by Dr. Małgorzata Marciniak
Małgorzata Marciniak received a broad education in pure and applied mathematics. She completed her PhD from the Missouri University of Science and Technology after receiving a Masters Degree from the University of Warsaw, Poland. Presently, Dr. Marciniak is interested in theories of creativity and in mentoring math projects for students of engineering and computer science.
My journey into the topic of solar energy began during Spring 2016 when two students from my Calculus 3 class, Delfino Enriquez-Torres and Maria Ignacia Serey-Roman, sparked my interest in the question of efficient geometry of flexible panels. We immediately started searching for a mathematical model that expresses such efficiency of geometrical shapes, but we failed to find an advanced mathematical model. All that we found was a simplified model that used an approximation \(\sin\alpha \approx \alpha \) for \(\alpha\) sufficiently small, which I found to be quite dissatisfying. Thus, we proceeded together with my students to build a model based on a Calculus 3 concept, which is called a mathematical flux. To be more precise, the project uses a modified version of the flux density (which is the ratio of the flux to the area of the surface of the panel) The flow through the surface \(S\) at time \(t\) can be represented by the integral \(\iint_S \vec{F}(t)\cdot d\vec{S}\), where \(\vec{F}(t)\) describes the vector field that expresses the position of the sun relative to the center of the earth at time \(t\). Thus, the Daily Energy can be obtained by integrating the previous formula with respect to \(t\), i.e.,
\[\mbox{Daily Energy} = \int \left[\iint_S \vec{F}(t)\cdot d\vec{S}\right] dt = \iiint_E \vec{F}(t)\cdot d\vec{S} dt.\tag{1}\]
The region of integration \(E\) is defined among these values of \(t\) so that \(\vec{F}(t)\cdot\vec{n}>0\). This assumption is necessary since the panels do not lose the energy if the Sun is shining from behind the panel. The Annual Energy is the sum of the Daily Energy calculated for each day throughout the year. The efficiency is determined in terms of the ratio of the Annual Energy to the surface area of the panel.
Numerical values of the annual flux density express the efficiency of particular shapes, their positions or their segments. High values of the flux density characterize efficient shapes, however the information about the efficiency of alternative shapes is valuable since the circumstances (for instance the shape of the roof) may prevent from using the most efficient shapes.
The formulas for the simulation are set up with care and attention but they must be validated by multiple approaches. The first attempt is made by hand calculations, but more complicated shapes may not be computed that way. Often a correct syntax runs for a long time without providing numerical answers due to software and computer limitations. Then the algorithms must be optimized to make the process more efficient. When the software syntax is developed, verified and provides numerical answers, the entire procedure needs to run a few times, preferably on different computers and with different programs.
After running such a simulation and receiving results for three locations of the Earth: the North Pole, the Equator and New York City, we developed a taste for experimenting with locations in outer space. Additional motivation came from our collaboration with NASA-GISS Summer Outreach Program and the NASA interns that I was mentoring.
The very first outer space location that sparked our interest was the location of geostationary satellites. These satellites remain above a fixed location of the Equator. The model was very similar to the one prepared for the Equator, but the time parameter was significantly different due to a longer exposure of such locations to the sunlight. In general, if the satellite is located at altitude \(h\) on a planet with radius \(r\) then its exposure to the sunlight remains within the following range of time \(t\) (in radians):
\[-\frac{\pi}{2}+\sin^{-1}\left(\frac{r}{r+h}\right)\lt t\lt\frac{3\pi}{2}-\sin^{-1}\left(\frac{r}{r+h}\right).\tag{2}\]
Plugging, for example, \(r=6,378\) km and \(h=35,786\) km one obtains
\[-1.419\lt t\lt 4.561\tag{3}\]
making a satellite day roughly \(5.98\,rad\) long. Just to recall, the entire revolution is \(2\pi\approx6.28\,rad\). Thus, the satellite experiences
\[t \approx \frac{5.98}{6.28}\times 100\% = 95.17\%\tag{4}\]
of the day filled with sunlight.
The second outer space location interesting to us was Mars. We analyzed the optical data and simulated the actual weather to be able to create a statistical model. We were expecting and were able to prove that due to the sandstorms, the efficiency of the panels on Mars were about 50% of these on Earth.
The third location we worked with, was the Earth's satellite, the Moon. That model was significantly different and way more challenging due to Moon’s additional movement called "libration". The Moon’s wobbling and nodding creates a quite unusual path of the Sun in the Moon’s sky, which motivated a new approach and resulted in us using the concept of the etendue rather than the flux.
Etendue quantifies the relationship between a light source (surface \( S \) parameterized by \( \vec{S}(\theta,\phi) \) with normal vector \( \vec{N} \)) and a light collector (diaphragm, surface \( r \) parameterized by \( \vec{r}(x,y) \) with normal vector \( \vec{n}\)). To be precise, the etendue measures the agreement between the directions of the normal vectors (\(\vec{N}\) and \(\vec{n}\)) from the two surfaces and is defined as a double surface integral of the dot product between the two normal vectors:
\[\mbox{Etendue} = \iint_S \iint_r (\vec{N}\cdot\vec{n})(d\vec{r}\cdot d\vec{S}).\tag{5}\]
Since the solar panel has only capacity of absorbing the light, the integration does not take place over the entire surfaces \(S\) and \(r\) but only for the range of the parameters, where \(\vec{N}\cdot\vec{n}\gt 0.\)
In our model the solar panel is the diaphragm and a selected segment of the sky is the light source. As in the previous model, efficiency is a unit less value and its meaning is defined through comparison among the obtained results. A single value of efficiency for a panel is virtually useless, but with a collection of calculated efficiencies we can compare and contrast using ratio analysis.
The project is far from being complete. We did not analyze multiple locations on Earth, only the North Pole, the Equator and NYC. Similarly, on the Moon, we only provided calculation for the North Pole, and the equator. Other locations are untouched and waiting for skillful researchers to discover.