This contribution provides in the following the short cuts of some examples of mathematical problems concerning future energy issues. The problems presented here deal with the topic ‘Solar Thermal Energy’. They are suitable for lessons in secondary schools. Our presentation is focused on the basic structure of the problems, in order to give an impression of the didactical concept and the general principles.

1.1 Example 1

> This problem can be treated in lessons to the topic of percent calculation and the rule of three. It requires the understanding and usage of data representations.

In private households the required warm water can be partly heated up by solar thermal collectors. They convert the solar radiation energy in thermal energy. It helps us to decrease the usage of fossil fuels, which lead to environmental problems. Hot process water needs preferably to have in private households about 45°-55°C. In our region, the usable thermal energy from the sun is not sufficient to reach this temperature permanently, because of the seasonal behaviour. Thus, an input of supplementary energy is necessary.

Figure 1: A solar thermal energy plant (DGS)

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The following table shows how much of the needed energy for heating up water to a temperature of 45°C in private households can be covered by solar thermal energy, respectively how much supplementary energy is needed.

a)

Energy Coverage in % Month March (1) to February (12) Figure 2: Usable Solar Energy and Additional Energy

How many percents of the needed thermal energy for one year can be covered by solar thermal energy?

Energy is measured by the unit kWh. An average household in Germany consumes nearly 16.000 kWh thermal energy per year. 1 l fossil oil provides approximately 10 kWh thermal energy. The combustion of 1 l oil produces nearly 68 l CO2.___________________________________________________

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Assume in the following, that the used water is heated up to a temperature of 45°.

b) How many kWh may be covered in one year by solar thermal energy?

c) How many litres of oil have to be bought for the needed supplementary thermal energy for one year for a private household?

d) How many litres oil would be needed without solar thermal energy?

e) How many litres CO2 could be saved by an average household in Germany during one year if using a Solar Collectors?

1.2 Example 2

> This problem deals with linear functions. The understanding and usage of graphical representations is performed.

Energy is measured by the unit kWh. An average household in Germany consumes nearly 16.000 kWh thermal energy per year. 1 l fossil oil provides approximately 10 kWh thermal energy. The combustion of 1 l oil produces nearly 68 l CO2.________________________________

Figure 3: Dimensioning Diagram

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The diagram in figure 3 provides data for planning a solar collector system for a private household. It shows the dependence of the needed collector area on the part of Germany where the house is situated, on the number of persons living in the respective household, on the desired amount of warm water per day and person, as well as the desired coverage of the needed thermal energy by solar thermal energy (in per cents).

Example: In a household in middle Germany with 4 persons and a consumption of 50 l warm water per day for each one, follows for a reservoir of 300 l and an energy coverage of 50%, that a collector area of 4 m[46] is needed.

a) What would be the needed collector area for the household you are living in? Which assumptions do you need to make first? What would be the minimal possible collector area, what the maximal one?

b) On a house in southern Germany there is installed a collector area of 6 tT that provides 50% of the produced thermal energy. How many persons could be supplied in this household with warm water?

c) Describe by the term of a linear function the dependence of the storage capacity on the number of persons in a private household. Assume first a consumption of 50 l warm water per day and person, and second a consumption of 30 l. Compare the two function terms regarding also their graphical representation.

d) Show in a graphical representation the dependence of the collector area on a chosen storage capacity, assuming a thermal energy coverage of 50% for a house in middle Germany.

1.3 Example 3

> This problem can be integrated in lessons to quadratic parabola and uses their focus property.2

Direct solar insolation may be concentrated in a focus by means of parabolic sun collectors. These use the focus property of quadratic parabola. Special sun collectors are figures with rotation symmetry, they evolve by rotation of a quadratic parabola. Their inner surface is covered with a reflective mirror surface, that is why they are named parabolic mirrors. Sun beams may be assumed as being parallel. Thus, if they fall on such a collector, parallel to its rotation axis, the beams are reflected that way, that they all cross the focus of the parabola. The thermal energy insolation may be focused this way in one point. The temperature of a heating medium, which is lead through this point, becomes very high, relatively to the environment. This is used for heating purposes, but also for the production of electric energy.___________________________________________________________

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a)

Figure 4: Parabolic Sun Collectors (DLR)

A parabolic mirror was constructed by rotation of the parabola y = — x2. Determine its focal length (x and y are measured in meter).

b) A parabolic mirror has a focal length of 10 m. Which quadratic parabola was used for its construction?

c) Has the parabolic mirror with y = 0,05×2 a greater or a smaller focal length than that one in b)?

d) A parabolic mirror shall be constructed with a width of 2,40 m and a focal length of 1,25 m. How great is its arch, i. e. how much does the vertex lay deeper than the border?

e)

Figure 5: EuroDish System (DLR Almeria Spain)

In figure 5 you see a parabolic mirror, the EuroDish with a diameter of 8,5 m. Determine out of the figure approximately, neglecting errors resulting from projection sight, its focal length and the describing quadratic parabola.

Other focussing sun collectors are figures with length-symmetry, they evolve by shifting a quadratic parabola along one axis direction. They are named parabolic trough solar collectors.

Figure 6: Parabolic Trough Solar Collectors

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f) The underlying function of a parabolic trough solar collector is given by y = 0,35×2 (1 unit = 1 m). Where has the heating pipe to be installed?