Solar Energy

Directionality of Sunlight

Because the Sun is so small in the sky, the sunlight that reaches Earth has an almost uniform direction: its orientation is the same as the vector pointing from the Sun to the Earth. However, due to the Earth’s orbit, tilt and rotation, the Sun’s apparent position in the sky is always changing.

In this quiz we will explore how we can design a solar collector to collect as much sunlight as possible, despite changing incidence angles throughout the day and year. We'll find that these design considerations can improve the performance of all solar collectors, and are especially important for systems which use concentrating optics to increase the intensity of sunlight.

Directionality of Sunlight

If the Sun is perfectly overhead in the sky (corresponding to a solar altitude of \(90^\circ\)), the intensity on a horizontal solar collector is the same as the total solar intensity normal to the Sun.

If the Sun is at a lower altitude, the projected area of a horizontal solar collector becomes smaller with respect to the Sun, so it collects less energy. What is the intensity (in \(\si[per-mode=symbol]{\watt\per\meter\squared}\)) reaching a horizontal solar collector when the solar altitude is \(45^\circ\)?

Use \(\SI[per-mode=symbol]{1000}{\watt\per\meter\squared}\) as the solar intensity normal to the Sun.

Directionality of Sunlight

Ideally, we'd like our collector to be oriented so that its surface is normal to the Sun, which maximizes the incident intensity and therefore the collected energy. However, since the Sun's position in the sky is always changing, it's not possible to have a stationary collector always point towards the Sun.

Suppose we are installing a stationary solar collector at a latitude of \(40^{\circ}\) N. This refers to a location North of the equator, such that the angle formed between a line from the center of the Earth to the location and a line from the center of the Earth to the equator would form a \(40^{\circ}\) angle. How should we orient our collector to maximize the incident solar energy over the course of a year?


                   

Directionality of Sunlight

An equinox occurs when Earth's axis of rotation is perpendicular to the vector from the Earth to the Sun. We have two equinoxes per year: the equinox in March marks the beginning of spring and the equinox in September marks the beginning of fall (for the Northern hemisphere. In the Southern hemisphere, these seasons are swapped).

During an equinox, sunlight hits the equator (with a latitude of \(0^{\circ}\)) at normal incidence, while the incidence angle of sunlight reaching a higher or lower latitude will be equal to the latitude angle.

You install a \(\SI{1}{\meter\squared}\) collector at a latitude of \(40^{\circ}\) N, oriented such that it is tilted \(40^{\circ}\) South.

How much energy does it collect over the course of the equinox, in \(\si[per-mode=symbol]{\kilo\watt\hour\per\meter\squared}?\)


Recall that the equation we derived for available daily solar resource is \(I_\textrm{surf} \times \cos(\phi)/\pi\), where \(\phi\) is the location's latitude, and \(I_\textrm{surf}\) is the solar intensity at the Earth's surface. For this problem, you can use \(I_\textrm{surf} = \SI[per-mode=symbol]{24}{\kilo\watt\hour\per\meter\squared\per\day}\).

We derived this equation by taking the projected area of a certain latitude (the area intercepting sunlight) and dividing by the total surface area of that latitude. This accounts for daily averaging, addressing the fact that the sun will not be up for all \(\SI{24}{\hour}\) of a day.

Directionality of Sunlight

When sunlight reach the Earth's surface with a large incidence angle, it reduces the solar resource, since the same amount of sunlight is spread over a larger surface area compared to the case of normal incidence. This is why daily solar resource (\(I_\textrm{surf} \times \cos(\phi)/\pi\)) decreases at higher latitudes - high latitudes correspond to large solar incidence angles.

The reduced solar resource at higher latitudes can be addressed by tilting solar collectors towards the equator. In the northern hemisphere, collectors can be tilted south and in the southern hemisphere, collectors can be tilted north. By tilting a collector in this way, sunlight no longer has a large incidence angle on the collector. For example, if we place a collector at a location with a latitude of \(40^{\circ}\) N and tilt the collector \(40^{\circ}\) south, then the incident sunlight the collector sees would be the same as if it was located at the equator.

It's worth noting that while you can increase the effective available solar resource per collector area by tilting it, you cannot change the effective available solar resource per land area. This is because a tilted collector will shadow some of the land area behind it, making it unusable by other collectors.

Directionality of Sunlight

You decide that the \(\SI{1}{\meter\squared}\) stationary collector at \(40^{\circ}\) N doesn't collect enough energy, so you decide to give it an upgrade: you raise the collector onto a tracking system that can rotate the collector so it always points towards the Sun

How much energy would the new and improved system be able to collect on the equinox (the day which is exactly \(\SI{12}{\hour}\) long, at all latitudes)? Answer in units of \(\si{\kilo\watt\hour}\).

Assume constant atmospheric absorption so the solar intensity is always \(\SI[per-mode=symbol]{1000}{\watt\per\meter\squared} = \SI[per-mode=symbol]{24}{\kilo\watt\hour\per\meter\squared\per\day}\).

Directionality of Sunlight

Tracking the Sun allows us to collect more solar energy with the same amount of collector area. This is because a collector facing towards the Sun always sees the direct solar intensity, rather than a reduced value.

Tracking is also required for concentrating solar intensity at large concentration ratios, which we learned about in the previous quiz. Because concentrating optics only accept incident radiation within their input angle \(\theta_\textrm{in}\), the concentrator inlet always needs to be pointed towards the Sun. This is done by rotating the concentrating optics so that they always face towards the Sun.

Directionality of Sunlight

After you mount your solar collector on a tracking frame, you realize you can add a concentrator to increase the intensity of sunlight incident on your solar collector.

Suppose your tracker has a precision of \(1^\circ\), which means that if you program it to point at the center of the Sun, it will point within \(1^\circ\) of the center of the Sun.

What is the maximum achievable concentration ratio of concentrating optics used in conjunction with that tracker?

Recall the equation for conservation of etendue: \(A_\textrm{out} \sin^2(\theta_\textrm{out}) = A_\textrm{in}\sin^2(\theta_\textrm{in}),\) and that the half angle of the Sun is \(0.27^{\circ}\).

Directionality of Sunlight

Next you want to try using a long, curved mirror to focus sunlight on your collector. This type of concentrator can be paired with a single-axis tracker (which only has one axis of rotation), which is simpler and cheaper than a full tracker.

If your single-axis tracker has a precision of \(1^{\circ}\), and the surface of the curved mirror has surface roughness which leads to reflections being up to \(0.5^\circ\) off from the intended direction, what is the maximum concentration ratio that can be achieved?

For a single-axis system, conservation of etendue is given by \(A_\textrm{out}\sin(\theta_\textrm{out}) = A_\textrm{in}\sin(\theta_\textrm{in})\), and remember that the half angle of the sun is \(0.27^{\circ}\).

Directionality of Sunlight

Tracking can increase the amount of energy collected by a solar collector, and tracking is necessary for using concentration, since the Sun must stay within the acceptance angle of the concentrator. Imprecision in solar tracking and optical surfaces can severely limit the achievable concentration ratio, and these sources of imprecision are why we can never achieve the maximum solar concentration ratio in real systems.

However, it is expensive to build very precise trackers and optics, which limits their use in solar energy systems. Instead, it is most common to see moderately precise trackers and optics, with accuracies of around \(0.5^{\circ}\).

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