Hello, and welcome back to chapter 2.2. In this chapter we want to deal with the calculation of the efficiency and the output of parabolic draft collectors.

Let's start with the basic equation, the basic heat balance for a thermal collector. The completely absorbed solar radiation equals the sum of the useful thermal power and of the thermal losses. To calculate the absorbed solar power we want to look at the corresponding equation

that is known for the low temperature collector. Since both systems are thermal systems one can use the same principles and the same calculation approach but it has to be modified slightly. The absorbed solar radiation, Qsol absorbed, can in case of a flat plate collector be

expressed as the product of the transpacivity of the glass tau and the absorptivity of the absorber alpha. The global radiation on the absorber E global ups and the collector area. The expression tau times alpha can be summarized to the optical efficiency eta ops.

We now want to check all factors if we have to modify something if we want to apply them to a parabolic draft collector. Let's start with the global radiation. First we must take here into account that in case of a parabolic draft collector or

more general for all concentrating systems we cannot use the global radiation. As you already know, we have discussed this several times, global radiation cannot be concentrated completely and only the direct fraction can be used.

Therefore, the global radiation must be replaced here by the direct beam radiation on the collector surface GBT. The useful thermal power can then be expressed with the equation here on the bottom. Let's now look in detail into GBT.

GBT can be expressed as a function of the DNI if the cosine of the incident angle is considered. The cosine theta is also referred to as cosine losses because it reduces the DNI per square meter.

For a better understanding this is illustrated in the next slides. Let's look here at the aperture area of a collector. We will not look at the curved mirror but at the projected area. If the radiation now occurs perpendicular to this collector then the angle of incident

theta is zero and the cosine of this angle is one. So there are no losses. In this case the beam radiation relative to the collector surface corresponds to the direct normal insulation. So if the DNI is 1000 watts per square meter then here the beam radiation relative

to the collector is 1000 watts per square meter too. This will change if the sun is not directly vertical but shines at a certain angle onto the collector. Here for example at an angle of 50 degree.

Then the radiation that was previously distributed on one square meter is now distributed over a larger area. That is pure geometry and can be calculated by applying the cosine of the incident angle. This means for this example what was previously distributed over a lens of the collector

of one meter is now distributed exactly over a lens of 1.56 meter and the radiation intensity onto this larger surface decreases accordingly. The DNI of 1000 will then decrease to a beam radiation of only 642.8 watts per

square meter. As we already know the angle of incident depends on the position of the sun relative to the collector surface and can be described as a function of the solar height alpha and the solar azimuth angle psi.

These angles we have discussed in more detail in the lecture on solar radiation. Both variables depend on the time of the day, on the season and of course also on the geographical position. The equation for calculating the incident angle for a north-south oriented parabolic trough collector as a function of these two angles is given here.

The angles of course change over the day and also over the year. The incident angle then can assume values between 0 and 90 degree. The following graph shows how the angle of incident can change over the day and the year.

This graph shows curves for a summer day here in June and for a winter day in December. In June we have relatively small angles during noon. This is shown by the blue solid line. And in the morning, the evening the angles are larger. In winter shown by the red solid line it is a little different because the sun is

deeper and we observe the largest angle at noon. The dashed lines show the corresponding cosine losses. For the summer the losses are small at noon, here the loss is almost zero and in winter it is correspondingly larger. Losses of about 40% can occur at noon due to the steep angle of incident.

Now however the cosine losses are not the only losses that occur due to the angle of incident but we have additional losses that are related to the fact that such a collector is not infinitely long. Therefore for incidence angle higher than zero, losses here at the end of the collector

occur. On this illustration we see that the first ray that hits the collector here at the edge is only reflected on the absorber tube at a certain distance from the edge. Consequently the first pass of the collector here does not see any radiation and is useless

under these conditions. The losses can be calculated as a function of the incident angle and the focal length of the concentrator F. For a row of collectors it must be considered that the second collector in the row does not experience the same complete losses because it might see reflected radiation from the

collector before. So we have now discussed the extent to which we have to adapt the radiation input for parabolic troughs. We now know that we have to use instead of the global radiation the direct radiation

but not the direct normal radiation but the direct radiation relative to the collector surface. We therefore have to take into account the angle of incident which is required to calculate the corresponding losses. Next let's look at the optical efficiency.

We know the optical efficiency of the flat plate collector is the product of the transmissivity of the glass cover and the absorptivity of the absorber tube. In case of a parabolic trough collector the collector consists of a few more optical components and therefore the optical efficiency is a bit more complex as you can see here

in this equation. I will now walk you through the individual factors. Let's start with the first element, the reflector. When the radiation hits the reflector it reflects not all 100% of the incoming solar radiation but a few percent will get lost.

We have discussed this in the previous chapter. Then the reflected radiation hits the glass tube. The transmissivity of the glass tube tau is also not 100% but a little lower. Then we have to consider the absorptivity of the absorber tube that also does not absorb

100% of the radiation. These are the three important optical properties but there are a few more factors than these. In addition for example we have to consider also the so called intercept factor IC. The intercept factor takes into account the degree of accuracy of a parabolic trough collector.

It describes the fraction of reflected radiation that finds its way to the absorber tube. Due to inaccuracies of the collector some of the radiation fails the receiver. So the intercept factor considers how good the parabolic shape is and how perfect the

concentration is. Several factors play a role here. On one hand the surface is not always perfectly parabolic. Then the surface has a certain roughness so that the radiation may be scattered. Then a misalignment of the absorber is also taken into account.

So if the absorber is not installed perfectly in the focal line but deviates slightly from it then we have additional losses. Manufacturing tolerances in the support structure on which the mirrors are mounted have to be included as well. Then the tracking of the collector can also play a role.

If the tracking is not perfectly then the focus is also not perfect. And finally the intercept factor also takes into account the so called sun shape. We have already discussed that the solar radiation is not perfectly parallel and additionally it can be expanded further by dust in the atmosphere.

You can observe this for example in a sand desert when you have a lot of dust in the air then the edge of the sun is not as clear and not as sharp as for example in the morning on a clear winter day in the mountains. Under dust conditions the sun seems to be a little larger.

If the edge is not clear that means that the rays are a little bit more scattered which makes it more difficult to concentrate then. And this is what we call the sun shape. The intercept factor takes into account all these different errors and effects. In principle the intercept factor can assume values between one and zero

but good collectors have values over 90%. The next factor to consider for parabolic draft mirror is the fact that a mirror is not always perfectly clean and that dirt can deposit on the mirrors or on the glass

group. Dirt can have two effects. The first effect is of course that radiation is blocked by the dust particles. Secondly the radiation can be further deflected and scattered so that it may miss the focal point. Therefore the factor of cleanliness must also be taken into account.

In order to minimize all these effects the reflectors will be cleaned from time to time. Special machines have been designed for this purpose. One example for such a cleaning vehicle is shown here on the photo. The last factor that needs to be taken into account is the incident angle correction factor,

the so-called incident angle modifier. All optical properties depend on the angle of incident. This you can observe on your own for example at a glass window. If the radiation is vertical the transmissivity of the window is very high and you see hardly

any reflection. But if you look at the window under a steep angle you see more reflection in the glass which means that the transmissivity is lower and the reflectivity is higher. This kind of dependency you have for all optical properties and this is considered

by the incident angle modifier. For vertical radiation this factor is 1 and then it decreases with increasing incident angle. An example for an empirical correlation for the incident angle modifier chiteta is given at the bottom of this slide.

So we have dealt now with the optical efficiency and the last term missing here are the heat losses. Heat losses occur at the receiver tube. We have radiation losses at the absorber tube itself and we have radiation and convection

losses at the outer glass tube. The total heat losses are composed of the convection losses and the radiation losses. If we have to consider both depends on where we draw the boundary for the heat balance. If we consider just the absorber tube itself only radiation losses have to be considered

because as already mentioned the space between the absorber and the glass is evacuated. And hence the convection losses from the absorber tube to the glass can be neglected. Convection losses occur only on the outer glass envelope because here the wind can lead to force convection and may cool the glass.

Therefore if we make the heat balance around the glass we have to consider both. In most cases it is more practical to make the heat balance just around the absorber tube and then it is sufficient to look at the radiation losses only which are calculated according to the well known Stefan-Boltzmann law.

The equation considers the emissivity which is an optical property of the absorber tube. The emissivity is expressed by epsilon, the Stefan-Boltzmann constant sigma, the receiver diameter D and the receiver length L. And then we got the temperature of the receiver

and the temperature of the sky both to the power of 4. The temperatures where you have to take care have to be inserted here in Kelvin and not in Celsius. Since the sky temperature is very low compared to the receiver temperature the ambient temperature can be used instead which is usually a little higher than the sky temperature.

But this makes not much a difference in the result. The equations on this slide describing the heat losses are based on the physical process itself. But also empirical equations for the heat loss can be found in the literature and

can also be used. However it has to be emphasized that such a correlation is only valid for a specific product. In this graph here you can see the heat loss determination for a specific brand of absorber tube. The curve shown here was determined from measurements and an equation was derived to

describe this curve. The advantage of such empirical functions is that the specific product is described very precisely but the disadvantage is that the result cannot be transferred so easily to another product. Thus both approaches, the physical and the empirical, have their advantages and disadvantages.

We have now discussed all effects and parameters required to determine the useful power of a collector. We have discussed the direct radiation on the collector GPT where we have to take into account the cosine effects. We have talked about the optical efficiency and all the parameters that have to be considered

there. And we have looked at heat losses that results from radiation losses of the absorber tube. Convection losses of the absorber tube can be neglected as mentioned before. With this we are now able to calculate the useful solar heat collected by the absorber.

We can then also determine which temperature or temperature increase we can reach with such a collector for a specific fluid and a specific mass flow. For this calculation I need to know the specific heat capacity Cp of the heat transfer

fluid as seen in this equation here on the top. Knowing the useful absorbed heat we are also able to calculate the efficiency of the collector. For this we divide the useful absorbed heat by the solar radiation GPT on the collector's

surface a collector. We can then insert the first equation into the second and then get the following correlation. The efficiency equals to the optical efficiency minus the heat losses divided by the incoming radiation on a collector.

And then we can insert here our parameters that we have just discussed in the previous slides. Here in this equation you can see again what we have already discussed at the beginning of this chapter, the higher the concentration factor, the smaller this part of the equation and the higher the collector efficiency.

So here you got the overview of the most important equations to calculate the useful heat and the efficiency of a parabolic trough collector. And this is at the same time a good summary of what we have discussed in this subchapter.