Heliostats have a potential of being the truly lowest cost solution to all the power one needs. These systems can provide for domestic heating, electricity, and lighting.

The realization that all flat plate solar collectors have lots of expensive collector area and still deliver only low grade temperatures compelled me to look for a better solution.

Many years ago I noticed that cheap glass mirror tiles cost less than $1.00/ft² ($.59 at the time). Flat plate collectors generally need two layers of transparent insulation that each cost more than $2.00/ft². Flat plate collectors also need a layer of insulation on the back side. Flat plate collectors usually are not optimally oriented to the sun.

Lets see; can I think of a system that uses lots of cheap mirrors and a small expensive collector?

"AHA! A heliostat."

You may consider that all the light from the sun is parallel, because it is so far away. Each mirror is initially adjusted so they each reflects light onto the same spot. When the sun moves in any direction all mirrors are adjusted by exactly the same amount to track the sun and they still reflect light onto the same spot. This is mathematically exact and works anywhere in the world when reflecting on any spot. This also works with any arbitrary arrangement of mirrors. (I won't discuss the proof here just trust me for now.)

I have been thinking about these concepts since 1972. The germ of the idea occurred during one of the energy crisis. I was thinking about tracking thermal collectors then the "AHA!"thought popped into my head. The thought that mirrors are much cheaper than flat plate collectors was very powerful. The secondary thought the actual collector can be stationary and on the ground was a plus.

Hey, this ain't so hard.

Heliostats are mirrors that track the sun and reflect the sunlight onto a central receiving point. Most heliostat solar power systems consist of arrays of heliostats.

If the mirrors are inexpensive enough then they can collect solar energy at a cost that is less expensive and with higher efficiency than flat plate collectors. The arrays of heliostats concentrate the energy on a relatively small collection point that, though more expensive per ft² than flat plate collectors, are smaller and thus the total cost is lower.

If used for the collection of heat the temperatures can be quite high, over 600°F. If used to generate electricity the cost of the precious solar cells are only a fraction of the cost of cells in flat plate systems.

Note! As of 980306 I have abandoned the idea of using PV cells in concentrator heliostat applications. There are some subtle problems with the cells that I don't know how to overcome. See:
up980301.htm#pv

The heliostats I am describing consist of many small elements. Each of these elements is an individual heliostat. When directed to a common collection point there will be a concentration proportional to the number of elements used. If you use a 200ft² array of 10 elements the concentration factor will be about 10 to 1. With this much concentration factor the relative thermal collection efficiency is about double that of a tracking flat plate collector. If compared to stationary flat plate collectors the relative collection efficiency may be about 4 times better.

I have coined a term, (I think?), called the "Stasis Temperature". This is the temperature a system will attain with no power extracted from the system.

The plane of the collector is normal to the sun at noon on January 8th. This is approximately the day that the earth is closest to the sun. I have chosen winter as the standard time as this is when homes need the most solar energy. The day must be very clear and generally with no wind or in a sheltered area such as a south facing open garage. I have done this measurement in Minnesota. I assume that the temperature would be higher in lower latitudes as the distance traveled through the atmosphere is shorter when the sun is at a higher altitude. (Others have suggested that I do this measurement at the spring or fall equinoxes as this would be a better average throughout the year. I will stick with the January 8th day because it will give me a worst case reading. Besides, winter is the time of the year that heat is needed most)

I have chosen this model as it represents the most general type of collector on the market. Other models can be devised for other purposes. Others may have higher or lower stasis temperatures depending mainly on the quality of the absorber, the transparent insulation R value, or time of year.

Triple air layer insulation is used in higher temperature flat plate collectors where the added insulation value allows higher stasis efficiency and temperatures. If higher useful temperatures are required then the lowered efficiency is acceptable.

All materials have absorption and emissivity factors. Absorption refers to the ability of a material to absorb radiation. Emissivity refers to the ability of a material to emit radiation.

In physics absorption factors can refer to the full spectrum of radiation. We in the solar field are referring to absorption over a specific spectral range of sunlight here on earth. The absorption factor is referenced to a true black body surface which by definition is 1. The radiation that is not absorbed is lost due to reflection.

Emissivity usually refers to the radiation of power from a surface at a certain temperature. The emissivity factor is referenced to a true black body surface which by definition is 1.0. The published emissivity factors, I think, are measured at a temperature of 25°C. The radiated power is normally greater at higher temperatures.

Most solar collectors are operated at elevated temperatures well above that which was used to measure the emissivity factor. The result is that the amount of emitted radiation is higher than expected.

Classically flat black paint, Parsons black, was used on solar collectors. While it is cheap to apply it is not necessarily the best material to use. The black paint has an absorption factor of .98. This means that 98% of the available solar energy is converted into heat in the collector. The collector gets hot. Some of the heat is emitted as radiation. In this case 98% of the possible black body energy is radiated.

Look at it this way. If the emissivity factor is lower we get to keep more of our hard earned energy. Ideally the emissivity should be zero. Of course that's impossible. The best we can do is to look for materials that have high absorption and low emissivity.

This is a table of absorption and emissivity for some solar materials. (?)
(I don't have the citation but it was a good one.)

There is another table from:
Department of Agricultural Engineering, University of Missouri-Columbia. (U of Miss)

A good reference on Radiant-Heat Transfer. (Marks (1))
Marks Standard Handbook for Mechanical Engineers ninth edition
Eugene A. Avallone & Theodore Baumeister III
Page 4-66 to 4-70

* Sandia National Labs, Sun Lab, Environmentally Friendly Solar-Selective Absorber Material is the Result of Laboratory-Industry Collaboration. Thermafin uses Black Crystal coatings.

This is a table of reflectivity and emissivity for some building materials. (Marks (2))
Marks Standard Handbook for Mechanical Engineers ninth edition
Eugene A. Avallone & Theodore Baumeister III
Page 12-85, Table 12.4.21b

Nielsen Enterprises
They sell reflective Mylar films.
Hydroponic Gardening/REFLECTIVE FILMS
3019 S 256th St.
Kent, WA. 98032
(253) 941-7281

Reflectivity and Emissivity table 2.

	%	E
		Average
Surface	Reflectivity	Emissivity	Reference
Aluminum foil, bright	92 - 97	0.05	Marks (2)
Reflective Mylar Film	90 - 93	0.05	Nielsen
Aluminum sheet	80 - 95	0.12	Marks (2)
Plate glass mirrors coated with aluminum on back	85
Aluminum-coated paper, polished	75 - 84	0.20	Marks (2)
Steel, galvanized, bright	70 - 80	0.25	Marks (2)
Rhodium	70 - 80
Stainless Steel	62 - 65
Chromium	60 - 63
Aluminum paint	30 - 70	0.50	Marks (2)
Building materials: wood, paper, glass, masonry, nonmetallic paints	5 - 15	0.90	Marks (2)

Stasis temperature is simply the temperature that the standard model, or other model, collector reaches compared with the ambient temperature. Stasis efficiency is proportional to the inverse of the actual collector temperature rise divided by the potential temperature rise with no power extracted from the collector. When any collector temperature is the same as the ambient temperatures then this is the maximum solar heat flow out of the system. This is the 100% stasis efficiency temperature, (although the temperature is so low as to make it not useful). It stands to reason that at higher temperatures there will be less heat available for transfer to a load as the remainder of the heat is lost through radiation or conducted through the insulation. At some higher temperature, the stasis temperature, there will be no heat available for transfer and all is lost to radiation or conducted through the insulation. This is the 0% stasis efficiency temperature, (although the available heat is so low as to make it not useful for power generation). This temperature may be useful for such tasks as baking or firing ceramics.

I have derived the equation myself from actual measurements that I have made. If I have made an error, such as using dirty glass, have extra unaccounted heat losses, have air leaks, or have un-calibrated thermocouples I will only be off in the stasis temperature value. This error, even if in error by 25%, would not significantly alter the conclusions gained from knowing that concentrator solar collectors nearly always have higher efficiencies than flat plate collectors of any type.

The stasis temperature measurement is based on a standard model called the "Stasis Model". The stasis model is a flat plate collector with a black body type of absorber under 2 layers of borosilicate glass with a spacing of 1/2 inch and spaced 1/2 inch from the absorber. Thus there are two 1/2 inch air spaces in front of the absorber. This is double air layer insulation. There is also 6 inches of polyisocyanurate foam insulation on the back. A thermocouple is attached to the absorber for measuring the stasis temperature and another outside in the air for measuring the ambient temperature.

This equation is not rigorous as it is only the first order linear approximation of the true stasis efficiency equation. I have not needed to find the nonlinear equations yet as this one has served me well.

 /     / (Tc - Ta)      \ \ 
| 1 - | ---------------- | | * 100 = ST% 
 \     \ (Ts - Ta) * CF / /

ST% = Stasis efficiency of a particular theoretical solar collector system.
Tc  = Collector temperature as measured in a particular collector system.
Ta  = Ambient air temperature around the collector in °F.
Ts  = Stasis temperature. I use (Ts - Ta) of 340°F. You could use 188°C.
CF  = Solar Concentration Factor.

Stasis temperature can be converted from °F to °C at will. Stasis efficiency is dimensionless as it's a percentage.

I have found that a flat plate collector normal to the sun with 2 layers of glass and perfect insulation will have a differential temperature rise of about 340°F. If you extract enough heat from the system so that the temperature rise is 170°F then the system is 50% efficient theoretically.

When sun light is concentrated the stasis temperature rise is multiplied by the concentration factor. In my example of 10 to 1 the stasis temperature rise would be 3400°F theoretically. (Of course there are other losses not described here which limits this temperature.) If the extraction temperature is 340°F above the ambient then the system would be about 90% efficient. This has the effect of reducing the required area by about half for the same heat extraction compared to a flat plate collector.

Of course the flat plate collector could never reach the temperatures obtained by the concentrator. They don't even try.

I assume that stasis efficiency is linear from 0% to 100% between stasis temperature and ambient temperature. I think this is reasonably valid at temperatures below 300°F. Stasis efficiency, if taken rigorously, is certainly not linear with high temperatures. I have not needed to calculate stasis efficiencies at temperatures more than 600°F yet.

The stasis efficiency is only the starting point for total efficiency calculations. It describes the theoretical maximum efficiency at a particular temperature that a collector can attain. This is an efficiency goal that cannot be exceeded.

Other efficiencies to be considered are:

1.	Mirror light Reflection efficiency	MR%=90% Dirty mirrors.
2.	Mirror Cosine angle efficiency	MC%=90% Mirrors set at an angle to the light rays.
3.	Mirror Scattering efficiency	MS%=98% The mirrors are not flat.
4.	Solar Light Divergent efficiency	LD%=96% Generally light from the sun diverges 1/2°. If the light is sent a long distance then the concentration factor is reduced as the spot the mirror reflects upon will be larger than the size of the mirror.
5.	Atmospheric Scattering efficiency	AS%=96% Flat plate collectors are pretty good with overcast clouds. The heliostat is not very bad at collecting scattered light.
6.	Receiver Reflection efficiency	RR%=95% Light reflected from the receiver.
7.	Receiver Cosine angle efficiency	RC%=95% Receiver set at an angle to the incident rays.
8.	Receiver Absorption efficiency	RA%=95% Light absorbed by transparent insulation in the receiver.
9.	Non black body radiation absorption efficiency	BB%=150% This can be greater than 100% as selective absorption coatings that absorb short and long wave radiation differently.
10.	Thermal losses in Insulation	(TI%=95% Heat loss through insulation in receiver).
11.	Solar Cell efficiency	(SC%=15% Light to electric conversion losses).

These efficiencies must all be multiplied together along with the stasis efficiency at a particular temperature to get the final total efficiency. Multiply solar influx by the total efficiency.

The rule of thumb for tracked flat plate collectors is about:

1000Watt/m², 900Watt/yd², or 100Watt/ft²

The rule of thumb for concentrators is about:

 800Watt/m², 700Watt/yd², or  80Watt/ft²

Tot%=	ST%*	MR%*	MC%*	MS%*	LD%*	AS%*	RR%*	RC%*	RA%*	BB%*	TI%
Tot%=	95%*	90%*	90%*	98%*	96%*	96%*	95%*	95%*	98%*	130%*	95%

Power out thermal  = 61 Watts / ft² = 80 * 76%      

Power out electric =  9 Watts / ft² = 80 * 76% * 15%

You may think that the heliostats are expensive to make. DOE's Solar I and Solar II, the solar power towers at Barstow California, and others are very expensive systems. They don't have to be low in cost as they were powered not by the sun but by "Government Dollars". For any system to be truly cost effective they need to deliver power at a cost that is less than that delivered by the local energy utilities.In my area this is $.06/KWhour. Another rule of thumb is the installed cost of the system needs to be less than about $1.00/Watt. None of the many systems I have studied come close to this performance.

In Minnesota there is a law that requires that "solar collectors" have a stand alone pay back period of less than 40 years in today's dollars. Those small tracking concentrating trough water heaters that produce about $80 of hot water per year are not permitted to be called "solar collectors" in Minnesota. Without government tax incentive money and research grants none of the systems I have studied can be called "solar collectors" in Minnesota if installed near a utility power source. The energy obtained from the utilities is just too cheap to compete effectively. If a system can produce power at a low enough cost it should be able to stand alone in the marketplace and compete with the utilities on an even playing field.

The heliostats that I have been working on are very low in cost. They are constructed of low cost materials and use no high tech construction methods. I think they are about $5.00/ft² but this is only a guess at this time. The use of a personal computer to compute the orientation of the mirrors is the only high tech component. This computer performs high precision calculations that compensate for all of the mechanical construction errors. El Cheapo 8088 XT class computers are fast enough to compute the orientation math for at least 20 individual heliostats arrays. (I have been using an obsolete Zeos Pocket computer as the controller.)

Note! Gangs of heliostats are linked together in such a way that they are controlled by a pair of calculation. Theoretically, if all heliostats are constructed to close mechanical tolerances such that they all move synchronously only a few calculation are made for them all. As a practical matter smaller sets of arrays are linked. Each set should have its own calculations as there may be orientation errors when widely spaced. All errors are accounted for in my software.

In my prototype I have purposely installed everything crooked to demonstrate the ability of my software to fully compensate for these errors.

Generally I have not seen my simplified descriptions of collector operation using stasis temperatures described in the literature. An understanding of stasis temperature and stasis efficiency can enhance ones understanding of solar collector operation.

I think understanding of solar collector operation has been hampered by solar insolation maps based on solar radiation on the flat ground. Solar collectors don't work this way. Solar collectors are not placed flat on the ground. They are at least raised to an angle that tries to make them normal to the sun. This gives the impression that there is far less solar power in the winter than in the summer. This impression is false as we are closer to the sun in the winter by about 3.3% than in the summer and generally the air is clearer.

Since the power received is the square of the apparent surface area of the sun the actual power received is about 7% more in winter than in summer.

Of course there is less duration of light in the winter day and the light has to pass through more air so the total energy availabe to solar projects will be about 1/2 that of the summer.

Solar Radiation Data Manual for Flat-Plate and Concentrating Collectors.

In general heliostat arrays should use the 2 axis tracking data set. Be aware that heliostat performance is a little less than this data set. Primarily the error is caused by cosine angle errors in the mirrors.

My proposed 200ft² array produces about:

2.2	KW peak electric.
9	KW peak thermal.
36	MBtu/hour peak thermal.
20	KWhours/day electric.
.26	MBtu/day thermal.
81	KWhours/day thermal.

This is a serious amount of energy and power!

Assumptions I used for the folowing calculations.

1.	Storage capacity is about .6 MBtu for a single days heat storage for my house.
2.	When latent heat storage is used I ignored the sensible heat content because I want to keep the temperature near the melting point. (Besides sensible heat data for these materials is hard to find.)

When used for domestic heat a heliostat array can deliver the energy at a high temperature. I have investigated the use of phase change materials that melt at temperatures up to 611°F. This is the melting point of Potassium Nitrate. Latent heat storage is 11968 Btu/cubic foot of heat storage fits in 50 cubic feet. This is dense compared to the massive storage needed for the low grade storage used by flat plate collectors.

Another material is Ferric Chloride III at 20625 Btu/cubic foot is only 29 cubic feet. This may have problems as the chemical dissociates at 599°F.

Another use of heliostats is in the operation of energy efficient greenhouses. The greenhouse I envision would be a heavily insulated all metal pole barn with a relatively small solar window on the north side. Light passes through this window and is distributed to the plants by way of a secondary curved mirror inside the barn. The barn is fully insulated except for the window and has little heat loss. With thermal storage there would be an ample amount of heat for the night time. I estimate that the heliostat should be 75% of the floor area of the greenhouse. Others I have talked to have suggested that plant growth could be accelerated by doubling or tripling the amount of sunlight.

Just think of the savings in fuel over a conventional greenhouse.

(Note! There will still be some burning of a clean carbon based fuel to produce carbon dioxide as a nutrient for the plants.)

This is a serious amount of heat and light.

References

Lighting Fundamentals
LIGHTING UPGRADE MANUAL
US EPA Office of Air and Radiation 6202J
EPA 430-B-95-003, January 1995
U.S. EPA Green Lights Program

Grainger's
1997 Catalog #388

World Products Inc.
Makers of Electro Luminescent Light Sources
Their made by NEC.

Lumitex, Inc.
Tech Note 1-1: Light Energy/Brightness Units and Conversions

Ilumination. (Marks (3))
Marks Standard Handbook for Mechanical Engineers ninth edition
Eugene A. Avallone & Theodore Baumeister III
Page 12-121, 122

<Clare Snyder>
Data received in an email from Clare.

By definition there are 673 lumens in 1 Radiometric Watt at 555nm wavelength.

1 Watt       = 673 lumen

Another reference in German: http://www.biologie.uni-erlangen.de/botanik1/html/body_einheit_1.htm

I calculated the luminous efficiency by:
lumens/Watt / (673 lumens/Watt) * 100%= n Luminous Efficiency at 555nm
This is not a rigorous calculation as one is supposed to integrate at other wavelengths with other conversion factors. For our purposes the 673 lumens/Watt is close enough.

Illuminance of a surface is expressed by footcandles and lux.

1 footcandle = 1 lumen / ft^2
1 lux        = 1 lumen / m^2

When using a home type heliostat some mirror tiles are positioned to reflect light into north facing windows of the house.

Conventional incandescent filament lamps have an electric to visible light conversion of less than 3%.
1ft² of sunlight from a single mirror contains about 80W of light.
Small 1ft² mirror tiles deliver the equivalent light output of:
80W / 3% = 2700W of light from incandescent bulb.

This calculation shows the electric power input of an equivalent light output from an incandescent lamp. The sunlight is pure light. Eventually all the power in the light will be converted to heat, 80W in this case. If incandescent lamps were used the heat load would be 2700W instead of 80W.

If one compares sunlight with halogen or fluorescent lights the equivelant electric power will be less but they can never come close to the efficiency of real sunlight.

Of course, light is light. But some light is nicer than others. The light from the sun is dazzlingly white and bright. It has been reported that some depressions in the winter can be reduced simply by being exposed to sufficiently bright white light. I would think that bright sunlight, the real thing, would be beneficial.

This is a serious amount of light.

Steel Melter.

Use a heliostat to melt metal. A heliostat reflects light on a parabolic mirror reflector for a final concentration factor of about 50 to one. This combination will produce temperatures in excess of 3000°F. At these temperatures the total efficiency will be fairly low. Even so, there is a large amount power available. Melting aluminum is easy and steel is only a little harder with an influx of 5 to 10kW of power.

(I still haven't melted steel yet. Maybe this summer.)

Aluminum Melter.

One could use a heliostat system to melt aluminum beverage cans into aluminum ingots. The melt is cleaner with the pot sealed from the air. There are significant savings of fuel costs. This is a value added enterprise as the ingots are worth more than the raw cans. They are much smaller in size.

Ceramics Kiln.

Use a heliostat to fire ceramics. An array of heliostats reflects light on an insulated box with a window in the bottom. A mirror set at 45° reflects light up and into the box. There should be a large amount of thermal mass inside the box. This mass allows a slow cool down after the sun sets. The box is insulated with vermiculite. This combination will produce temperatures in excess of 2000°F. At these temperatures the total efficiency will be medium. Firing ceramic pottery or ceramic bricks is easy with an influx of 20 to 40KW of power.

Note! I have been told that this doesn't work as the time needed for ceramic firing is much longer than can be reliably obtained is a single solar day. Too bad :(

Resarch Furnace.

Solar research furnaces use a single heliostat to reflect light into a stationary paraboloid with a stationary receiver point. If the paraboloid and heliostat are accurately constructed these systems have concentration factors of about 1000 to 1 or more or higher. They can directly produce high purity plasmas.