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Claims  |
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Having described my invention, I claim:
1. A radiant energy concentrator comprising: a prismatic lens having a
plurality of prisms arranged to direct incoming light to a common focal
area, each of the prisms having a front face, a back face, and a bottom
face which joins the front and back faces, each prism further having the
front and back faces of the prism oriented such that the angle of
incidence of the light passing through the front face is equal to the
angle of incidence of the light passing through the back face; an energy
receiving means spaced from said prismatic lens, said energy receiving
means coinciding with the focal area of said prismatic lens; and support
means for supporting said prismatic lens, and energy receiving means.
2. The combination called for in claim 1 wherein the energy receiving means
comprises a tube, said tube having a passage formed therethrough such that
a fluid may flow therethrough.
3. The combination called for in claim 2 with the addition of: a
transparent tube concentrically arranged about the tube and spaced
therefrom.
4. The combination called for in claim 3 wherein a vacuum is formed between
the transparent tube and the tube to insulate the tube.
5. The combination called for in claim 1 wherein the energy receiving means
comprises a photovoltaic cell; and means to cool said cell.
6. The combination called for in claim 1 wherein the prisms of the lens are
arranged along a curved surface, each prism being arranged so as not to
obstruct light passing through the adjacent prism.
7. The combination called for in claim 1 wherein the bottom face of each
prism over extends outwardly between a position beyond the path of the
rays of light passing within the prism to a position less than the path of
the rays of light passing out of the adjacent prism.
8. The combination called for in claim 1 wherein the prisms are arranged to
focus the light along the longitudinal length of the energy receiving
means.
9. The combination called for in claim 1 wherein the prisms are arranged to
focus the light on a single spot on the energy receiving means.
10. A solar energy collector comprising a support means having an opening
formed in the upper end thereof, said support means adapted to support the
collector; an energy receiving means positioned in the lower portion of
the interior of said support means; a curved prismatic lens spaced from
said energy receiving means, said lens being secured to the upper end of
said support means, said lens comprising a plurality of prisms arranged to
direct light to a common focal area, each prism having a front face which
forms the outer surface of the lens, a back face and a bottom face which
form the interior surface of the lens, the improvement comprising: the
front and back faces of each prism being oriented such that the angle of
incidence of the light passing through the front face is equal to the
angle of incidence of light passing through the back face.
11. The combination called for in claim 10 wherein the prisms of the lens
are arranged along a curved surface, each prism being arranged so as not
to obstruct light passing through the adjacent prism.
12. The combination called for in claim 10 wherein the bottom face of each
prism extends outwardly between a position beyond the path of rays of
light passing within the prism to a position less than the path of the
rays of light passing out of the adjacent prism.
13. The combination called for in claim 10 wherein the support means and
lens are linear and adapted to focus light along a longitudinal area
parallel to the longitudinal axis of the lens.
14. The combination called for in claim 10 wherein the lens is spherical
and the support means is rounded to focus light in a common spot on the
energy receiving means.
15. A Fresnel prismatic lens comprising: a plurality of prisms arranged to
direct light to a common area, each of the prisms having a front face, a
back face, and a bottom face which joins the front and back faces, each
prism further having the front and back faces of the prism oriented such
that the angle of incidence of the light passing through the front face is
equal to the angle of incidence of the light passing through the back
face.
16. The combination called for in claim 15 wherein the prisms are arranged
along a curved surface, each prism being arranged so as not to obstruct
light passing through the adjacent prism.
17. The combination called for in claim 15 wherein the bottom face of each
prism over extends outwardly between a position beyond the path of the
rays of light passing within the prism to a position less than the path of
the rays of light passing out of the adjacent prism.
18. The combination called for in claim 16 wherein the bottom face of each
prism over extends outwardly between the position beyond the path of the
rays of light passing within the prism to a position less than the path of
the rays of the light passing out of the adjacent prism.
19. The combination called for in claim 18 wherein the bottom face of the
prism is oriented such that the angle between the lens optical axis and
the bottom face of the prism is an angle at least greater than one-half
the turning angle of the light passing through the prism but less than the
full turning angle of the light passing therethrough.
20. The combination called for in claim 15 with the addition of: a support
means adapted to support said prismatic lens; and an energy receiving
means positioned by said support means and coinciding with the focal point
of the prismatic lens such that sunlight is concentrated on the energy
receiving means.
21. The combination called for in claim 15 wherein the prisms are arranged
to direct light to a common area comprising a line disposed parallel to
the longitudinal axis of the lens.
22. The combination called for in claim 15 wherein the prisms are arranged
to direct light to a common area comprising a concentrated spot.
23. The combination called for in claim 15 wherein the focal length of the
lens is less than the width of the lens such that the F-number of the lens
is less than 1.0.
24. A curved Fresnel prismatic lens comprising: a plurality of prisms
arranged along a curved surface to direct incoming light to a common focal
point, each of the prisms having a front face, a back face, and a bottom
face which joins the front and back faces, each prism further having the
front and back faces oriented such that the angle of incidence of the
light passing through the front face is equal to the angle of incidence of
the light passing through the back face, and wherein the bottom face of
each prism extends outwardly from a position beyond the path of the rays
of light passing within the prism to a position less than the path of the
rays of light passing out of the adjacent prism so as not to obstruct any
light passing through the prisms.
25. The combination called for in claim 24 wherein the bottom face of the
prism is oriented such that the angle between the lens optical axis and
the bottom face of the prism is greater than one-half the turning angle of
the light passing through the prism but less than the turning angle of the
light.
26. The combination called for in claim 2 wherein the prisms are arranged
to direct light to a common area comprising a line parallel to the
longitudinal axis of the lens.
27. The combination called for in claim 16 wherein the prisms are arranged
to direct light to a common area comprising a concentrated spot.
28. The combination called for in claim 17 wherein the prisms are arranged
to focus along a line.
29. The combination called for in claim 17 wherein the prisms are arranged
to focus on a spot.
30. The combination called for in claim 15 wherein the lens is linear.
31. The combination called for in claim 15 wherein the lens is spherical.
32. The combination called for in claim 15 with the addition of: a support
means; and energy receiving means spaced from said lens by said support
means such that the light passing through the lens is focused on the
energy receiving means.
33. A method of concentrating energy from light rays emitted from a source
of energy comprising the steps of: providing a plurality of prisms having
a front and back face joined by a bottom face; refracting the light
through the front and back faces of the prisms to a common focal area such
that the angles of incidence with the rays refracted through the prisms is
equal with respect to the front and back faces of the prism.
34. The method called for in claim 33 with the additional steps of:
arranging the prisms along the curved surface such that each prism does
not obstruct light passing through the adjacent prism.
35. The combination called for in claim 33 with the additional steps of:
over-extending the bottom face of each prism beyond the path of the rays
passing therewithin.
36. The combination called for in claim 1 with the addition of: means
pivotally securing said support means such that the longitudinal axis of
said support means is aligned with a polar axis; means to move said
support means for diurnal tracking of the sun; means to move the energy
receiving means relative to the lens and align said energy receiving means
at the sharpest focus of the lens as the sun changes angles of declination
relative to the lens.
37. A radiant energy concentrator comprising:
energy receiving means disposed along a receiving axis;
a prismatic lens means comprised of a plurality of juxtaposed prisms
arranged to form a generally curvilinear outer surface to direct incoming
light to said energy receiving means; and
means for supporting said prismatic lens means relative to said energy
receiving means such that said prismatic lens means is disposed from and
curved about said energy receiving means, said prismatic lens means being
formed in such a manner that the radial distance from the receiving axis
to any one of said prisms varies in accordance with the relative position
of said one prism within the said prismatic lens means.
38. The energy concentrator as set forth in claim 37 wherein said prismatic
lens means is symmetrical relative to a central plane passing through the
receiving axis, said radial distance being a function of the angle the
central plane makes with a plane passing through the receiving axis and
said one of said prisms.
39. The energy concentrator as set forth in claim 38 wherein said radial
distance to any one of said prisms decreases as the distance from said one
prism to the central plane increases.
40. The energy concentrator as set forth in claim 39 wherein said prism
includes a front face, a back face, and a bottom face joining the front
and back faces, each prism being formed such that a light ray impinging
upon said front face at a given incidence angle will pass through said
prism and emerge at substantially the same given angle relative to said
back face. |
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Claims |
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Description |
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BACKGROUND
Heretofore, Fresnel lenses have been of two types, a flat type and a curved
convex type. A three-dimensional or spherical Fresnel lens is designed to
focus on a point, and a two-dimensional or cylindrical lens is designed to
focus on a line.
Previously, Fresnel lenses have been primarily designed for use with a
point source of light to create a wide collimated beam of light such as
those used in a lighthouse or studio stagelight.
An example of the beam focusing lens is disclosed in the patent to
Pascucci, U.S. Pat. No. 1,504,970, which discloses a Fresnel lens having
on one face concentric zones united by miters, the miters being cut
whereby they are parallel to the path of the rays passing through the
lens. The juncture of the miters between the bottom face and the rear face
of the prism is assumed to be a perfectly pointed configuration. However,
in the manufacture of these types of lenses out of material such as glass
or acrylic plastics the surface tension of the material placed in the mold
at the juncture of these faces will cause the point to be rounded. If the
mitered face is constructed, as taught by Pascucci, and aligned parallel
to the path of the rays, and if the lens is used to focus sunlight, the
rounded point will cause divergence of the light rays along that portion
away from the focal point as the light passes through the prism. In
addition, errors occur in manufacturing the prisms and in aiming of the
lens toward the sun. These errors will cause a portion of the light to
intercept the bottom face and thus refract away from the focal point. The
loss due to light striking the rounded corner and bottom face could be
anywhere from 20% when the lens is glass to 5 to 10% when the lens is an
extruded plastic type lens.
Fresnel lenses have been used in solar collectors as disclosed by the
patent to Sleeper, U.S. Pat. No. 3,125,091, which discloses an inflatable
cylindrical type lens having prisms formed therein.
Prisms in the Sleeper patent are superimposed on a circular arc which
imposes restrictions on the optical efficiency of this type of lens. The
light passing through the edges of this type of circular lens is so
severely deflected that it is completely lost. In addition, the Sleeper
collector teaches a flexible type solar collector which is incapable of
reaching high temperatures and withstanding exterior weather conditions
for 20 years or more.
Heretofore, most prior art Fresnel lenses had longer focal lengths with
F-numbers of 1.0 or greater. This required more material, more insulation
and a greater volumne of space to construct the lens and solar collector
to collect the proper amount of heat. Furthermore, much of the material
content (and thus the cost) of the collector is due to the housing and
structural support system, which can be greatly reduced by reducing the
lens focal length to make a more compact collector unit. Unfortunately,
conventional Fresnel-type lenses suffer great losses in transmittance as
the focal length is reduced. This is the reason that prior-art Fresnel
lens collectors have utilized long focal length lenses with F-Numbers
(focal length divided by lens width) of 1.0 or greater.
It should be readily apparent that solar collectors must be highly
efficient in order to utilize the full potential of converting the sun's
energy into a useful form of energy. The loss of 10 or 20% of the energy
transmitted through the lens is often times critical as to whether the
system may be used for merely heating purposes or for conversion of light
into energy useful for air conditioning, generation of electricity, or
other processes. In the past, the flat plate collectors have been used due
to the simplicity of construction. However, flat plate collectors need a
large area and a large heat absorber, and have very low collection
efficiencies. Heat absorbers are often constructed from stainless steel or
copper to minimize corrosion and are therefore extremely expensive to
manufacture.
Fresnel lenses in use now have a high loss of light transmittance through
the lens due primarily to reflections at the prism surfaces. This high
reflection loss causes a large decrease in collection efficiency.
Therefore, it is highly desirable to produce a solar concentrator which has
the highest transmittance and the shortest possible focal length. Such a
concentrator will achieve the highest collection efficiency at minimum
cost. The new concentrator described below has these beneficial
characteristics.
SUMMARY
I have devised a new curved prismatic Fresnel-type lens for use in a solar
energy collector. Generally, the collector comprises a housing of
insulating material having an energy receiver positioned at the bottom of
the housing and the prismatic lens secured across the open top of the
housing to direct light toward the energy receiver which may be a heat
absorber or other device such as a photovoltaic cell.
The lens comprises a substantially smooth, convex outer surface having a
plurality of prisms arranged side by side along the curved inner surface
of the lens to direct the parallel incoming sunlight to a common area or
target coinciding with the energy absorber.
Each of the individual prisms has a front and back face joined by a bottom
face. The front and back faces of the prisms are oriented on each prism
such that the angle of incidence of the incoming light on the front face
of the prism is equal to the angle of incidence of the outgoing light on
the back face of the prism. The equal angles of incidence minimize the
reflection loss of the light components and therefore affords the highest
transmittance of light through each of the prisms and therefore through
the lens. Each prism is further arranged so as not to block the light
passing through the adjacent prism, and the bottom face of each prism is
over extended to a position beyond the path of light passing within the
prism to a position short of the path of the light passing out of adjacent
prisms to compensate for aiming errors and rounded points between the
bottom and back faces of the prism.
The lens has a higher light transmittance than any other lens with the same
F-number and material composition. The lens can be designed with an
F-number (focal length divided by width) of significantly less than 1.0
such that the overall depth of the units is thus minimized to reduce the
materials needed to build the collectors. The shorter focal length of the
lens reduces the effect of errors in aiming. The lens is capable of use in
a line-focus collector to produce temperatures of up to 500.degree. F, and
can be designed in a point-focus collector to produce temperatures up to
1,000.degree. F.
The primary object of the invention is to produce a prismatic lens for use
in solar collectors which minimizes reflective losses as the light passes
through the individual prisms of the lens thereby maximizing transmittance
of light therethrough.
A further object of the invention is to provide a prismatic type lens
having a high efficiency and which has a shorter focal length capable of
having F-numbers less than 1.0.
A still further object of the invention is to produce a collector which is
capable of producing the maximum amount of energy for a minimal amount of
space and which is lightweight for use on roofs without excessively
loading the support structure and which is capable of producing a highly
efficient energy collection device.
A still further object of the invention is to produce a lens having
individual prisms which are over extended beyond the path of light within
said prisms to reduce the effect of errors in manufacturing the lens and
in aiming the lens toward the sun, and to prevent light losses due to
blockage by the rounded point between the back and the bottom face of the
prisms.
A still further object of the invention is to provide an energy collector
which maximizes energy concentration along an energy receiver by a lens
constructed of prisms having front and back surfaces which have equal
angles of incidence with the light passing therethrough.
Other and further objects of the invention will become apparent upon
referring to the detailed description hereinafter following and by
referring to the drawings annexed hereto.
DESCRIPTION OF DRAWINGS
Drawings of preferred embodiments of the invention are annexed hereto so
that the invention may be better and more fully understood, in which:
FIG. 1 is a perspective view with parts broken away to more clearly
illustrate the details of construction of the collector;
FIG. 2 is a diagrammatic view illustrating a typical solar concentrator;
FIG. 3 is an enlarged cross-sectional view of the energy receiver;
FIG. 4 is an enlarged cross-sectional view of a first modified form of the
energy receiver illustrated in FIG. 3;
FIG. 5 is an enlarged cross-sectional view of a second modified form
thereof;
FIG. 6 is an enlarged cross-sectional view of a third modified form
thereof;
FIG. 7 is an enlarged cross-sectional view of a fourth modified form
thereof;
FIG. 8 is a partial end elevational view of the lens disconnected from the
collector;
FIG. 9 is an enlarged end view of the left outermost prism of the lens
illustrated in FIG. 8;
FIG. 10 is a graph diagrammatically illustrating an end view of the lens
constructed by polar coordinates;
FIG. 11 is an elevational view of a modified form of the collector with
parts broken away to more clearly illustrate the details of construction;
FIG. 12 is a cross-sectional view taken along line 12--12 of FIG. 11;
FIG. 13 is a graph illustrating the overall transmittance of light through
the lens for various F-numbers of the new lens; and
FIG. 14 is a graph illustrating the concentration of light in sun's versus
the spread angle of the light for the new lens with an F-number of 0.5,
for three different solar collectors.
DESCRIPTION OF A PREFERRED EMBODIMENT
Referring to FIGS. 1 and 2 of the drawings, the collector 20 is utilized to
concentrate light energy by use of a prismatic linear lens 22. The linear
lens 22 directs the incoming sunlight to a common focal area 76 generally
lying in the focal plane of the linear lens 22. It is general practice to
use an energy receiving means which coincides with the common focal area
of linear lens 22 to convert light energy to heat for various uses. In a
solar collector the energy receiving means is generally an absorber such
as the one illustrated in FIGS. 1 and 3 comprising a hollow elongated tube
24 having fins 26 diametrically secured thereto which may be coated with
dark coating such as chrome black to more efficiently absorb sunlight.
Tube 24 has a passage 28 through which a heat exchange medium flows to
absorb heat.
The tube 24 may be resting on spacers 30 which space the tube 24 off of the
interior housing. The housing comprises a generally truncated V-shaped
inner housing I having sidewalls 36 and 38 which are joined by an inner
bottom 40 which are spaced from outer housing walls 42 and 44 and bottom
46. A layer of thermal insulation 48 is positioned between the inner
housing I and outer housing O to prevent transfer of heat away from the
energy receiving means. The insulation 48 may be fiberglass or some other
suitable type. A clear, concaved transparent cover 50 is positioned over
the upper side of tube 24 in longitudinal passages 52 formed in the lower
end of sidewalls 36 and 38 to decrease heat loss from the receiver tube
24.
A C-shaped channel 54 is formed at the upper end of the outer housing
sidewall 42 and 44 and joined with the inner housing sidewalls 36 and 38,
respectively, to connect the tops together. The channel 54 runs
longitudinally along the sidewalls of the collector 20. A resilient gasket
56 having a slot 58 for receiving the edges of lens 22 is disposed
longitudinally in C-shaped channel 54. The gasket 56 provides a weather
tight seal to protect the collector 20 interior from rain, dust and
corrosion. End panels 60 are positioned across each end of the inner
housing sidewalls 36 and 38 and outer housing sidewalls 44 and 42 to seal
and insulate the collector 20. Tube 24.sub.1 passes outwardly from the
lower end of end panels 60. Tube 24.sub.1 can be a flexible hose to allow
the collector 20 to rotate with sprocket 98 to track the sun.
As best illustrated in FIG. 2, the connection 62 on the end of tube
24.sub.2 is attached to pipe 64 which is connected to the outlet of pump
66. The pipe 68 is connected between the inlet of pump 66 and the outlet
of heat exchanger 70 having an inlet. Pipe 72 is connected between the
inlet of exchanger 70 and flexible tube 24.sub.1. A heat exchange medium
is pumped through pipe 64 and 68 and through tube 24 of collector 20 for
heating the heat exchanger 70. Heat exchanger 70 has an inlet pipe 74 and
an outlet pipe 76 through which another heat exchange medium is passed for
connection to apparatus to use the solar heat collected for air
conditioning or other applications. It should be readily apparent that
various types of arrangements for using the heated medium are possible and
are well known in the art and the embodiment shown in FIG. 2 is merely an
example of such an arangement. A typical heat exchange medium is water.
The axis of rotation of the collector will usually be parallel to the
Earth's polar axis to facilitate tracking.
As best illustrated in FIGS. 1, 8, and 9, the prismatic linear lens 22
comprises a curved outer surface formed by the front face 74 of a
plurality of prisms 78. Prisms 78 are arranged as shown in FIG. 8 from the
outer most prism 78.sub.1 to the innermost prism 78.sub.11. The prisms 78
are arranged to focus the light to a common focal area 76 which
corresponds to the intersection of the X-Y axes. The inner surface of the
lens 22 is defined by the remaining surfaces of the plurality of prisms
78.sub.1 -78.sub.11 each of which has the front face 74 corresponding to
the outer surface of the lens, a back face 80 and a bottom face 82 to form
a linear prism element 78 which has a substantially truncated triangular
cross-section as best illustrated in FIG. 9. As illustrated in FIG. 9, the
prism base line 84 shown in dashed outline is parallel to the front face
74 of lens 22 and has an equal length thereto. It should be readily
apparent that the curved outer surface of the lens may be comprised of
short, straight sections of front face 74 which give the appearance of the
curved surface. The length of surfaces 74 is ideally minimized to be as
short as practical and may in reality be actually curved in the
construction of lens.
The front face 74 is positioned at an angle B.sub.1 with the incoming
parallel rays 86 and 88 of sunlight diagrammatically illustrated by dashed
lines in FIGS. 8 and 9. Each prism 78 is oriented such that the incidence
angle B.sub.1 with the incoming light relative to the front face 74 is
equal to the angle of incidence B.sub.2 formed by the outgoing sunlight
relative to the back face 80 of the prism 78 such that eacn prism 78 is
oriented along the inner surface of the lens 22 to refract the light at a
specified turning angle A.sub.F, this angle being an angle between the
axis 23 of lens 22 and the ray of light 88, to orient the light passing
through each of the prisms 78.sub.1 - 78.sub.11 to a common focal area 76
which preferably coincides with the upper surface of the energy receiving
means. The equal angles of incidence B.sub.1 and B.sub.2 of the sunlight
with the front face 74 and back face 80 as it passes therethrough each
prism 78 provide for a maximum light transmittance and therefore maximum
efficiency of the lens 22 and collector 20.
The second prism 78.sub.2 has a turning angle A'.sub.F which is slightly
less than A.sub.F. The angles of incidence B.sub.1 ' and B.sub.2 ' with
front face 74 and back face 80 are equal but the angles have a different
value from the angles of incidence B.sub.1 and B.sub.2. It should be
readily apparent that each set of angles change for each prism 78.sub.1 -
78.sub.11 in order to maximize the transmittance of light therethrough.
In the center of lens 22 where the prisms become small it is preferable to
provide a double convex linear lens 90 which is oriented perpendicular to
the axis 23 of lens 22. A substantially rectangular edge 92 is provided
adjacent the outermost prisms 78.sub.1 which fits into slot 58 of gasket
56.
Lens 22 is symmetrical about the axis 23 and therefore the second half of
lens 22 is a mirror image of the first half shown in FIG. 8.
As best illustrated in FIG. 9, the tip 94 of each prism 78 of lens 22 is
slightly rounded due to the fact that the lens is molded of transparent
material such as glass or acrylic in which the surface tension of the
material as it cures causes tip 94 to round off where it is joining
surfaces 80 and 82. Therefore the tip 94 will not be perfectly pointed as
illustrated by the dashed outline for construction purposes only.
As will be more fully explained hereinafter, the design of this lens allows
for the maximum transmittance T of energy through the lens while allowing
focal lengths of the lens 22 to be designed having an F-number in the
range of slightly negative to positive infinity. The lens 22 has a design
such that an F-number of 1.0 or less may be produced while maintaining
transmittance levels of light about 90%. F-number is herein defined as
focal length divided by lens width, for example, x.sub.1 /2y.sub.1, for
the lens of FIG. 8, wherein the focal length is the distance between the
focal point and an imaginary chord passing from one edge of the lens to
the other, such chord passing perpendicularly through the axis 23.
It should be readily apparent that in actual usage the prisms 78 would be
much smaller and more numerous than shown in FIG. 8, to conserve material
and to maintain a light weight lens.
In order to prove that maximum transmittance occurs when angle B.sub.1 =
angle B.sub.2 relative to the respective front face 74 and back face 80, I
have formulated the following proof:
Consider the refraction phenomenon for an individual prism, wherein:
B.sub.1 = the angle of incidence at the first prism face,
B.sub.2 = the angle of incidence at the second prism face,
A.sub.f = the total turning angle of the light ray,
n = the prism material index of refraction.
From Snell's law, one can show that: (1) A.sub.F = B.sub.1 - sin.sup.-1
(sinB.sub.1 /n) + B.sub.2 - sin.sup.-1 (sinB.sub.2 /n). For a thin Fresnel
lens made of non-absorbing material, such as acrylic plastic, light
absorption losses will be small compared to light reflection losses.
Furthermore, for non-polarized light such as solar radiation, one can show
from basic electromagnetic theory that the single-surface reflectances for
the prism are governed by the following equations:
##EQU1##
wherein P.sub.1 = the reflectance at the first prism face, and wherein
P.sub.2 = the reflectance at the second prism face. The overall prism
transmittance (T) is: (4) T = (1-P.sub.1) (1-P.sub.2), assuming no light
absorption within the prism. When T is maximized, its differential should
vanish:
##EQU2##
For a fixed, desired value of A.sub.F, equation (1) can be differentiated
to yield:
##EQU3##
Now, when B.sub.1 = B.sub.2, equation (6) becomes: (7) dB.sub.1 =-
dB.sub.2. Furthermore, from equations (2) and (3), when B.sub.1 = B.sub.2,
P.sub.1 = P.sub.2, and
##EQU4##
therefore, equation (5) becomes: (8) dT = 0. Thus, when B.sub.1 = B.sub.2,
an extremum of transmittance T exists. To ensure that this extremum is a
maximum, one performs the second differentiation and makes the appropriate
substitutions for B.sub.1 = B.sub.2, dB.sub.1 =-dB.sub.2, P.sub.1 =
P.sub.2, dp.sub.1 /dB.sub.1 = dP.sub.2 /dB.sub.2, d.sup.2 P.sub.1
/dB.sub.1.sup.2 = d.sup.2 P.sub.2 /dB.sub.2.sup.2 to obtain: (9) d.sup.2 T
= -2(dP.sub.1 /dB.sub.1).sup.2 (dB.sub.1).sup.2 -2(1 -P.sub.1) (d.sup.2
P.sub.1 /dB.sub.1.sup.2) (dB.sub.1).sup.2. Therefore, T is maximized if
and only if: (10) (dP.sub.1 /dB.sub.1).sup.2 + (1 -P.sub.1) (d.sup.2
P.sub.1 /dB.sub.1.sup.2) > 0.
Since the curve defined by equation (2) is concave upward for every B.sub.1
value, equation (10) prevails and therefore each prism 78 should be
constructed such that the angle of incidence B.sub.1 with the solar ray at
the front face 74 equals the angle of incidence B.sub.2 with the solar ray
at the back face 80 to achieve the highest transmittance T allowed by the
laws of physical optics. This is the basic principle for defining the
individual prisms in the new lens.
As shown in FIG. 9, the beam of sunlight for a particular prism is
diagrammatically shown by dashed lines 86 and 88 which define the width of
a single beam which passes through a single prism 78. As best illustrated
in FIG. 9, it can be seen that the bottom face 82 of prism 78 is
preferably extended beyond the path of the light within prism 78 defined
by line 88 but short of the path of the light passing out of the adjacent
prism 78.sub.2 defined by line 86'. It should be readily apparent that no
light passes through the triangular area defined by points
(X.sub.4,Y.sub.4), (X.sub.5, Y.sub.5) and (X.sub.6, Y.sub.6) (FIG. 9) and
therefore no light passes through the bottom face 82 or through the tip 94
formed between faces 80 and 82.
Point (X.sub.5, Y.sub.5) defined by the equations as the intersection
between back face 80 and bottom face 82 of the outermost prism 78 is only
a construction point because in practice the intersection will form a
rounded tip 94.
Tip 94 is rounded because the molded material usually used to form lens 22
such as glass or acrylic has a surface tension which causes the tip to
round off in the molding process. It is necessary to compensate for this
phenomenon by the over extension of the bottom face 82 of the prism 78 to
prevent any light from intersecting the rounded tip 94 which would cause
undue loss of light transmittance. In addition, the over extension of
bottom face 82 aids in reducing the effect of errors which occur in aiming
the lens and manufacturing same. The angles in FIG. 9 are exaggerated for
clarity, and the drawing is not to scale; thus, rays 86 and 86' would
intersect at the focal plane, although not clearly indicated in FIG. 9.
One method of defining the orientation of the front face 74 and back face
80 and bottom face 82 of prisms 78 which comprise the lens 22 is best
illustrated in FIG. 10. This method utilizes equations in polar
coordinates to define surfaces 74, 80 and 82 in the upper quadrants of a
standard X-Y coordinate system. The focal point 76' of the lens 22 is
positioned at the origin of the X-Y coordinate system.
The front face 74 is defined as follows: Define d = (n.sup.2 + 8).sup.1/2
/2 - n/2. Then
##EQU5##
defines the front face 74 curves, wherein r.sub.o is an arbitrary constant
of integration.
The back face 80 of the prism 78 is defined as follows:
##EQU6##
defines the back surface curves, wherein r.sub.1 is an arbitrary constant
of integration.
The bottom face 82 of prism 78 is preferably three-fourths of the turning
angle A.sub.F which may be expressed as:
##EQU7##
wherein r.sub.2 is an arbitrary constant of integration.
The three formulas indicated above representing the curves defining the
front face 74, back face 80, and bottom face 82, were derived by writing
and subsequently solving the governing differential equations
corresponding to Snell's law of light refraction subject to the
constraints that the incidence angles B.sub.1 and B.sub.2 at the front
face 74 and the back face 80 of each prism 78 are equal and that the
bottom face 82 of the prism is properly over extended. Each of these
differential equations has been derived and solved analytically by the
inventor and represented by the integral solution curves illustrated in
FIG. 10. It should be readily apparent that an infinite number of curves
may be developed depending upon the constants of integration desired.
Thus, prisms of any desired size and location may be defined by the above
formulas. For example, prism 478 represents such a prism. However, it is
desirable that the prisms comprising the lens do not block or shade one
another. Therefore, the prisms should be selected along a common outer
surface curve 74. With this configuration, each prism has maximum
transmittance with no blocking or shading by other prisms. Thus, such a
lens will have a higher transmittance than any other lens of the same
size, focal length and material composition.
Each prism 78'.sub.1 - 78'.sub.19 represents the intersection of these
curves defined by the equations hereinbefore given.
It should be readily apparent that by varying the distances between the
curves which are controlled by the arbritrary constants of integration
r.sub.o and r.sub.1 and r.sub.2 that the size and shape of each prism 78'
may be varied.
In addition in general practice it would be desirable to use only a portion
of the prisms generated by these curves such that a truncated form of the
lens would be formed.
In addition, as illustrated in FIG. 10 a ray of light diagrammatically
illustrated by the dashed line indicated as 86 could pass through either
of the prisms 78'.sub.5 or 478. Turning angle A.sub.F.sbsb.5 is different
from the turning angle A'.sub.F of each deflected ray as it passes through
either of the representative prisms 478 and 78'.sub.5. However, it should
be readily apparent that either prism 478 or 78'.sub.5 bends the ray
through a desired turning angle A.sub.F and each angle of incidence
B.sub.1 with either front face 74 or 74' is equal to the angle of
incidence B.sub.2 with the rear face 80 of either prism 478 and 78'.sub.5.
Utilizing the standard coordinate transformation of X = r cos u and Y = r
sin u, the curve is plotted on the X-Y coordinate system in which the
X-axis is parallel to the incoming light. As illustrated, there are an
infinite number of curves which define faces 74, 80, and 82 for different
integration constants r.sub.o, r.sub.1, and r.sub.2 from the formulas.
Each of the light rays will be turned toward the focal area 76'
corresponding with the intersection of the X-Y axes. This orientation of
each prism 78 results in equal angles of incidence at the front face 74
and the back face 80 thereby ensuring a maximum transmittance for the lens
22.
The bottom face 82 of the prism 78 is best positioned at an angle relative
to the Y-axis equal to 3/4A.sub.F. Although FIG. 10 only illustrates the
upper quandrants, the lower half of the lens 22 is a mirror image of the
upper half, assuming the x axis passes between the upper half and lower
half. Therefore, it is unnecessary to further compute the curves necessary
to define the lens.
As another method of defining the lens 22, the sequence of formulas set out
in Table 6 defines each individual prism 78 in rectangular coordinates.
The sequence of formulas in Table 7 defines a method of constructing an
entire lens 22 as is illustrated in FIG. 8 of the drawings. Each formula
defines a series of points along the cross-section which are connected to
define the lens 22.
Both methods of defining the lens are equally valid for point-focussing
lenses or line-focussing lenses. For line-focussing lenses, the variable Y
used above is a standard rectangular coordinate. For point-focussing
lenses, the variable Y is a radial coordinate in an axisymmetric lens. The
cross sections of the linear and point-focussing lenses will be identical,
and the calculation sequence identical for both.
TABLE 6
1. (x.sub.1, Y.sub.1) is the point chosen to define the desired prism
location.
2. l is set equal to the desired slant hight of the prism.
3. t is set equal to the desired slant base thickness of the prism.
4. a is calculated as the root of the following implicit equation:
##EQU8##
7. b is calculated as: B= a/2 + v
8. n is the index of refraction of the prism material.
These coordinates, distances and angles totally define the prism
configuration, location, and orientation.
It should be appreciated that no light passes through the portion of the
prism forming a triangle with vertices (x.sub.4,y.sub.4),
(x.sub.5,y.sub.5), and (x.sub.6,y.sub.6). Therefore there are no optical
transmittance losses due to light impinging on the bottom face of the
prism or due to light hitting the prism point (x.sub.5,y.sub.5).
TABLE 7
1. Choose a desired focal length for the lens, f.
2. Choose a desired lens width (for line - focus lens) or diameter (for
point-focus lens), w.
3. For outermost prism, set x.sub.1 = f, and set y.sub.1 = w/2.
4. Calculate the outermost prism design for this (x.sub.1,y.sub.1)
according to the procedure of Table 6.
5. After completing this outermost prism design, calculate the next prism
design by using the point (x.sub.4,y.sub.4) in place of (x.sub.1,y.sub.1)
in the calculation sequence of Table 6.
6. After the second prism 78.sub.2 design has been calculated, (FIG. 8)
point (x.sub.10,y.sub.10) will be known. Calculate the third prism design
by using the point (x.sub.10, y.sub.10) in place of (x.sub.1,y.sub.1) in
the calculation sequence of Table 6.
7. Continue calculating the lens design prism by prism until the optical
axis is reached. If desired, a simple bi-convex lens can be used in place
of the last few prisms near the optical axis, as shown in FIG. 8.
8. The lens design below the optical axis is the exact mirror image of the
lens design above the optical axis.
9. Refer to FIG. 9. The convex outer surface of the lens is made up of
straight line segments such as the line from (x.sub.2,y.sub.2) to
(x.sub.3,y.sub.3). However, the performance of the lens is not adversely
affected but in fact improved if a smooth curve is faired through these
straight line segments to provide the smooth, continuous convex outer lens
surface, such as that presented in FIG. 8. In practice, the prisms are so
small that it is nearly impossible to distinguish between the
pieced-together line segments and a continuous curve.
10. This lens design definition is valid for any reasonable focal length
and lens width, and is equally suitable for linear Fresnel lenses and
circular Fresnel lenses.
Utilizing Tables 6 and 7, one calculates each point by going through the
computations in Table 6 and utilizing trigonometric functions to locate
each point of intersection of the lens. As illustrated in FIGS. 8 and 9,
each point of the lens 22 is mathematically constructed up to the X-axis.
As explained heretofore, a double convex lens 90 may be formed in the
center of lens 22 perpendicular to X-axis.
The lens material preferably comprises a material such as methyl
methacrylate, commonly known as acrylic plastic having an index of
refraction of approximately 1.491. This material is available in exterior
grades and is sold under such registered trademarks, as Lucite
manufactured by E. I. DuPont de Nemours of Wilmington, Delaware of
Plexiglas by Rohm and Hass Company of Philadelphia, Pennsylvania and other
companies. Other materials which might be used to construct this lens
would be polystyrene having an index of refraction of 1.590 or
polycarbonate having an index of refraction of 1.586 or methacrylate
styrene copolymer sold under the tradename NAS. These materials may be
extruded from a die using conventional molding methods to produce the lens
in a linear from or the material may be calendered and bent to the desired
curved surface under conventional method of manufacture well known in the
art. The desired widths of lens 22 would be subject to various factors but
may be in a range anywhere from about 1 to 4 feet (30 - 120cm.) and
F-numbers anywhere from about .2 to 1. For the Fresnel lens collector 20
to operate effeciently throughout the daylight hours, it will generally be
ncessary to track the sun's apparent motion across the sky, thereby
keeping the lens always pointing in the general direction of the sun. It
is well known in the art that the sun's apparent motion is characterized
by a diurnal motion from East to West during the day at a rate of about
15.d | | |
