Robustness of a Reflective Solar Cooker

&nb= sp;

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 at a substantially constant pow= er for two hours without adjustment. This rob= ust design is based on three techniques: a tem= plate for aiming ahead of the sun [2]; a dark pot with suitable depth; and a reflector optim= ized for angular tolerance. The last two techniques will be explained in this article.

The angular tolerance of an infinitesimally small reflecting surface is the acceptable range of its orientation. It characterizes the robustness of a reflecting surface against manufacturing tolerance and solar movem<= /st1:PersonName>ent. The angular tolerance is not unifo= rm over the entire reflector. It= is usually most stringent at the rim. Obviously, the tolerance can be increased by making the reflector smaller, or making the cookware bigger. In either case the gain, de= fined as the ratio of cooking power with and without the reflector, is reduced. Our objective is to maximize the gain while achieving the desired angular tolerance.

This objective will be accomplished by optimizing the reflector sh= ape, analytically. First we shall introduce the concept of angular span. Next we shall note directionality = of the angular span, and obtain a contour map of the angular span i= n each of the two critical directions. Finally we shall optimize the pot depth so that a good compromise can be made between these two contours maps.

Angular Span and Angul= ar Tolerance

Assume that the coo= kware is a cylindrical pot, with radius r and height h. Let XYZ be a Cartesian coor= dinate system whose origin coincides with the cen= ter of the cookware, and the Z-axis coincides with the axis of the cookw= are (fig. 1). The Z-= axis is usually within a few degrees from the vertical direction, so that food can be distributed uniform= ly at the bottom of the pot.

Projection of the cookware on a spherical surface ce= ntered at an arbitrary point A is generally not circular (fig. 2). Thus angular span 2b of the cookwar= e with respect to point A is directional. Consider a circle inscribed in the projection. Ideally, nominal reflection from point A strikes the center= of this circle. Then the angular tolerance is ±b, where b is also directional. The minimum of 2b is either or , where T_i are tangent points on the inscribed circle. T₁T₂ and T₃T₄ define two critical directions of the angular tolerance. In the first direction, reflection= from point A spread across the Z-axis.&n= bsp; In the second direction, reflection from A spread along the Z-axis.

In either critical direction the angular tolerance ±b consists of thre= e components: structural misalignment (±b₁), surface waviness (±b₂), and movement of the sun (±b₃). The first two com<= /st1:PersonName>ponents double due to reflection; the third does not. Assume that all three components of b are in the same direction, then

. = &nb= sp; = &nb= sp; (1)

Every 1° error in surface orientation, either du= e to approximation in the design, tolerance in = the construction, structural deformation, or surface waviness, causes 2° reduction in the tolerance to solar movement, which is equivalent to a 8-minute loss in unattended cooking.

Fig. 1 A cylindrical cookware and an infinitesimally small reflecting surface at point A

<= /td>

Fig. 2 Projection of the cookware on a spherical surface centered at point A.

The bottom surface appears larger than the top because it is closer to point A.

Arrows indicate critical directions at which the angular tolerance is minimum.

However in reality, accuracy of the surface orientat= ion is not as crucial. This is becau= se that over the entire reflector, three comp= onents of angular tolerances are not coplanar in general. Their directions are quite randomly distributed. Statistically, <= /p>

. = &nb= sp; = (2)

Minor deviation in local alignment (b₁ and b₂) has very little effe= ct on the tolerance to solar movement (b₃). A high-precision reflector is not required for unattended cooking, so long as the reflector is designed with adequate angular tolerance.

Angular Tolerance in T= wo Critical Directions

In each of the two critical directions (across and along the Z-a= xis), we first outline a “sweet spot” in which the desired angular tolerance is achievable. Then= we construct a reflector in this sweet spot to achieve the desired angular tolerance, and to maximize the cooking power. Not surprisingly, reflectors optimized for the= se two different directions won’t be congruent. We shall minimize their differences by optimizing the pot de= pth, so that a good compromise can be achieved.

Angular Tolerance Across the Z-Axis

In this case reflection from point A spreads over T₁T₂ in figs. 2a and 2b. Angular span of the cookware_{<=
!--[if gte vml 1]>} is difficult to evaluate. For simplicity we approximate the angular span by , where D₁D₂=3D 2r is a dia<= st1:PersonName>meter of the cookware. The error in= volved is small, especially if A is far from the cookware. The locus of eq= ual angular span, abbreviated as the locus of equal-span, is given by:

. = &nb= sp; = (3)

A slightly= better approximation exists near the XY-plane: =

= ;

. = &nb= sp; = (4)

Equations (3) and (4) together describe a locus of equal-span, which is roughly a sphere concentric with the cookware (fig. 4a). A family of such loci constitutes a contour map of angular span. Provided tha= t the reflecting surface can be designed properly that nominal reflection bisects the angular span, this family of loci is also a contour map of the angul= ar tolerance. The angular tolera= nce increases toward the cookware (fig. 4b).&n= bsp; The “sweet spot” is outlined by a contour (i.e., locus) associated with the desired angular tolerance. For maximum cooking power, the rim (edge, or opening) of an optimal reflector consists of tangent points= of the contour with respect to the incident sunlight (fig. 4a). It is a great circle of the sphere= .

= ;

Fig. 3= Top: Side view of a cookware and a reflecting point A; and

Bottom: A cross-section in a plane containing= the tangent points.

= ;

Fig. 4 For angular tolerance acro= ss the Z-axis, a) A locu= s of equal-span (and equal-tolerance); and

b) A contour map of the angular span= (or angular tolerance)

= ;

To cook wi= thout adjustment for two hours, let b=3D15°. Then radius R_o of the great circle is either rcot15° or rcsc15°, depend= ing on whether the rim is close to the XY-plane. The gain of the cooker, defined as=

, = &nb= sp; = &nb= sp; = (5)

is between= 14 and 15. Assume that the solar cooker is 60% efficien= t, and that the sunlight intensity is 300 W/m². Then approxima= tely 50 mm (2 inch) deep of dense food (soup, ric= e, etc.) can be cooked unattended, irrespective of the radius of the pot.

= ;

Angular Tolerance along the Z-Axis

Analysis in this directi= on is more complicated than the previous one. For pedagogic purpose we shall fir= st consider the simplest case of cooking with= a flat pan (or a shiny pot with one dark end). Then consider a dark pot with an arbitrary depth.

Cooking wit= h a Flat Pan

By elementary geometry, the locus of equal-span is a pair of intersecting circles in which the pan = is the common cord (fig. 5a). A family of such loci constitutes a contour map= of angular span, which is also regarded as a contour map of the angular tolerance (fig. 5b). The “sweet spot” is outlined by a contour associated wit= h the desired angular tolerance. Fo= r maximu<= st1:PersonName>m power, the optimal rim again consists of tangent points = of the contour with respect to the incident sunlight. Two distinct solutions exist. If the sunlight is parallel to the Z-axis, each optimal rim is axisymmetric (fig. 6). <= /p>

At any giv= en point, angular tolerance is ma= ximized when the no= minal reflection bisects the correspondi= ng angular span. By elementary geometry, nominal reflection should be directed at a= Z-intercept (F= or F^*) of the contour containing the reflecting point (fig. 6).

Fig. 5 For angular tolerance alon= g the Z-axis, a) A locu= s of equal-span (and equal-tolerance), w= here j=3D4b= is the measure o= f the cord; and b) A contour map of the angular span= (or angular tolerance)

&nbs= p;

Fig. 6 Reflectors optimized for a flat pan and for sunlight parallel to= the Z-axis

According to figs. 5b and 6, angular span is not uniform<= /st1:PersonName> over the reflector. On contours associated with greater angular span, Z-intercepts F and F^{*
are farther away fro}m<= /st1:PersonName> the pan. Thus a reflec= tor optimized for angular tolerance at every point does not h= ave a well-defined focal point. I= t is definitely not a parabola.

However in practice, angular tolerance neednR= 17;t be optimized over the entire reflector. Po= ints closer to the cookware can tolerate more systematic misalign<= st1:PersonName>ment. Thus a fixed focal point is acceptable. For simplicity the reflector is almost always a parabola.

When the sun is far off the Z-axis, the previ= ously designed reflectors must be turned to foll= ow the sun. A first solution is to pivot the reflector about its focal point F, and point its principal optical axis directly at the sun (fig. 7a). The focus remains sharp. However a portion of t= he reflector protrudes outside of the contour, hence unable to achieve the des= ired angular tolerance.

A second solution is to pivot the reflector about its apex, and turn its principal axis halfway toward the sun (fig. 7b). The reflecting surface protrudes much less tha= n in the first solution. However t= he focal point F becomes blurred. The aberration can be very large, especially if the sun is far off the Z-axis. In order to reduce aberration, eit= her the reflector must be trim= med severely around its rim, or the pan must remain facing the sun. Neither option is desirable. Thus a flat pan (or a shiny pot wi= th a dark bottom) is unsuitable for unattended solar cooking.

<= /td>

Fig. 7 Alignment schemes when = the sun is off the Z-axis

a) Pivoting about the focal point; b) Pivoting a= bout the apex.

Cooking wit= h a Dark Pot

Consider now a more general case of cooking with a dark pot.&n= bsp; Let q be the angle between its diagonal and its bottom surface (fig. 8). A reflecting point may face the top, bottom, or diagonal of the pot. In any cross-section through the Z-axis, locus of equal-span consists of ei= ght circular arcs. Each arc is part of a circle containing the top, the bottom, or a diagonal of the cookware as a cord.&n= bsp; For clarity these circles are not shown in their entirety.

Also for clarity, let th= e Z-axis be fixed vertically. <= span style=3D'mso-bidi-font-weight:bold'>A cooker can only be optimized for a specific nominal elevation e of the sun. The design is relatively si= mple when e=3D90°. The = sunlight is parallel to the Z-axis. Two pairs of optimal rim locations (EE and E^*E^*) exist. Both have the sam<= /st1:PersonName>e radius R, hence identical gain G:

. = (6)

At the same angular tolerance (±b), the locus associate= d with a deeper pot (with larger q) is wider. Thus a higher gain can be achieved.= As a concrete exa= mple, let b=3D15° at the rim. By ra= ising q from 0 to 30°, the gain is raised from 4 to almost 11.

.    =             &nb= sp;            =       (7)

At our residence, L=3D37.5°N. Thus e_m_ax=3D75°. The minimum operating elevation e_m= _in depends on weather and terrain.&nbs= p; It is 25° at our residence. We chose the average of ma= ximum and minimum elevation as the nominal elevation. Therefore e=3D50°.

&nb= sp;

When the sun is at the n= ominal elevation, principal optical axis of the reflector should be parallel to the sunlight (fig. 9), so as to maintain a sharp focus. For maximum power, the optimal rim again should consist of tangent p= oints of the contour with respect to the incident sunlight.   H= owever no parabolic surface fits such a rim, unless e=3D 90°. T= hus we only demand that the rim intersects the locus of equal-span at a point most distant from the optical axis.&n= bsp; Of two such points T and T*, the former is chosen to yield a shorter focal length f.

=

Let r be complementary of the nominal elevation e; R₊ = be the reflector radius at intersection T; and H be the height of point T, = measured from the focal point along the principal axis (fig. 9). T= hen

=

,    =             &nb= sp;            =        (8)

.    =              (9)

.    =             &nb= sp;            =             &nb= sp;   (10)

=

As a concrete examp= le, let r=3D100 mm; h=3D115 mm; e=3D50°; b=3D15= °; and q=3D30°. Then H=3D 69 mm, R₊=3D418 mm, and f=3D178 mm.

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

&nb= sp;

Fig. 9 A parabolic surface optimized for a predetermined nominal elevation e

The optimal rim is the intersection between the parabolic surface and the appropriate locus= of equal tolerance. It is neither axisymmetric nor coplanar.   It is best described point-by-point, as follows:

= 1.      Let XYZ be a Cartesian coordinate system fixed to the cookware, where Z is the axis of symmetry. Let xyz be a Cartesian coor= dinate system fixed to the parabolic surface, whe= re z is the principal optical axis. X and x are congruent. <= i>Z and z are at an angle r= .

= 2.      Obtain an equation of the parabolic surface in the xyz coordinate system. Then transform the equation into the XYZ coordinate system.

= 3.      Consider a series of cross-sections perpendicular to the Z-axis. In each cross-section the contour = becomes a circle; the parabolic surface becomes an ellipse. Both are both quadra= tic in X and Y. Eliminate X = by subtraction. A quadratic equa= tion in Y is obtained. Of t= he two real solutions, take the one with lower value. Now that both Y and Z are known, X can be solved in the equation of the circle. Two points (±X,Y<= /i>,Z) are found on the rim.

= 4.      A series of points (±X,Y,Z) describes the optimal rim in the XYZ coordinate system. It should be transformed into the xyz coordinate system.

For the example cit= ed above, a projection of the optimal rim in the xy-plane is shown by a solid curve in fig. 10. It consists of four lopes. It is symmetric about the y-axis, but not about the x-axis.

Compromise on the Rim Shape

The great circle in fig. 4a is duplicated in fig. 10= , as a dashed curve. It is not congr= uent with the four-loped curve. In= order to match maximum radius R₊ of the four-loped curve with R_{o
of the great circle,}

.    =             &nb= sp;            =             &nb= sp;            =             &nb= sp;   (11)

In the example abov= e, e=3D50°. Thus q 25°. In= order to match minimum radius R_- (near the x-axis in fig. 10) with R_o, q 45°. A good compromise on q is about 30°. A deeper pot loses more heat through natural convection. A shallower pot makes the four-loped curve narrower near the x-axis.

In addition to angular tolerance in two critical directions, many other factors should be considered, e.g., power, raw-material cons= umption, bulk and weight of the product, mechanical stability, ease of assembly, and appearanc= e. As a rule of thumb, the ri= m should be free of sharp kinks, which are present in the four-loped curve. The kinks reduce power without sav= ing raw material or reducing the bulk. However they do cause stress-concentration which promotes distortion and crack.

The projection of a good solution is shown by a dott= ed curve in fig. 10. It is an el= lipse slightly offset from the x-axis. It is tangent to the four-loped cu= rve on the y-axis; and halfway between the four-loped curve and the great c= ircle on and near the x-axis.

&nbs= p;

Fig. 10 Projections of an optimal rim in the xy-plane

i) for tolerances across the Z-axis (dashed curve);

&nb= sp;            ii) for tolerances along the Z-axis (solid curve); and

            = iii) a good compromise (dotted curve) of the two solutions.

A simpler, and slig= htly less robust alternative can be obtained with plotting the great circle and = the four-loped curve. The project= ion of this solution on the xy-plane is an ellipse centered at the origin:<= /p>

,    =             &nb= sp;            =        (12)

where

.    =             &nb= sp;            =             &nb= sp;           (13)=

A family of still s= impler, and less robust solutions features an axisymmetric rim. The rim passes through (R_a<=
/sub>,H) in the yz-plane, where R_a is between R_=
o and R₊, H is given by eqn. (8). The focal length f is calcu= lated by substituting R₊ by R_a in eqn. (10).<= span style=3D'mso-spacerun:yes'>

Many other solutions are possible. Once projection of the rim is determined, three-dimensional description of the rim can be obtained by:=

.    =             &nb= sp;            =             &nb= sp;            = (14)

Summary

The robustness of a reflective solar cooker is characterized by angular tolerance of its reflector, especially along the rim.&= nbsp; The angular tolerance usually increases toward the cookware. Thus the reflector should cuddle a= round the cookware. A parabolic sur= face is not ideal, but very suitable for unattended cooking. With the focal point fixed at the = center of the pot, any point on the reflecting surface determines the parabola completely (through equations= 10 and 14). Therefore shape and position of the rim cannot be arbitrarily assigned.   To optimize the reflector, we consider angular tolerance in two critical directions, and obtain two conflicting solutions. Their difference is minimized by optimizing the depth of the cookware. In making the final compromise, many other factors must be consider. The rim of a robust reflector is a three-dimension= al curve resembling the edge of a slightly be= nt potato chip.

References

=

“A Parabolic Solar Cooker for Unattended Cooking,” by Li-Yan Zhu and Yun K. K= im, http://solarcooking= .org/unattendedparabolic.htm April, 2002

“An Alignment Template for Unatten= ded Solar Cooking,” by Li-Yan Zhu and Yun K. Kim, http://solarcooking.org/UnattendedAlignment.ht= m June, 2002<= /o:p>

“Making a Parabolic Reflector Out of a Flat Sheet,” by Li-Yan Zhu, http://= solarcooking.org/parabolic-from-flat-sheet.htm April, 2002 <= /span>

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Robustness of a Reflective Solar Cooker        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;   Revised February 15, 2003=

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Li-Yan Zhu & Yun K. K= im

Second Revision February 15, 2003

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