 44 is fixed by guide pin 48. Pinion 46 meshes with rack 50
attached to cam 52. Cam follower 54 of linkage 56 is guided by cam 52
against the reaction force of spring 58 as the cam 52 is moved. This
results in the movement of pin 60 which describes spiral 62 in a manner
similar to the device of FIG. 8. The pinion gears 36 and 40 can move along
splined shaft 38 so as to accommodate spirals of different sizes.
Once the developed spiral is wound into the resulting spiral Fresnel
reflector, it is necessary to stabilize the arms and maintain them in
their proper position with the desired amount of winding. Preferably, this
may be done by the use of radial connecting arms which radiate from the
center of the spiral and attach to the arms of the spiral at radial
points. Such radial arms 70 may be seen in FIG. 10. The spiral arms 2 may
be attached to the connecting arms 70 at attachment points 72. Although
four connecting arms are shown in FIG. 10, any number can be used.
According to a preferred method, the attachment points may be plotted on
the developed spiral and the Fresnel spiral reflector may be wound up from
the developed spiral simply by the attachment of the attachment points 72
to their appropriate connecting arms 70. The plotting of the appropriate
attachment points on the developed spiral may be done by determining the
number of connecting arms to be used, calculating the angle .beta. between
the connecting arms, and utilizing equation 2 or 8 to calculate the angle
.psi. on the developed spiral for each connecting point.
The connection of the spiral arms 2 to the connecting rods 70 at the
connecting points 72 must be done in such a way that the spiral arms are
not distorted at the connection points. According to a preferred
embodiment, this may be done by creating two holes 74 in the innermost
portion of the spiral arms at each connecting point. A U shaped length of
aluminum or stainless steel wire may then be slid through the holes so
that the base of the U connects the two holes as shown in FIG. 11. The
arms of the U which extend past the bottom of the connecting rod may then
be bent inwards as shown at 76 in FIG. 11. This provides a pivot for the
spiral arm to pivot about its innermost edge.
FIG. 10 shows the connecting rods as being formed of a single piece of
material. Alternatively, the connecting rods may be fixedly attached to a
separate center piece 80 at 82 as shown in FIG. 12. The center piece may
include a central bore 84 in which may be positioned a dowel plug 86 and
turning handle 88. The end of the developed spiral may be placed in the
dowel plug 86 and the dowel plug 86 turned for winding up the developed
spiral into the spiral Fresnel reflector. Using such a technique, one end
of the developed spiral is fixed a predetermined diameter from the dowel
plug while the other end is inserted into the dowel plug and wound until
the fixing points 72 are aligned with their appropriate connecting rods
70.
The center piece need not include a central bore and dowel. In such a case,
the spiral may be wound up by first attaching the outermost point of the
spiral to one of the connecting rods. The spiral is then wound at its next
innermost strip until its fixing point falls into line with the connecting
rod and this fixing point is then fixed to the connecting rod. The winding
is continued and succeedingly inner arms of the spiral are attached to the
connecting rod along a radial line reaching the center of the spiral, and
then outwards to the opposite edge. Once a full diameter of the spiral has
been fixed to the connecting rods, the other fixing points are mounted.
The frame for stabilizing the wound spiral need not be in the form of
radial connecting arms but, rather, may be in the form of a frame defining
a grid. Such a grid would be easier to manufacture and have greater
strength the the radial arm frame arrangement.
One form of such a grid is shown in FIG. 14. This is a grid compose of
equilateral triangles. The arms of some of the equilateral triangles form
intersecting rods 90 which could be the primary mounting points for the
spiral. The spiral could also be attached at other mounting points where
it crosses the grid.
A second form of grid is illustrated in FIG. 15. This is a grid composed of
a plurality of squares. The grid has a center point 100 from which radial
rods 102, which are defined by the arms of some of the squares, extend.
The spiral could be mounted at fixing points 104 to these radial arms, as
well as to other points on the grid where the edge of the spiral crosses
the grid. Other grid shapes are, of course, also possible.
EXAMPLE
It was desired to construct a spiral Fresnel reflector having a radius of
22.5", a focal length of 42.75", a maximum theoretical concentration ratio
of 1000 and 8 mounting arms. These parameters were introduced into
equations 1-6 which provided an arm width of 1.078", a flat radius of
23.175" and a minimum focal width of 1.21". A computer was used to plot
the above developed spiral as shown in FIG. 13. The spiral was cut along
its spiral line and holes for the mounting wire were cut into the spiral.
The connecting arms were then constructed which were made from square
tubing connected at a central point to a plywood center piece by square
nuts and washers. The spiral arms were then connected to the connecting
rods by the wires 76 as described above. The resulting apparatus was
tested at 5:00 in the afternoon on a sunny day and a concentration ratio
of 500 suns was measured.
The wound spiral need not be in the form of a flat plane. The wound Fresnel
reflector can also be in the form of a hollow cone or a hollow truncated
cone. This may be done by providing a frame in the form of a hollow cone
or the frustum of a cone. As with the flat spiral Fresnel reflector, the
spiral could be cut from a single sheet of flat material and wound up onto
the conical frame. The angle of inclination of any point on the wound
conic spiral would be set so as to reflect light through a chosen focal
point f.
There are at least four different types of conical spiral Fresnel
reflectors possible. A first type of conical reflector has an entirely
conic support frame having a frame angle .tau. which is greater than the
angle of inclination of the spiral arm at a given radial point. This is
referred to as a type (A) reflector. Such a type (A) reflector is shown in
FIG. 17. As seen in FIG. 17, which is a cross-sectional view through the
reflector, the angle of inclination of the arms at any point is less than
the angle of inclination .tau. of the frame up to the transitional point
T. Therefore, the arms are fixed to the frame at their outermost radial
point. If the reflector were to continue radially outwards beyond the
transitional point T, the angle of inclination of the arms would be
greater than the angle .tau.. This portion of the reflector would be a
type (B) reflector.
FIG. 16 shows a conical spiral Fresnel reflector which is a combined type
(A) and type (B) reflector. As can be seen in FIG. 16, the reflector
extends radially outward beyond the transitional point T where the angle
of inclination of the spiral arms is greater than the angle .tau.. In such
a type (B) portion of the reflector, the arms are fixed to the frame at
their radially innermost edges.
It should be noted that in a pure type (A) conical reflector, the
transitional point T need never be reached. That is, the reflector may
terminate radially at a point short of the point where the angle of
inclination of the reflector arms equals the angle .tau.. In the case
where the outermost arm of the spiral does have an angle of inclination
which equals the angle .tau., so that the transitional point is reached,
the outermost arm may be fixed flat to the support arm. Such an
arrangement is shown in FIG. 18. In this Figure, the frame is in the form
of a plurality of radially extending rods 112 having a quadrilateral cross
section. The arms of the spiral are secured at fixing points 114 located
at the outermost edges of the spiral arms. As can be seen at 116, the
outermost arm of the spiral, which is at the transition point, is secured
at both it | | |
