Concentrated solar thermal is one approach to combating the problem of intermittency in solar power generation. The idea is to focus the sun’s rays from a broad area onto a single collector to heat molten salt (or another material). This heat can then be stored for later use to generate steam and drive turbines when the sun isn’t shining, complementing the use of photovoltaics.

This assignment will explore how the sun’s rays can be focused onto a collector using an array of plane mirrors (as shown in the image below; a similar plant is being developed near Port Augusta, SA, and others are planned around Australia).

By  National  Renewable  Energy  Laboratory  -  http://earthobservatory .nasa .gov/Features/RenewableEnergy/Images/solar two .jpg, Public  Domain,  https://commons.wikimedia.org/w/index.php?curid=1455806

Part A: Mirror laws

The physical laws for reflection in a mirror are that

(A) The reflected light ray is in the same plane as the incoming ray and the normal to the mirror.

(B) The angle between the reflected ray and the normal to the mirror (the angle of reflection) is the same

as that between the incoming ray and the normal to the mirror (the angle of incidence). This is usually stated as “the angle of reflection is equal to the angle of incidence”.

For a plane mirror, these laws are mathematically expressed by the formula

ar  = −i + 2(i · ),

where  is a unit normal to the plane of the mirror, i  is the unit vector pointing in the opposite direction of the incoming ray and ar  is a vector pointing in the direction of the reflected ray.

1. Sketch the vectors and label the angles.                                                                                 [1 mark]

2. Show that ar  is also a unit vector.                                                                                        [2 marks]

3. Show that the formula above follows the two physical laws. For the first physical law you can do this with a couple of sentences or alternatively using appropriate dot and cross products. For the second, you should calculate an appropriate dot product to find an expression involving the angle of reflection and hence show that it is equal to the angle of incidence.                                                     [3 marks]

Part B: Heliostat

As the sun travels across the sky, the mirrors need to be rotated to ensure that they continue to reflect the light onto the collector.  These rotations are performed by pairs of motors that allow the mirrors to rotate around two perpendicular axes. We will calculate what the rotation angles for a mirror should be, given the location of the sun in the sky and location of the collector.

1. Assume that the x-axis is aligned due North, the y-axis due West and the z-axis vertically upwards. The origin is located at the centre of the mirror. The “zero”position of the mirror (with no rotation) has its normal pointing due North, aligned with the x-axis.  Sketch this coordinate system.  Include the mirror in its zero position and label the normal of the mirror by n0 .                             [2 marks]

2. Write down n0  in this coordinate system.                                                                              [1 mark]

3. If the mirror is now rotated by an angle ψ about an axis aligned with the z-axis and then an angle θ about the horizontal axis of the mirror (this axis coincided with the y-axis in the zero position of the mirror, but has now been rotated), then the new normal to the mirror is

n = Rz Ry n0

where

Rz  =  l

−sin(ψ)

cos(ψ)

0

and       Ry  =  l 

siθ)」

cos(θ)

are the rotation matrices around the two axes respectively.1   Note that ψ can take any value, but θ must be negative for the mirror to rotate towards the sky.  Perform the matrix operations to write

down the new normal to the mirror n in terms of ψ and θ .                                                  [3 marks]

4. Let S  be the unit vector pointing in the direction from the mirror to the sun. This vector makes an angle α with the positive z-axis. The projection of this vector onto the surface of the Earth (the plane z = 0) makes an angle β with the positive x-axis. Sketch the vector S  and these angles.2   Using this information, write down an appropriate vector S  in terms of α and β .                               [3 marks]

5. Using the expressions above together with the formula in part A, write down an expression for the unit vector pointing in the direction of the reflected ray ar  in terms of ψ , θ , α and β .                [2 marks]

6. It is always useful to check whether your formulæ are correct. One way to do so is to use a few examples for which you can work out the answer by other means and compare your results. For this assignment, we will give you some numerical values for you to check your formula: If ψ = 40◦ , θ = −70◦ , α = 50◦ and β = 20◦ , show that the reflected ray points in the direction (−0.2743, 0.1118, 0.9551).    [1 mark]

7. Let the location of the collector in this coordinate system be given by the vector c.  Write down a (simple) vector equation between the vector for the reflected ray ar  and c if the reflected ray is to hit the collector. Hence write down a vector equation between c, S  and  .                             [2 marks]

8. By manipulating this equation, find an expression for  in terms of S  and c. Note:  You should work with  the  vectors  and not substitute  their values  in  terms  of ψ , θ , α and β .   To  begin with,  you might consider taking a dot product with S .                                                                                                     [4 marks]