It's Alive! It's Alive!
Wiring the Dish & Commanding it to Move
September 2005
 Last month (August) we talked about wiring up our solar reflector
actuators and moving the "dish". You can see in Photo 1 the
orange colored #16 electrical cable wired from the utility box to the
two actuators.
After powering up the lines with one of our actuator controllers, the
results were mixed - both positive and negative. What do you want first
- the good, the bad, or the ugly? Let's just take it in order. The
good: The east-west swing actuator powered up and worked flawlessly; with
a slight modification in the pivot or swing point, we can achieve
somewhere around a 120 degree swing -- which is enough for about 8 hours
of direct sunlight during the summer. The bad: The elevation actuator
powered up, but did not raise the dish. Apparently the weight of a heavy
fiberglass dish is too much for the plastic gear train. In the future we
will need to use a stronger actuator or a lighter dish. The ugly: It
looks like it's time for a new paint job on the dish mount. Aiming the
Dish based on Predicted Solar Position
To
complete this year's research on the solar collector portion of our solar
turbo-generator system, I am including the following equation study and
computer program for calculating the sun's position relative to any point
on the earth's surface, and any time of day.
The following "Solar Calculation Details" were found on
the NOAA website (National Oceanic & Atmospheric Administration) at:
www.srrb.noaa.gov/highlights/sunrise/calcdetails.html
General Solar Position Calculations
First, the fractional year y is calculated,
in radians.
y = (2*Pi/365)*(day_of year - 1 +
(hour-12)/24)
From y, we can estimate the equation of time
(in minutes) and the solar declination angle (in radians).
eqtime = 229.18*(0.000075+0.001868*cos(y)-0.032077*sin(y)-0.014615*cos
(2*y)-0.040849*sin(2*y)
declin = 0.006918-0.399912*cos(y)+0.070257*sin(y)-0.006758*cos(2*y)+
0.000907*sin(2*y)-0.002697*cos(3*y)+0.00148*sin(3*y)
Next, the true solar time is
calculated in the following two equations. First the time offset is
found, in minutes, and then the true solar time, in minutes.
time_offset = eqtime - 4*longitude +
60*offset
where eqtime is in minutes,
longitude is in degrees, timezone is in hours from UTC (Mountain
Standard Time = +7 hours).
tst = hours*60 + minutes + time_offset
where hr is the hour (0-23), mn is
the minute (0-60), sc is the second (0-60).
The solar hour angle, in degrees,
is:
ha = tst/4 - 180
The solar zenith angle (Phi) can
then be found from the following equation:
cos(Phi )= sin(lat)*Math.sin(declin)+cos(lat)*cos(declin)*cos(ha)
And the solar azimuth (Theta,
clockwise from north) is:
cos(180-Theta) = -(sin(lat)*cos(Phi)-sin(declin))/(cos(lat)*sin(Phi))
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Sunrise/Sunset Calculations
For the special case of sunrise or sunset,
the zenith is set to 90.833° (the approximate correction for
atmospheric refraction at sunrise and sunset), and the hour angle
becomes:
ha = +/- arccos (cos(90.833/(cos(lat)*cos(declin))-tan(lat)*tan(declin))
where the positive number
corresponds to sunrise, negative to sunset.
Then the UTC time of sunrise (or
sunset) in minutes is:
sunrise = 720 + 4*(longitude-ha)-eqtime
where longitude and hour angle are
in degrees and the equation of time is in minutes.
Solar noon for a given location is
found from the longitude (in degrees) and the equation of time (in
minutes):
snoon = 720 + 4*longitude - eqtime
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And here is their Suncalc program to calculate the sun's
position relative to your location and time:
10
' SUNCALC
20 'Program to compute sun position, sunrise, solar power
collection, etc.
30 '
40 'general comments
50 '
60 'To use, adjust the constants in the next section to reflect
your earth
70 ' position, the day and time on which you want to find the sun,
and
80 ' the direction you want to face a solar collector. Default
values are
90 ' noon on the solistice in DeKalb, IL, USA with sunlight
falling on
100 ' a horizontal stretch of ground.
110 'Program runs in UBASIC, available in Simtel archives
www.simtel.net
120 ' On a PC, type 'ubibm', then 'load "suncalc'; type
'system' when
130 ' done. Other forms of BASIC will probably work with some
fiddling.
135 ' You'll need to know: #pi=3.1416; atan(x)=arctangent
function.
140 'Note: 'energy collection' is a multiplier. It's the fraction
of the
150 ' energy received by an equal-sized collector in constant
full-frontal
160 ' sun. Assuming no air or shading, this is 1.40 kilowatt/m^2,
spread
170 ' among all wavelengths. I don't know what it reduces to with
air
180 ' absorption. If your collector is wavelength-sensitive, there
is a
190 ' different number which depends on the absorption spectrum of
air.
200 'Times in this program are relative to local solar max (here,
"noon")
210 ' This moment is actually (12+xx) o'clock GMT, where xx=Longitude/15.
220 ' (Of course this is not usually an integer, e.g. might be
17:45 GMT)
230 ' Convert to local time by subtracting the appropriate number
of whole
240 ' hours (ROUGHLY the integer nearest to xx) -- don't forget to
adjust
250 ' for daylight savings time as well. I decided not to get
involved
260 ' with the conversion; you'll see Longitude is not really used
below.
270 'Notes on the model used (see accompanying text file): We
ignore
280 ' the finitude of sun's distance to earth (23491 earth radii)
290 ' eccentricity of earth's orbit (dist=3.40% greater July4 than
Jan4)
300 ' non-sphericity of earth (radius=0.34% larger at equator than
poles)
310 'Written by dave rusin, rusin@math.niu.edu, on 4/10/94
320 ' Released to the public domain. No warranties, of course.
330 ' This program is not intended to be beautiful, fast, robust,
or
340 ' amazingly accurate, but is mathematically sound. For
improvements,
350 ' or for details of calculations, please contact the author.
360 '
370 'user input section
380 '
390 point 4:' specifies extra accuracy in UBASIC computations.
400 Slant=23.45:' angle of inclination of axis to earth orbit
410 Latitude=+42:' data for DeKalb, Illinois, USA
420 Longitude=+89:' consult an almanac or atlas to get your
coordinates
430 Days=91:' days since March equinox
440 Timex=12.0:' hours since midnight at which you want to find
the sun
450 Nx=0:Ex=0:Ux=1.0:'outward normal on solar collector (North,East,Up)
460 ' "up"=(0,0,1);"northwest"=(1,-1,0), etc.
470 '
480 'now solve the problems at hand
490 '
500 print
510 print "Earth computations, latitude";Latitude;",";
520 print Days;"days after March equinox."
530 print using(2,3),"Solar collector facing ";Nx;",
";Ex;", ";Ux
540 print "Times relative to local noon (=solar max)"
550 gosub 630:' internal calculations
560 gosub 1400:'calculations for a moment
570 gosub 1640:'calculations for a day
580 gosub 1960:'calculations for a year
590 end
600 '
610 'subroutines follow
620 '
630 'internal data conversions
640 '
642 'need to normalize so nx^2+ex^2+ux^2=1.
645 Length=sqrt(nx*nx+ex*ex+ux*ux)
646 Nx=Nx/Length:Ex=Ex/Length:Ux=Ux/Length
650 Alpha=Slant*#pi/180:'convert to radians
660 Theta=Latitude*#pi/180:'note: we don't actually use Longitude
670 Psi=Days*2*#pi/365.25
680 Noon=atan(tan(Psi)*cos(Alpha))
690 if cos(Theta)*cos(Psi)*cos(Noon)<0 then Noon=Noon+#pi
700 'glossing over cases where phi, psi, or theta = +- pi/2
710 Phix=Noon+(Timex-12)*2*#pi/24
720 return
730 '
740 'compute apparent solar position:
750 ' (N,E,U)=cartesian coords, (Ang1,Ang2)=spherical.
760 ' Also compute energy flux here.
770 '
780 N1=-sin(Theta)*cos(Psi)*cos(Phi)
790 N2=-cos(Alpha)*sin(Theta)*sin(Psi)*sin(Phi)
800 N3=+sin(Alpha)*cos(Theta)*sin(Psi)
810 N=N1+N2+N3
820 E=-cos(Psi)*sin(Phi)+cos(Alpha)*sin(Psi)*cos(Phi)
830 U1=cos(Theta)*cos(Psi)*cos(Phi)
840 U2=cos(Alpha)*cos(Theta)*sin(Psi)*sin(Phi)
850 U3=sin(Alpha)*sin(Theta)*sin(Psi)
860 U=U1+U2+U3
870 if E=0 then Ang1=#pi/2
880 if E<>0 then Ang3=atan(abs(N/E))
890 if E>0 then Ang1=atan(N/E)
900 if E<0 then Ang1=atan(N/E)+#pi
910 if U=1 then Ang2=#pi/2
920 if U=-1 then Ang2=-#pi/2
930 if abs(U)<1 then Ang2=atan(U/sqrt(1-U*U))
940 'now u=sin(ang2),e=cos(ang2)*cos(ang1),n=cos(ang2)*sin(ang1)
950 if U<0 then Energy=0:goto 980:'you get no energy with sun
below horizon
960 Energy=(N*Nx+E*Ex+U*Ux)*100:if Energy<0 then Energy=0
970 ' (assuming collector is one-sided: no energy if sun is behind
it).
980 return
990 '
1000 'Technical problem: how to solve A*cos(phi)+B*sin(phi)+C=0
for phi.
1010 '
1020 F=sqrt(A*A+B*B)
1030 if F=0 then Nroots=0:goto 1150
1040 'actually the equation with a=b=0 DOES have solutions if c=0
too
1050 'but having inf many solutions is in our setting like having
none
1060 A=A/F:B=B/F:C=-C/F:' now a^2+b^2=1
1070 if B=0 then X=A*#pi/2:goto 1100
1080 X=atan(A/B):' then a=+-sin(x),b=+-cos(x)
1090 if B*cos(X)<0 then X=X+#pi:'now what we need to solve is
sin(x+phi)=c
1100 if abs(C)>1 then Nroots=0:goto 1150
1110 if abs(C)=1 then Phi1=C*#pi/2:Root1=Phi1-X:Nroots=1:goto 1150
1120 Phi1=atan(C/sqrt(1-C*C)):' =arcsin(c)
1130 Root1=Phi1-X:Root2=#pi-Phi1-X:Nroots=2:goto 1150
1140 'difference is 2*arccos(-C/sqrt(A^2+B^2)). This gives day
length.
1150 return
1160 '
1170 'calculate day's energy accumulation
1180 '
1190 'probably possible to get a closed form integral of sun*normal,but
1200 ' seems easier just to do numerical integration:
1210 Total=0
1220 Numpoints=100
1230 for I=1 to Numpoints
1240 Phi=2*#pi*I/Numpoints
1250 gosub 740
1260 Total=Total+Energy
1270 next I
1280 Total=Total/Numpoints
1290 return
1300 '
1310 'ho-hum routine to print times nicely
1320 '
1330 print using(3,0),int(X);":";
1340 X=X-int(X)
1350 X=int(0.5+60*X)
1360 if X<10 then print " 0"; print using(1,0),X;:goto
1380
1370 print using(3,0),X;
1380 return
1390 '
1400 'show data for the moment
1410 '
1420 print
1430 print "Observations about this moment"
1440 print "******************************"
1450 print "At ";:X=Timex:gosub 1310:print
",";
1460 print " sun appears to be at coordinates: ";
1470 Phi=Phix
1480 gosub 740
1490 print using(2,3),"N=";N;" E=";E;"
U=";U
1500 if U=1 then print "straight overhead":goto 1600
1510 if U=-1 then print "straight underneath":goto 1600
1520 print "Sighting is ";
1530 if E=0 then print "due ";:goto 1550
1540 print using(4,2),abs(Ang3)*180/#pi;" degrees ";
1550 if N>0 then print "north, and ";:goto 1570
1560 if N<0 then print "south, and ";
1570 print using(4,2),abs(Ang2)*180/#pi;" degrees ";
1580 if U>=0 then print "up.":goto 1600
1590 if U<0 then print "down ... but you can't see
it."
1600 print "Relative energy collection rate is
";using(4,2),Energy;
1610 print " percent maximal"
1620 return
1630 '
1640 'show data for the day
1650 '
1660 print
1670 print "Observations about this day"
1680 print "***************************"
1690 A=cos(Theta)*cos(Psi)
1700 B=cos(Alpha)*cos(Theta)*sin(Psi)
1710 C=sin(Alpha)*sin(Theta)*sin(Psi)
1720 gosub 1000
1730 'note:similarly one can find e.g. when sun is due west, etc.
1740 if Nroots>0 then 1780
1750 print "No dawn or dusk that day -- sun is all day
";
1760 if sin(Psi)*cos(Alpha-Theta)>0 then print
"shining" else print "hidden"
1770 goto 1920
1780 Dawn=(Root1-Noon)*12/#pi+12:Dawn=Dawn-24*int(Dawn/24)
1790 Dusk=(Root2-Noon)*12/#pi+12:Dusk=Dusk-24*int(Dusk/24)
1800 if Nroots=2 then 1830
1810 print "sun touches horizon just at";:X=Dawn:gosub
1310:'nroots=1
1820 print ".":goto 1880
1830 if Dawn>Dusk then X=Dawn:Dawn=Dusk:Dusk=X:Root1=Root2
1840 print "This day has";using(3,2),Dusk-Dawn;"
hours of daylight: ";
1850 print "Dawn at ";:X=Dawn:gosub 1310
1860 print ", dusk at ";:X=Dusk:gosub 1310
1870 print "."
1880 Phi=Root1:gosub 740
1890 print "Sunrise at bearing
";using(3,2),Ang1*180/#pi;" degrees."
1900 Phi=Noon:gosub 740
1910 print "High sun at angle
";using(3,2),Ang2*180/#pi;" degrees"
1920 gosub 1170
1930 print "Energy collected is
";using(4,2),Total;" percent maximal"
1940 return
1950 '
1960 'show data for the year
1970 '
1980 print
1990 print "Observations about this place - energy collection
per day"
2000 print
"*********************************************************"
2010 print "Solistices:";
2020 Days=0:gosub 630:gosub 1170:print using(4,2),Total;" %
maximal"
2030 print "June equinox:";
2040 Days=91:gosub 630:gosub 1170:print using(4,2),Total;" %
maximal"
2050 print "December equinox:";
2060 Days=273:gosub 630:gosub 1170:print using(4,2),Total;" %
maximal"
2070 return |
This information will prove
to be valuable to those of the who prefer to aim a solar collector based
on predicted solar position rather than an active sensor feedback system.
While an active sensor system may prove to be slightly more accurate, a
purely predictive control mechanism will provide more than sufficient
results for high efficiency solar heat conversion. The method you utilize
is strictly up to you. That wraps up our focus on the collector end of
things for this year. It's already September up here in the north country
and snow is just around the corner. Next month we are going to take a
look at the new turbine we have designed and have begun building for
several of our club members. Until then, keep your projects active and
moving forward -- let us and the world know how you are progressing.
Ken Rieli
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