** **

#
robustness of a reflective solar cooker

** **

*li-yan zhu and yun k. kim*

introduction:

in the spring of 2002, we described a parabolic solar
cooker for unattended cooking [1]. it cooks continuously for two hours without
adjustment. this robust design utilizes three techniques: a template for aiming
the reflector ahead of the sun, thus doubling the duration of in-focus
operation [2]; a large and tall cookware absorbent in all directions; and a
reflector optimized for *angular tolerance*.� the third technique will be
explained in this article.

the angular tolerance of an infinitesimally small
reflecting surface is the range of its acceptable orientation.� it
characterizes the robustness of a reflecting surface against manufacturing
tolerance and solar movement. �for two-hour unattended cooking,� �15� angular
tolerance is required throughout the reflector.� this requirement is most
difficult to meet at the rim of a reflector.� we will show how to optimize the
rim location, such as to meet the tolerance requirement, and to maximize the
cooking power.

the optimization is
complicated by asymmetry, which is caused by elevation and movement of the
sun.� the rim location will be optimized first in a cross-section where the pot
is symmetric with respect to the optical axis.�� a parabola is determined
accordingly, and used to generate an axisymmetric reflecting surface.� the
surface is trimmed such that the desired angular tolerance is attained around
the rim.

**angular span and angular
tolerance:**

** **

the robustness of a reflector at an arbitrary reflecting
point a is quantified by the *angular tolerance *(�*b*)
of its normal vector. this tolerance is provided by *angular span* (*g*) of the cookware, with respect to the
reflecting point a (fig. 1).� it is anisotropic (directional), unless the
cookware is spherical.� the direction of interest varies throughout this
article.� it will be specified as needed.

to maximize the angular tolerance, the *nominal
reflection* (*i.e*., reflection under ideal conditions, in the absence
of manufacturing tolerance and misalignment) must bisect the angular span *g*.� under this condition,

g=2*b*.�������������������������������������������������������������������������� (1)

the angular tolerance consists of structural misalignment
(�*b*_{1}), surface
waviness (�*b*_{2}), and
movement of the sun (�*b*_{3}).�
the first two components double due to reflection; the

fig. 1� cross-section of an axi-symmetric reflector

third does not.� assume that all three components of *b* are coplanar.� in order for all
reflection to strike the cookware,

_{
}.�������������������������������������������������� (2)

to maximize *b* (hence *g*), the cookware should be large, close to and facing
the reflecting point.� given *b*, if the
structure is made more precise (*b*_{1} reduced) or if the surface is made smoother (*b*_{2}
reduced), the reflector will be able to accommodate greater solar movement (*b*_{3}
increased).

**locus of equal-span:**

the cookware shown in fig. 1 is
a flat *pan*, or the black bottom of a shiny pot.� for simplicity assume
that the combination of pan and reflector is axisymmetric, and that the sun
moves in the plane of this cross-section.� we shall examine how angular span *g* of the pan (hence angular tolerance *b *of the reflecting surface) varies with
location of the reflecting point **a**.

by elementary geometry, the locus of equal angular span
(abbreviated as *equal-span*) is a pair of intersecting circles sharing
the pan as a common cord (fig. 2a).� let *j *be
the measure of the cord (fig. 2a). �then

_{
}.�������������������������������������������������������������������� (3)

combine equations (1) and (3):

_{
}.��������������������������������������������������������� (4)

fig. 2� a) locus of equal-span and equal-tolerance;
and�� b) contour map of angular tolerance *b*

thus the locus of equal-span is also the locus of equal
angular tolerance (abbreviated as *equal-tolerance*).� a family of such
loci with successively increasing radius exhibits successively decreasing
angular span (*g*) and angular
tolerance (*b*).� it constitutes a
*contour map* of *g* or *b* �(fig.
2b).�

**optimal rim location:**

** **

a locus of equal-tolerance is not a valid reflecting
surface, because its focus falls off the pan.� thus the angular tolerance
varies over a reflector.� angular tolerance at the outer edge (*i.e*., the
*rim*) of the reflector is most important, because the rim has the
greatest radius (hence circumference).� it has most influence on the cooking
power.� angular tolerance also exhibits a local minimum at the rim, which is
most distant from the cookware.� thus the rim location should be optimized
first during the design process.�

the optimization may take one of the two paths.� in the
first case the power, hence radius of the reflector, is specified *a priori*.�
to maximize the angular tolerance, the rim can move only in parallel with the
optical axis.� in the second case angular tolerance is specified at the rim.�
the rim can only move alone the locus of equal-span to maximize its radius,
hence power of the reflector.� in either case the optimal rim location is
farthest from the optical axis, among all points with equal-tolerance (fig.
3a).�� since the locus of equal-tolerance consists of two intersecting circles,
there are two pairs of optimal rim locations (e or e^{*} in fig. 3a).

thus there are at least two distinct reflectors with
identical radius (and power).� angular tolerance is identical on their rims,
but not in their interior surfaces.� therefore these two reflectors do not
perform equally well in unattended cooking.

(a)������������������������������������������������������������������������������������������ (b)

fig. 3� a) the rim of a reflector should be farthest
from the optical axis on any circle of equal-span;

b)� nominal reflection from the rim strikes the
optical axis but not the center of the pan.

** **

recall that nominal
reflection from an arbitrary point on the reflector must bisect the angular
span.� reflection from the rim is of no exception.� by elementary geometry, the
nominal reflection must be directed at the bottom (or top) of the circle of
equal-span.� it does not pass through the center of the pan (fig. 3b).� since
the angular span varies over the reflector, circles of equal-span change their
sizes.� the bottom (or top) of the circle slides along the optical axis.�
therefore a reflector optimized for angular tolerance does not have a
well-defined focal point.� it cannot be a parabola.

angles between the reflecting
surface and the optical axis (*a* or *a*^{*}), also
angles between the reflection and optical axis (*f *or -*f*), will be given
later in a more general case.

** **

**angular span of a pot:**

** **

a shallow *pan* is unsuitable for use in a
concentrating solar cooker, because its angular span is rather low.� in
practice a deep pot darkened over the entire outer surface is

fig. 4� a pot with diagonal angle *q*.� portions of the reflector facing its bottom, wall,

�or diagonal are shown in matching color and pattern

preferred.� the depth of a pot is characterized by angle *q *between its diagonal and its bottom
surface (fig. 4).� a pan may be considered as a special pot (_{}).

depending on location of reflecting point, angular span of
the pot may be determined by its bottom surface, wall, or one of its diagonals
(fig. 4).� correspondingly, the reflecting point is said to *face* the
bottom surface, wall, *etc*. of the pot.

with a pot, the locus of equal-span consists of seven
circular arcs (fig. 5).� however the optimal rim location e (or e^{*})
usually falls on four longer arcs.� three shorter arcs are irrelevant.� for
simplicity, we will outline the locus of equal-span by four intersecting
circles, each containing a diagonal of the pot as its cord (fig. 6).

fig. 5� locus of equal-span
extended by a pot consists of seven arcs

� optimal rim locations are located on four longer
arcs.

fig. 6� locus of equal-span outlined by four
intersecting circles

���

���

fig. 7� nominal reflection bisects an arc which is
part of the circle of equal-span.

�it does not pass through the center of the pot.

nominal reflection should bisect the arc which is part of
the locus of equal-span and whose cord is a diagonal of the pot.� it does not
bisect the diagonal itself (fig. 7).� at optimal rim locations, the angle
between nominal reflection and the optical axis is:

_{
}.�������������������������������������������������� (5)

at optimal rim locations, the angle *a* or *a*^{*}
between the reflecting surface and the optical axis is called the *semi-apex*
angle of the reflector.� this term is adopted from studies of a funnel
(conical) reflector.� depending on the rim locations, it is given by:

�

_{
}��������� (on the
deeper reflector);��������������������� (6)

or�������� _{
}� (on the shallower
reflector).������������������ (7)

** **

** **

**locus of optimal rim location:**

** **

given the pot depth *q*,
the optimal rim location varies with angular tolerance *b*.� let *r *be the pot
radius, *xy* be a cartesian coordinate system such that the *y*-axis
coincides with the optical axis, and the origin coincides with the center of
the pot (fig. 6).� the *locus of optimal rim
location* e(*x*,*y*) is shown in
fig. 8 and given by:

_{
}.����������������������� _{
}

since the reflector is
substantially larger than the pot (*x*>>*r*), the locus of
optimal rim location can be substituted by its asymptotes in preliminary design
works.� let *b* �0 in equations (8) and (9).�� a pair of asymptotes is
obtained:

_{
}.������������������������������������������������������� (10)

the optical gain of reflector is defined below and derived
from equation (8):

_{
}.�������������� (11)

with the same angular tolerance,
a deeper pot can accommodate a larger reflector, hence achieving a higher
gain.� as a concrete example, unattended cooking typically requires no more
than two hours of full power.� by aligning the reflector one-hour ahead
of the sun [2], the required angular tolerance *b*
is reduced from 30� to 15�.� by raising *q *�from 0 to 30�, the
maximum reflector size *r*/*r* is raised from 2 to 3.31.� the gain is
raised from 4 to 10.95.� then the cooker works very well without a plastic bag
around the pot.

�

fig. 8 �locus of the optimal edge location, shown in
the first quadrant only.

(the spherical cookware will be discussed in the
sequel.)

** **

**asymmetry of the angular
tolerance:**

** **

we assumed previously that the sun moves in the plane of
cross-section.� obviously this must be false in most other cross-sections.� due to axisymmetry of the pot, we may remain in the present
cross-section and assume that the sun moves in a direction associated with
minimum angular tolerance.�� this direction is coplanar with the cross-section
if *q* <41�, and perpendicular to the cross-section if *q*>42�.�
the exact value of *q* at which the transition occurs depends on the angular
tolerance.� however for practical purpose it can be considered as a constant (*q*=41.5�).

most pots are shallower than 41�.�
thus in most design works in-plane solar movement is assumed as before.�
to design a reflector for an exceptionally tall cookware, the solar movement is
assumed to be perpendicular to the cross-section.� the
locus of optimal rim location e(*x*,*y*) becomes:

_{
},���������������������������������������������� (12)

where *f*
is given by equation (5), and

_{
}.��������������������������������������������������������� (13)

the ordinate *y* has an
optimal range instead of a unique value.� we shall take advantage of this
(limited) freedom, and select its value such as to maximize the angular
tolerance for in-plane solar movement.� as a concrete example, let *q*=60�,
*b*=15�.� since *q*>42�, *b*
is associated with out-of-plane solar movement. it will be denoted *b*_{o
}for clarity.* *�by
equations (5), (12) and (13), *x*/*r*=3.99.� by equation (8), angular
span *b*_{i }associated
with the in-plane motion is 25.4�. finally by equation (9), the optimal rim
height is *y*/*r*=0.81.�

**angular span of a
spherical cookware:**

** **

in addition to the direction of solar movement, elevation
of the sun also contributes to asymmetry of the cooker. while the reflector
faces the sun, the cookware remains horizontal.� the angle *h *between their axes may be as low as 0
near the equator, and as large as 60� in a winter noon in san francisco, usa,
or along the great wall, china.�

axi-symmetry of the cooker could be preserved, if the
cookware were spherical.� in reality we do not make and use a spherical
cookware.� however an ordinary cookware can be modeled mathematically as an
imaginary sphere, for the ease of analysis.� obviously this treatment is
approximate. the radius of this imaginary sphere *s* depends on the
elevation and movement of the sun, as well as the size and the depth of the
pot.� for a reasonably deep pot (30�<*q*
<45�), and unpredictable solar position/movement, we chose *s* such
that angular span *g *is exact on
the *x*-intercept of the reflector (fig. 9).�� thus,

_{
}.���������������������������������������� (14)

the optimal rim location e(*x*,*y*) is:

_{
}.���������������������������� _{
}

the resulting reflector is smaller than that given by
equation (8), as has been suggested by fig. 5. continuing with the previous
example, let *b*=15� and *q*=30�.� then by equations
(11) and (15), the gain of this reflector is only 9.95, instead of 10.95.� however
this smaller reflector achieves or exceeds specified *b* over a wide range of cooking conditions, whereas the
larger reflector achieves specified *b *only
if the pot is perpendicular to the optical axis.

a spherical cookware is not a special case of a deep-pot.
with a spherical cookware, the locus of equal-span is a circle concentric with
the cookware (fig. 9).� the locus of optimal rim location lies on the *x*-axis.
nominal reflection from the rim also lies on the *x*-axis. the reflecting
surface is inclined 45� from the optical axis at the rim.

fig. 9� the locus of equal-span when a deep pot is
modeled as an imaginary sphere.

there is only one pair of optimal rim locations in
this case.

**a parabolic reflector**:

when the cookware is modeled as a sphere, the nominal
reflection should always be directed at its center.� thus the reflector has a
well-defined focal point, which is at the origin in fig. 9.� the reflector is
parabolic.� it is completely determined by the rim coordinates (*x*_{0},
*y*_{0}).� to obtain its equation, first calculate slope *k*_{n}
of the nominal reflection, and slope *k *of the reflecting surface,
both at the rim:

_{
};���������������������������������������������������� (17)

_{
}.����������������������������� (18)

the coefficients, including the focal length *f*,
are:

_{
};����������������������������������������������������������������� (19)

_{
}.��������������������������������������������������������� (20)

equation of the parabola is:

_{
}.�������������������������������������������������������������� (21)

�����������

geometrically, a properly aligned parabolic reflector is
100% efficient.� incident sunlight strikes the cookware either directly, or
through one reflection.�

a **non-parabolic reflector**

** **

unless the cookware is spherical, a reflector optimized
for robustness is not parabolic.�� this was clearly stated early with reference
to a pan, and is true for a pot as well.� unfortunately the optimal reflector
shape depends on shape, size, and orientation of the cookware.� it also depends
on the direction of solar movement.� therefore it is difficult to design a
truly optimal reflector for a versatile solar cooker.

fortunately an optimal reflector does not differ
significantly from a parabola.� as a concrete example, consider a pot located
axisymmetrically with respect to the optical axes.� two optimal reflectors and
two parabolic reflectors are shown in fig. 10. their differences are small
indeed.� furthermore, angular tolerance of a reflector increases toward the
optical axis.� the benefit of optimizing the reflector shape, hence improving
robustness in the interior of the reflector, is not overwhelming.

therefore a parabolic reflector is often adopted for
ordinary cookware.� an exception occurs when the reflector is homemade out of a
flat sheet [3].� in this case maximizing the angular tolerance is important,
even in the interior of the reflector.

fig. 10�� optimal and parabolic reflectors for a pot
axisymmetric to the optical axes

** **

** **

**optimization of the rim shape **

** **

the optimal rim shape depends on the criterion chosen.� to
maximize the robustness against manufacturing tolerance and solar movement, the
rim should exhibit substantially equal angular tolerance.�� in most designs the
reflector is a portion of an axisymmetric surface, while the pot is not
axisymmetric with respect to the same axis.� therefore the optimal rim shape
will be neither axisymmetric nor planar.� it depends on size, shape, and
orientation of the cookware, as well as the elevation and movement of the sun.�
in practice the reflector and its rim shape are optimized for a �typical�
operating condition.

fig. 11� two cross-sections of a pot tilted with
respect to the optical axis

as a concrete example, we have a dark pot 255 mm in
diameter, 160 mm in height.� it will be used in n38�, where annual average of
the sun�s elevation is 90�-38�=52� at noon, and lower off the noon.� we chose to optimize the reflector for 50� elevation.� the objective is to maximize
the cooking power while achieving �15� angular tolerance around the rim.�

consider two cross-sections through the optical axis,
which is the *y*-axis in fig. 11.� since the pot is not axisymmetric with
respect to the optical axis, the shape of each cross-section is relevant to the
in-plane solar movement only.� this difference from an axisymmetrically located
pot does not simplify the derivation, because both cross-sections must be
considered now.

in the first cross-section the pot looks like a drum. it
is parallel to the *x*-axis.� for simplicity the drum will be modeled by
an inscribing rectangle whose width and height are 108 and 104 mm,
respectively.� the depth of this rectangle (*q*=43.9�)
exceeds the threshold (41.5�).� however as explained in the last paragraph,
only in-plane solar movement will be considered in each cross-section.� the
optimal rim locations (�482,�188) in this cross-section are given by equations
(8) and (9).�� the corresponding parabolas are given by equations (17) through
(21), and shown in fig. 12.

in the second cross-section the pot is tilted 40� from the
*z*-axis.� the locus of �15�span is tilted from the optical axis, as
illustrated by four intersecting circles in fig. 12.� to achieve desired
angular tolerance the reflector must be within the envelope outlined by these
four circles.� to maximize the cooking power the reflector must extend as far
away from the optical axis as possible.

in fig. 12, the upper, deeper parabola meets the above
requirements much better than the lower, shallower parabola.� equations of this
deeper parabola in the *xy*-plane, and in the three-dimensional space are:

_{
}mm;�������������������������������������� (22)

and ����� _{
}mm.���������������������������� (23)

fig. 12� optimal reflectors in the *xy*-plane and
the locus of �15� span in the *yz*-plane.

projection of the optimal rim shape in the *yz*-plane
is shown in thin solid line.

rim positions p_{1} and p_{2} shown in
fig. 12 are optimal in the *xy*-plane only.� in the *yz*-plane the
reflector should be trimmed on the left and extended on the right, so that the
rim falls on the locus of �15� angular-span.� when the reflector is projected
onto the *yz*-plane, its rim is curved and tilted with respect to the
optical axis (fig. 12).

optimal rim locations p_{3} and p_{4} in
the *yz*-plane are intersects of the parabola and circles of �15�
angular-span.� we will not indulge in the exact solution here, because the �typical�
operating condition was somewhat arbitrarily chosen to begin with.� furthermore
in practice, the optimal rim shape is influenced by many other factors, such as
simplicity, deformation, utilization of raw material, etc.� however it is
important to realize that to maximize the angular tolerance, the rim should be
extended where it faces the pot cover, and trimmed where it faces the pot
bottom.�

**summary**

the *robustness* of a reflective solar cooker is
characterized by *angular tolerance* of its reflector, especially along
the rim.� the angular tolerance is much greater with a deep dark *pot*
than with a shallow *pan*, or a shiny pot having a dark bottom.� the
angular tolerance usually increases toward the cookware.� thus the reflector
should cuddle around the cookware.� for simplicity a reflector is often part of
a parabolic surface.� the optimal focal length depends on the size, shape, and
orientation of the cookware.� typically a reflector is tilted to face the sun.�
in this case the optimal rim shape is not circular.� the rim is tilted with
respect to the optical axis such that the reflector extends farther in the
direction facing the pot cover than in the direction facing the pot bottom.�

**references**

** **

- �a parabolic solar cooker
for unattended cooking,� by li-yan zhu and yun k. kim, http://solarcooking.org/unattendedparabolic.htm
april, 2002
- �an alignment template for
unattended solar cooking,� by li-yan zhu and yun k. kim, http://solarcooking.org/unattendedalignment.htm
june, 2002
- �making a parabolic
reflector out of a flat sheet,� by li-yan zhu, http://solarcooking.org/parabolic-from-flat-sheet.htm
april, 2002�