Maths of Solar Panels - Positioning Solar Panels on … of Solar Panels 1 of 16 Maths of Solar...

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Maths of Solar Panels - Positioning Solar Panels on Buildings

Introduction Have you ever wondered why there are more daylight hours in summer than in winter? Do you remember what causes the seasons? Do you think there might be any practical need to know the answers to questions like these in real life? This activity and the other activities associated with it in this package, explore these questions and show how the answers are used in building houses and producing electricity from solar panels.

Terminology You will need to know the meanings of the following terms in order to make sense of this activity. By now you should have come across them in you study of Mathematics, Science and Geography. From your notes or a dictionary write down the meanings of each one.

True Bearings

Horizontal

Vertical

Angle of Elevation

Orientation

Direction

Angle of Inclination

Earth’s Axis

Check with others in your class to make sure you agree on the definitions of each of the terms and that all of you understand what they mean.

Inquiry questions Before starting the investigation make some predictions about each of the following: How does the position of the Sun (elevation and bearing) change through the day?

Elevation:

Bearing:

At what time of the day is the Sun at its highest point (maximum elevation) in the sky?

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How do you think this changes throughout the year? (Remember to allow for daylight savings.)

Make predictions about the best way to position solar panels What is the best orientation (direction and angle of inclination) for fixed solar panels where you live? You can use Google Earth or Maps to help with this. Find your house and zoom in. Set the image so that North is at the top of the picture. Make a quick sketch of the orientation of your house and then suggest which part of the roof would be best to put the panels on. Make sure you give a reason.

Which part of the roof:

Angle of inclination: (Choose one, or if you think the best angle is between two of the ones shown, add it in.)

Risks You must not look directly at the Sun. Give reasons why.



Background information The position of the Sun changes through the day as the earth rotates on its axis. It rises in the East each morning, and sets in the west every night. The path that the Sun appears to follow across the sky also changes throughout the year. This is caused by the tilt of the Earth on its axis. In winter, the Sun is lower in the sky and further North (in the southern hemisphere) while in summer it is higher. Your distance North or South of the equator determines exactly how high the sun will reach.



Practice The diagram below is a compass laid flat on the ground with 0o pointing North. Use the corresponding letters (a, b, c, d and e) to mark the lines that have the following bearings: a) 092o

b) 243o

c) 117o

d) 291o e) 000o



Activity Find the angle of elevation and the bearing of the Sun.

What you need

30 cm ruler

Bearing Alignment Circle Page. Download at http://www.stelr.org.au/maths-of-solar-panels/

Magnetic compass (either a physical one or a phone app are fine. Be careful not to stand near metal or electrical wires when doing this activity as both can pull the compass needle away from the correct position of North.)

Solar cell (from the STELR Renewable Energy Kit)

Multimeter (from the STELR Renewable Energy Kit)

What to do 1. Place the Bearing Alignment Circle on top of a book or a

folder so that it rests horizontally.

2. Place the compass on top of the Alignment Circle. 3. Turn the Alignment Circle until North on the paper is the

same as the needle on the compass. It should look like this: 4. Check with your teacher if you need to make a further

adjustment for true North. In most cases it should not matter but it is good to check.

5. Now take the compass away and place the ruler In the middle of the circle standing on

one end.

6. Turn the ruler about its vertical axis until the edges of the shadow are at right angles to the ruler. You have just lined up the bearing of the sun. You can read the bearing by looking at the angle value directly opposite the shadow.

7. Use the following diagram to practice on.

The bearing of the example is:

Now record your actual result The bearing of the sun is:



Now you are ready to line up the solar cell with the sun.

8. Connect the STELR solar panel to the STELR multimeter as shown in the photograph to the right. Connect the multimeter to one cell only.

Take your solar panel, ruler and Alignment Circle and set them up as in the diagram.

Lean your solar panel against the ruler and change the angle until the voltage reaches a maximum. Note the value on the multimeter.

The angle you have found is called the pitch angle.

9. Calculating the pitch angle.

Measure the height of the solar cell from the ruler. Remember to allow for the bottom part of the ruler where there are no markings.

Measure the horizontal distance from the base of the ruler to the solar cell.

Now use trigonometry to calculate the Pitch angle.

10. The angle of elevation for the sun is: 90o minus Pitch angle

11. Challenge: Prove the formula for angle of elevation given above.



Now record your all of your results in the space below Time: Maximum voltage: Bearing: Angle of elevation:

Discussion How did you make sure the ruler was vertical? How can you use these results to find the best orientation (direction and angle of slope) for fixed solar panels where you live? Describe how the ‘best’ angle of slope is likely to change depending on:

a. the time of year

b. your distance from the equator.

In this exercise you used a small solar cell and moved it up and down to align it properly.

Give two reasons why this is not a feasible approach when installing full size solar cells on

the roofs of houses and other buildings.

1

2



Extension (Level 10A) This activity is for those of you who want to learn more about the sun appears to move through the sky for different places on Earth’s surface.

Background Information The position where the sun rises and sets seems to change during the year due to the

Earth’s axis of rotation being tilted. This also means that the position of shadows at local

noon will change from day to day.

If you go outside at local noon in the middle of summer, autumn, winter and spring: 1. Where will the sun be?

2. How long will your shadow be?

Hints: In the middle of spring and autumn, people standing on the Equator have the sun directly overhead and their shadow is underneath them. In July, this is true for people standing on the Tropic of Cancer in the Northern hemisphere. In December it is true for people standing on the Tropic of Capricorn. 3. In Australia, Rockhampton and Alice Springs are just about on the Tropic of Capricorn.

What can you conclude about the position of the sun for people living south of these two towns?

Local noon is when the sun reaches the highest point in the sky. Brisbane, Sydney, Canberra, Melbourne and Hobart are all on Eastern Standard Time so 12 pm occurs at the same time in all of these cities. 4. What is the difference between 12 pm and local noon?



Activity

The position of the sun in the sky changes from day to day. It moves North and South throughout a year and produces a curve like a sine wave. An equation that models this motion and the graph that is plotted from it are given below.

Remember that the equation of a sine function is written in the form:

f(x) = A sin(n(x – h) + k where: A stands for amplitude n is a dilation or stretching factor and h and k move the curve along the x and y axes respectively. 1. Using a graphing calculator or graphing software, put in the equation and draw the graph

of the curve for x between 0 and 365. Does your curve match the one on this page?

2. On 21 March the sun is directly over the equator. 21 March is the 80th day of the year. a. Let x = 80 and find the value of y.

b. Now let x = 80.116 and find the value of y.

c. Explain what the difference in your answers means in terms of time and position.

3. The sun is again directly over the equator on 23 September. Work out what day of the

year this corresponds to and check that this value gives a value of zero or very close to it.



4. On what day of the year is the sun directly over the Tropic of Cancer?

5. On what day of the year is the sun directly over the Tropic of Capricorn?

6. What are the y values for these two days?

7. These values are angles measured in degrees. What do they tell us about the Earth?

Discussion Let us now try to relate these answers back to the task of placing solar panels on the roof of a building. 1. The best way to position a solar panel is so that the surface is perpendicular to the sun’s

rays. This is not possible in a practical sense. Explain why?

2. For someone on the equator, what is the best pitch angle to use when positioning solar

panels? Explain why.

3. For someone on the Tropics of Cancer and Capricorn, what is the best pitch angle to use

when positioning solar panels? (Hint: you calculated the values a few minutes ago.) Explain why.

4. The answers to question 3) are the same value, but the solar panels would not point in

the same direction? Why?

Research These angles that you have been working with are collectively known as Latitudes. You may have come across this term before, but they are not strictly mentioned in the curriculum until Year 11 in Mathematics. Find out what Latitude is and discuss with your class how it is connected to placing solar panels on roofs.



Notes for the teacher 1. Difference Between True North

and Magnetic North

True North and Magnetic North are

not the same thing. To convert from

Magnetic North to True North take

the compass value and rotate anti-

clockwise for all capital cities except

Perth. For Perth, rotate clock wise.

The size of the angle you need to

rotate by is given on the map for

each of the capital cities in Australia.

2. Circuit Setup

The solar panel should to be connected to the multimeter as shown in the picture below.

Start with the dial on the multimeter in the ‘OFF’ position. The black plug connects the

central COMM socket on the multimeter to the black (negative) socket on any one of the

four cells (labelled 1 to 4) In this picture we have connected to cell number 1. The red

plug connects the right-hand socket on the multimeter (labelled VΩmA) to the red

(positive) socket of the same cell.



When ready to commence measurements, turn the dial

anticlockwise to the 2V position as in the photo (right).

3. Testing solar cells

A students use only one of the cells in the STELR solar panel,

they may like to test all four cells one at a time to see which one

is most efficient, and then use it. Students may also like to

speculate on why one cell might be more efficient than the

others. Students can adjust the voltmeter sensitivity depending

on the sunlight, however the 2V setting is recommended.

Students will be able to get a reasonable set of answers even

on a cloudy day, however it is recommended to do this activity in

full sunlight if possible.

4. Positioning solar panels on a cloudy day

On a cloudy day, students can get a reasonable set of results even when they cannot

see a shadow.

It is possible to determine both bearing and pitch angles just using the solar cell and the

voltmeter. Start by holding the cell perpendicular to the ground and swivelling it around

the points of the compass. When the voltmeter registers a maximum, you have

established the bearing. Next, start to swivel the cell vertically as described in the main

body of the task instructions.

5. Websites

Here are a couple of good website animations for helping students visualise the path of

the sun through the sky.

SunCalc at http://suncalc.net

University of Nebraska – Lincoln Astronomy Department at

http://astro.unl.edu/naap/motion3/animations/sunmotions.html

6. Latitude and Longitude

The Geography curriculum mentions the use of Spatial Technologies in each year from 5

to 10. Implicit in this is the assumption that students would come across the terms

latitude and longitude before senior school. There is, however, no specific mention of

them and the mathematics underlying them before year 11. The current task has been

written so as not to have to go into the details of the terms, but if teachers feel

comfortable going into some depth with the ideas with their own students, this would be

encouraged.


http://suncalc.net/

http://astro.unl.edu/naap/motion3/animations/sunmotions.html


7. Trigonometric Functions and Year 10 A

The 10A Mathematics curriculum includes content on Trigonometric functions. The

extension section for this activity has been written with this in mind.

For teachers who are new to the course and perhaps are not familiar with how sine

functions and their graphs are connected the following script is given as a guide when

introducing the extension material.

The graph of sin(x) looks like a wave.

If the units for x are in degrees, then one cycle

will be between 0 and 360.

The amplitude of the curve is 1.

Changing the amplitude is achieved by

multiplying at the front of the sin(x)

So 23.44 sin(x) will stretch the curve vertically.

The length of a year is 365 days, so the next

thing we need to do is change the period of the

function. This is done by multiplying x inside the

brackets by 0.9657.

(Note that this is NOT 360

365 . A sine function is

only a model for the sun’s position throughout

the year and the value of 0.9657 gives a closer

match to the equinoxes and solstices.)

Finally we have to shift the curve along to the

right. The March Equinox is when the curve cuts

the x-axis. We do this by subtracting 80.116

from x and putting brackets around this part of

the expression.

Note: the equinox is not just a day but a precise

moment in time, hence the decimal value

instead of a whole number)



8. Tilt of Earth’s Axis and the Seasons

The Earth is tilted on its axis by an angle of 23.44o. The axis, marked red in the

diagrams, always points toward the same direction in the sky.

As the Earth orbits the Sun during the year it seems to us that the Sun moves North and South of the equator. In March and September the axis is side-on to the Sun and the Sun can be seen in the sky from both the North and South Poles. In June, the North Pole points toward the Sun and the South Pole is in darkness for months. In December, the reverse is true, with the South Pole facing the Sun and the North Pole in darkness. (Diagrams adapted from Google Earth screen shots.)

On the next page is another representation you may prefer to use with students.



Australian Curriculum References

Science (Year 10) Use and influence of science Advances in science and emerging sciences and technologies can significantly affect people’s lives, including generating new career opportunities. (ACSHE195) Physical sciences Energy conservation in a system can be explained by describing energy transfers and transformations. (ACSSU190)

Geography (Year 10) Represent the spatial distribution of geographical phenomena by constructing special purpose maps that conform to cartographic conventions, using spatial technologies as appropriate. (ACHGS075)

Mathematics (Year 10) Formulate proofs involving congruent triangles and angle properties. (ACMMG243) Apply logical reasoning, including the use of congruence and similarity, to proofs and numerical exercises involving plane shapes. Present reasoned arguments and apply an understanding of relationships to deduce properties of geometric figures. (ACMMG244) Solve right-angled triangle problems including those involving direction and angles of elevation and depression. (ACMMG245)

Mathematics (Year 10A) Use the unit circle to define trigonometric functions, and graph them with and without the use of digital technologies. (ACMMG275)

Senior Curriculum

Year 11-12 Essential Mathematics Time and motion: interpret complex timetables, such as tide charts, sunrise charts and moon phases .(ACMEM081) Earth geometry and time zones: locate positions on Earth’s surface given latitude and longitude using GPS, a globe, an atlas, and digital technologies. (ACMEM159)

Year 11 General Mathematics Applications of trigonometry: solve practical problems involving the trigonometry of right-angled and non-right-angled triangles, including problems involving angles of elevation and depression and the use of bearings in navigation. (ACMGM037)

Year 11 Mathematical Methods Trigonometric functions: Identify contexts suitable for modelling by trigonometric functions and use them to solve practical problems. (ACMMM042)


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