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an alignment mark for unattended solar cooking

li-yan zhu and yun k. kim

 

introduction

 

a solar cooker needs to face the sun, to receive maximum solar power.� with a pinhole made on the reflector, and an alignment mark made on a base-plate, it is quite easy to aim directly at the sun.� however to reduce the frequency of alignment, the cooker should face a direction where the sun will be sometime later.� the elapsed time from the time of adjustment to the time maximum power is received is called the advance.�� usually it is half of the expected duration of cooking, or half of the interval between adjustments.�

 

movement of the sun varies by latitude, season, and time in the day. it is difficult to predict by an ordinary user.� for convenience, an alignment template can be prepared for each latitude, at an interval of approximately 5�.� it shall be affixed to the base-plate.� it consists of a family of curves, each representing a month.� each curve contains several marks, each representing an hour on the clock.� the user selects an appropriate mark by the present month and time.� then let the sun shine through the pinhole onto the mark.�

 

a crude template was described in [1], for use with a specific cooker at approximately 38�n.� this article presents a general solution that accommodates season, latitude, time, and the bias angle of principal optical axis with respect to the base-plate.� the solution is presented in the vector form, programmed in excel spreadsheet, and shown graphically.

 

derivations

 

to an observer fixed to an inertial frame outside of the solar system, the planet earth obits around the sun about a fixed axis z, and spins about a fixed axis z (fig. 1).� both z and z are unit vectors. the angle between z and z is approximately 23.5�.� for convenience denote its complementary angle

g0=90�-23.5�=66.5�.��������������������������������������������������� (1)

 

 

 

 

 

 

 

 

 

 

 

 

fig. 1� a right-handed cartesian coordinate system fixed to an inertial frame

 

define two additional unit vectors

y=(z�z)/|z�z|,������������������������������������������������ (2)

x=yz.���������������������������������������������������������������������� (3)

xyz is a right-handed cartesian coordinate system, in which

����������� z=xcosg0+zsing0.�������������������������������������������������������� (4)

let the sun be at the origin of xyz coordinate system.� then the earth orbits in the xy plane.� its position is better described by its polar coordinates (x,b), where b=0 at the winter solstice and increases with time.� sunlight strike the planet earth in the direction:

����������� r=xcosb+ysinb.��������������������������������������������������������� (5)

the angle g between r and z varies by season:

����������� cosg=rz=cosg0cosb.��������������������������������������������������� (6)

the xyz coordinate system was established for the derivation of equation. (6).� it will not be used again in the sequel.

 

to an observer fixed to the earth surface, the sun orbits around him once a day.� the sunlight sweeps over a cone whose axis is z, and whose apex angle is 2g.� the observer can also identify the local latitude q, and an upward direction n.� let

y=(z�n)/|z�n|,������������������������������������������������������������� (7)

x=yz.������������������������������������������������������������������������ (8)

xyz is a right-handed cartesian coordinate system fixed to the earth (fig. 2).� the x axis points up and leans toward the equator; the y axis is horizontal and points to the east;� the z axis points to the north star. in this coordinate system, the normal vector is:

n=xcosq+zsinq.����������������������������������������������������������� (9)

establish a polar coordinate system (x,f) in the xy plane. the angular coordinate f is a measure of the sun�s movement around the observer.� it is zero at noon. the direction of sunlight depends on season (g) and time (f) in the day:

r=-(xsingcosf+ysingsinf+zcosg).������������������������������� (10)

 

 

 

 

 

 

 

 

 

 

 

 

 

fig. 2� a right-handed cartesian coordinate system fixed to the surface of earth

 

 

we will also need a coordinate system ijk fixed to the reflector (fig. 3).� let k be a unit vector pointing outward along the principal optical axis.� we want k to point at the sun at a given time f0, i.e.

k=-r0=xsingcosf0+ysingsinf0+zcosg.������������������������� (11)

to adjust the angle of elevation, the base-plate pivots about axis j which is orthogonal to both n and k.� let

j=(k�n)/|k�n|,������������������������������������� ����������������������� (12)

i=jk.������������������������������������������������������������������������� (13)

 

 

 

 

 

 

 

 

 

 


fig. 3� a right-handed cartesian coordinate system fixed to the reflector

 

 

for economy no dedicated screen will be erected in the ij plane.� sunlight coming through a pinhole illuminates a spot on the base plate tj.� thus we encounter yet another coordinate system tjp (fig. 4).� since the t axis is at an angle t from the i axis,

t=icost-ksint.������������������������������������������������������������� (14)

a vector normal to the base plate is

p=isint+kcost.������������������������������������������������ (15)

let h be the distance between the pinhole and the base plate.� then coordinates (t,j) of the spot are:

t=h�(r�t)/(r�p),�������������������������������������������������������������� (16)

j=h�(r�j)/(r�p),�������������������������������������� ����������������������� (17)

to locate the origin o of tj coordinate system on the base-plate, let� t=j=0 in equations (16) and (17).� thus r, which runs from the pinhole to the origin o, must be perpendicular to the tj plane.� for convenience we shift the origin to the �bull�s eye�, which is the spot location when k points directly at the sun.� the j coordinate remains unchanged.� however the t coordinate is replaced by

q=t-h�(k�t)/(k�p)=t-htant,������������������������������������������� (18)

 

 

 

 

 

 

 

 

 


fig. 4� a right-handed cartesian coordinate system tjp fixed to the base-plate

 

 

template design

 

using equations (6) and (9) through (18), an alignment mark can be located for any cooker, at any latitude, season, time, and for any advance.� for illustrative purpose let t=35�, h=100mm, q=40�n, and the advance be one hour.� each curve in the template (fig. 5, where the unit on both axes is mm) represents a month (beginning at approximately 7th of the corresponding calendar month, and ending at approximately 6th of the next calendar month).� each mark on the curve represents an hour on the clock, local time.� a longer (or shorter) advance can be achieved by moving the spot away from (or toward) the bull�s eye.� as a special case, the spot should always fall on the bull�s eye if no advance is desired.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


fig. 5� a sample template for one-hour advance, the curves crowd together

 

 

separating the curves on multiple templates will render each more legible.� however a single template is more practical.� to separate the curves, we increase the advance 5 minutes per month from the winter solstice (45 minutes) to the summer solstice (75 minutes).� the curves are now more readable (fig. 6).� the curves could be spaced more evenly if the advance were increased geometrically.� however this would result in advance with too many digits, for example 43 minutes and 28 seconds.

 

note that except for the solstices, each curve represents two non-consecutive months.� the symmetry is due to the fact that each curve is exact on approximately 22nd of the calendar month.� if each curve were exact on the 15th of the calendar month, there would be noticeable difference between curves representing january and november, etc.


 

 

 

fig. 6� template with uniformly increasing advances

(from 45 minutes in winter to 75 minutes in summer)

 

note also that the local time is that of the astronomical time.� it varies continuously by longitude.� since the �official time� varies discontinuously by zones, the local time can differ from the official time by half of an hour.� if the time zone is not partitioned at exact multiple of 15� longitude line, the difference can be even greater.� the situation is further complicated by the institution of �summer time�, also called the �daylight saving time�.� thus each location needs one or two correction constant, which can be tabulated and/or provided at each outlet of the solar cooker.

 

the template for 30�n (fig. 7) does not differ much from that for 40�n.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


fig. 7� template for 30�n


 

with a fixed advance, template for the southern hemisphere is exactly symmetric to that in the northern hemisphere. however to prevent curves from crossing each other, the advances should decrease from winter to summer on the template (fig. 8).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


fig. 8� template for the southern hemisphere

 

in the tropic, the sun may appear on either southern or northern part of the sky, depending on the season.� accordingly, the alignment curves appear on both left and right half of the template (fig. 9).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


fig. 9� curves appear in both left and right portion of the template in the tropic


 

the alignment curves are asymmetric about the horizontal axis.� the hour marks are spaced much denser on the positive direction of q axis than on the negative direction.� this asymmetry is due to nonlinearity of the tangent function.� recall that the bulls-eye is not located directly below the pinhole.� connection between the bulls-eye and the pinhole is tilted at an angle t from the normal direction of the base-plate.

 

spreadsheet

 

the attached excel spreadsheet can be used to calculate the coordinates of any alignment mark.� double click on the spreadsheet to open.� input area is highlighted in yellow. use the data, table function in excel to generate coordinates for all marks.

 

 

summary

 

equations are derived for drawing alignment marks on a template.� using the template, the frequency of tracking can be halved.� thus unattended cooking becomes much easier to accomplish.� the solutions are described in vector form, implemented in an excel spreadsheet, and displayed graphically.

 

reference

 

1. �parabolic reflector for unattended cooking,� by li-yan zhu and yun k. kim, http://solarcooking.org/unattendedparabolic.htm april, 2002

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