Robustness of a Reflective Solar Cooker
Li-Yan Zhu and Yun K. Kim
In the spring of 2002, we described a parabolic solar cooker for unattended cooking . 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 ; 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,
The angular tolerance consists of structural misalignment (±b1), surface waviness (±b2), and movement of the sun (±b3). 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,
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 (b1 reduced) or if the surface is made smoother (b2 reduced), the reflector will be able to accommodate greater solar movement (b3 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
Combine equations (1) and (3):
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.
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:
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:
The optical gain of reflector is defined below and derived from equation (8):
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 , 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:
where f is given by equation (5), and
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 bo for clarity. By equations (5), (12) and (13), x/r=3.99. By equation (8), angular span bi 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,
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 (x0, y0). To obtain its equation, first calculate slope kn of the nominal reflection, and slope k of the reflecting surface, both at the rim:
The coefficients, including the focal length f, are:
Equation of the parabola is:
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 . 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:
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 P1 and P2 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 P3 and P4 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.
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.