Improving solar power when the sun is low in the sky

Tests on my solar-powered Stirling engine determine it will run with a solar incidence angle of up to 59 degrees, but I would like the engine to work up to 66 degrees incidence or 24 degrees above the horizon. This number is selected because that is the angle that would allow the engine to run for 4 hours around noon on the shortest day of the year in Fremont, CA.

You don’t have to actually measure these angles, you can determine the solar elevation for a specific time anywhere on earth if you have the latitude and longitude using the NOAA solar calculator. All I have to measure is the time when the engine starts and stops.

Figure 1: The heat collector section of my Stirling engine schematically looks like this:

The Stirling engine displacer operates vertically just below the heat collector.

The ideal solution to improve solar power at high incidence angles would be to just aim the heat collector at the sun. This is not an attractive solution on my engine for two reasons:

First, this requires the complexity of either the engine on gimbals with solar tracking or at least a mirror that tracks the sun. This engine is meant to stay in a fixed position.

Secondly, especially in the case of a low-power engine, the friction is considerably higher with side-loads on piston and displacer which result when the engine is not oriented vertically. This friction would further reduce the operating envelope and decrease the life of an engine that is meant to run every day for years.

To come up with a satisfactory solution to this problem I wanted to first analyze the input solar power I was getting at 59 degrees incidence because that is the minimum the engine requires to run. With that information I want to come up with a design that will provide at least the same input solar power at 66 degrees incidence.

I should point out that a transparent cover of some type over the heat collector is absolutely necessary on this type of engine to minimize convection losses.

Analysis of the problem

There are three basic ways I’m losing solar energy before it arrives at the heat collector:

1. Transmission losses through the acrylic cover:

Energy is lost by both reflection and absorption of the acrylic. Using a light sensor, light, and both glass and acrylic cover plates I measured the transmission levels in both the visible and IR portions of the spectrum. I didn’t cover the full sunlight energy spectrum so these measurements should not be relied upon too heavily. Here’s what it looks like:

The glass and acrylic responded very similarly within the limits of my measurement accuracy. You can see the cover plate only transmits about 90% of the light even at zero incidence and is flat out to about 45 degrees incidence. By 59 degrees the transmission is down to about 80% and by 66 degrees it is down to about 74%. It’s obvious that you’d like to keep the incidence angle of light on the transparent cover at or below 45 degrees.

2. Reduced energy on the heat collector as a function of the sun’s incidence angle.

This coefficient is simply cosine of the incidence angle on the heat collector plate. The more nearly the collector is perpendicular to the sun the more energy you get.

3. Transmission losses as a function of the solar path through the atmosphere.

Wikipedia has an article that discusses and gives formulas for energy losses through the atmosphere.

I learned that it is not accurate to think the attenuation is proportional to the distance through the atmosphere, it’s actually better. See the article for more details including attenuation due to air pollution. I’m assuming here a basically clear sky and minimal air pollution.

The pertinent equations for my application are:
The air mass length (AM) is approximately
AM = 1/ cos z where z is the incidence angle.
This is essentially how much atmosphere the light goes through.

I = 1.1 * I0 * 0.7(AM.678)
I0 = 1353 w/m2 solar intensity just above the atmosphere
I = solar intensity in w/m2 on the ground at sea level.
When z = 0 degrees I = 1041 w/m2. I’ll use this value as 100% of maximum solar intensity.

Here’s a plot of the solar intensity vs incidence angle on a clear day at sea level:

Combining all of the above information the percentage of maximum solar intensity on the heat collector should be:

I% = 100 * acrylic cover transmission coefficient *
cos(incidence angle) * 1.1 * I0 * 0.7(AM.678)

Reference Incidence angle, deg Acrylic transmission coefficient Energy density coefficient on collector surface Transmission of solar energy through atmosphere Maximum solar intensity, % (1041w/m2 = 100%)
figure 1 0 0.9 1 1 90%
figure 1 59 0.8 0.515 0.82 34%
figure 1 66 0.74 0.407 0.74 22%
figure 2 66 0.9 (45 degree incidence) 0.407 0.74 27%
figures 2,3,4 66 0.9 (45 degree incidence) 0.707 (45 degree incidence) 0.74 47%
figure 1 facing directly at sun 66 0.9 (0 degree incidence) 1.0 (0 degree incidence) 0.74 67%

According to the above analysis I need at least 34% of the maximum solar intensity for the engine to run.

I can’t change the solar intensity losses caused by the atmosphere but I can work on the other two sources of loss.

Just sloping the acrylic cover so the incidence angle is 45 degrees or less will increase the maximum solar intensity at the collector (27%), but not enough for the engine to run. Figure 2 shows the design.


Figure 2: Sloping the acrylic cover and adding a reflective interior

One easy way to improve the energy density at the heat collector is to add the reflective vertical surface as shown in figure 2. This will cause up to double the energy falling on the heat collector when the sun is at low angles.

It’s easy to visualize this solar concentration. If you view the collector from the same angle as the sun at low elevation, you will see both the heat collector disk and also a reflected version of it on the reflective surface. When the sun elevation is higher, only a portion of the heat collector will be visible in the reflection. This vertical mirror acts to amplify the sunlight at low elevation angles (high incidence angles)
— just what is needed. It’s also underneath the acrylic cover so it won’t collect dirt.

I’ve tested the design shown in figure 3 with good results up to the target incidence angle of 66 degrees using just aluminum foil as the reflector. The face of the acrylic cover was at an angle of approximately 23 degrees.

Figure 3: Modified collector cover

Another way to increase the solar energy collected is shown in figure 4.


Figure 4: Collector with sloping face.

This design uses a collector plate with a sloped face. The collector could be made as a wedge-shaped section of an aluminum cylinder. This change requires a good heat insulating barrier around the wedge-shaped section to avoid high thermal losses. The larger heat collector could function as a heat reservoir too, although it would take longer to heat up before the engine could start in the morning.