We all have seen solar panels. They are mounted on traffic signs, light houses (see title photograph), and in our calculators. They are great green sources of energy. So why isn’t every roof in [Vancouver] covered with them? This would reduce our reliance on fossil fuels.
Let’s learn something about solar panels and do some simple calculations.
To create electric potential energy (or colloquially speaking electricity) we have to separate positive and negative charges. In batteries, the work required to separate positive and negative charges is done by an electrochemical reaction. In silicon-based solar cells, the work is done by the incoming solar radiation in a quantum process: a visible photon from the sun has enough energy to separate an electron from a silicon atom leaving behind a site that is positively charged, called a “hole” (Remember the duality of light, we can treat light as electromagnetic wave or as a stream of particles called photons). Electrons are attracted to the positive site and holes to the negative site, so once we connect a solar panel to some load (for example a light bulb, a motor or a heater) we have a current that can do work on such external load.
Let’s calculate how much power we can obtain from a solar panel in Vancouver. The solar constant (the amount of solar radiation per m2 at the top of the atmosphere) is about 1400 W/ m2. We have on average 12 hours of sunlight a day, which should give us 1400 W/ m2 times 12 hours = 16.8 kWh/ m2 per day. But according to Natural Resources Canada[fn]http://pv.nrcan.gc.ca/index.php?m=r[/fn] we can only expect on average 5.2 kWh/ m2 on a surface perpendicular to the direction from the Sun. This is due to atmospheric absorption and cloud cover.
There is another problem. As the Sun’s position changes during the day (differently in each day of the year) it is expensive to keep the panel perpendicular to the direction from the Sun. To do this requires directional alignment in two axes so the panels cannot be fixed to the roof.
As illustrated Figure 1 the power of radiation delivered to a given surface depends on the incident angle. Notice that the power contained in a 1 m2 column of radiation impacts 1 m2 of surface if it falls vertically but 2 m2 of surface if it is at an angle of 30 degrees. So at 30 degrees we are getting only ½ of power per m2! The angular dependence of the power delivered is:
$P_{\theta} = P_{90} \sin(\theta)$
where $ P_{90}$ is the power per unit area of radiation impacting the surface from the direction perpendicular to the surface, $\theta$ is the angle of the surface from the horizontal and $P_{\theta}$ is the power per unit area of the same radiation impacting the surface from the direction at the angle $\theta$ from the horizontal.
Figure 1. The radiation of the same intensity illuminating a surface at different angles
Notice that the angle is measured from the direction perpendicular to the direction of incoming radiation as described in[fn value=1][/fn]. So if we mount our solar panel at the angle equal to our latitude on a south facing roof we will only get on average 3.7 kWh/ m2 of energy per day impacting our panel.
Why should we tilt the panel at the angle equal to our latitude? Lets look at Figure 2. Twice a year at solstice the axis of rotation of the Earth is perpendicular to the direction of the incoming solar radiation (as shown). The panel tilted from the horizontal plane at the angle equal to the latitude of it’s position is at noon perpendicular to the direction of the incoming solar radiation. Over the year the angle between the axis of the Earth and the direction of the incoming solar radiation changes between -23.5° and +23.5°. Also the angle between the solar panel and the direction of the incoming solar radiation will change during the day. But such a position of the panel gives us a best yearly average of the power of the the solar radiation impacting the panel. How much do we loose if the tilt angle of the panel is different. Between the tilt angle equal to latitude and tilt angle equal to latitude -15° almost nothing. At tilt angle equal to latitude +15° about 10%.
Figure 2. An illustration explaining why we set up solar panels at an angle from horizontal roughly equal to latitude.
A two person family living in a small detached house uses about 40kWh per day for cooking, hot water, heating, lights, TV, laundry, computers, microwaves and so on at a cost of about \$850 a year. It would seem that a 10m2 solar panel should cover their needs. Unfortunately not! Only about 10% of energy hitting the panel is converted to AC electrical energy. This is because of the solar panel efficiency (15-25%), DC/AC conversion efficiency and resistive loses. So from a nominally 1kW panel (the panel which gives us 1 kW electrical power at noon in full sunshine, when oriented perpendicular to incoming solar radiation) we can expect on average only about 3kWh per day (1 gives us so called Photovoltaic potential in kWh/kW, which is how many kWh per month or year can we expect at a particular location from a 1kW panel). And such a panel has an area of 7-10 m2 depending on the technology! At the moment the complete 1kW solar system costs about \$7500. On a typical house and garage there is usually about 50 m2 of a south facing roof space to be covered by solar panels. So we could install 5-7 kW panels and get 15 – 20 kWh per day, almost half of the expected consumption. This installation would cost \$40-50K and would recoup the initial cost in 50-60 years; this is unfortunately not a good investment.
And there is an other problem: energy storage. Solar panels give us a peak production on a midsummer day. But peak consumption occurs in midwinter evenings. If the house is connected to the grid we can sell energy at peak production and buy it at peak consumption and all our previous analysis is valid. But if our house is located on a small island we have to think about energy storage. We would have to store about ¼ year’s energy = about 4 000 kWh! A big car battery (mass 55 kg) stores about 200Ah at 12 V = 2.4 kWh. We would need about 1700 such batteries!
On the other hand a solar panel and a modest battery bank would be a perfect solution for a summer cottage used only on weekends. It would be a good exercise to try to calculate the expected energy consumption, the necessary size of solar panels, and the capacity of the batteries.
It sounds pessimistic at the moment but the technology is improving. Printed “roll on the roof” solar cells are coming [fn]http://www.nanosolar.com/[/fn]. In a few years we might be able to save some money and more importantly the enviroment by covering our roofs with solar cells.
Reasonable quality coal of the type used in power stations has an energy content (higher heating value, or HHV) of around 30 MJ/kg. This number varies considerably depending on the source of coal. The chemical formula for coal is roughly (CH)n. Let’s calculate how much coal we have to burn in a typical large power station to produce 1 GWe for a year. For this we have to assume a conversion efficiency η for thermal to electrical energy. Really good modern steam generators have η ≈ 0.4, so to generate 1 GWe we’ll need a a thermal power of 1/0.4 = 2.5 GWth.
The total amount of thermal energy required to generate 2.5 GWth for one year is
(2.5)(3600 s/h)(24 h/d)(365 d/y)(109 W) = 7.9 x 1016 J.
The mass of coal that needs to be burnt to produce this energy is
(7.9 x 1016 J)/(30 x 106 J/kg) = 2.6 x 109 kg = 2.6 Mtonnes.
To convert this mass of coal to the mass of CO2 produced on burning, consider the chemical reaction:
4CH + 5O2 → 4CO2 + 2H2O
As, always when burning fossil fuels, every carbon atom in the fuel ends up in a CO2 molecule. The molecular mass of CH is 13; that of CO2 is 44. Thus 13 tonnes of coal produce 44 tonnes of CO2.
In other words, the 2.6 Mtonnes of coal burnt each year in a 1 GWe power station produces (2.6)(44/14) = 8.9 Mtonnes of CO2.
As we are using some rough numbers here and also ignoring CO2 emissions caused by mining and transportation of coal, let’s call our result 10 Mtonnes of CO2 per GWe per year. Its hard to know what to do with 10 million tonnes of anything, let alone a gas (which at STP[fn]Hyper Physics. Ideal Gas Law (online). http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/idegas.html [9 June 2010].[/fn] would fill 5 cubic km). However, this is the CO2 production from only one large coal-fired plant in one year. Two such plants are being opened in China every week[fn]BBC News. China Building More Power Plants (online). http://news.bbc.co.uk/2/hi/asia-pacific/6769743.stm [9 June 2010].[/fn]. Although ideas and plans abound[fn]Wikipedia. Carbon Capture and Storage (online). http://en.wikipedia.org/wiki/Carbon_capture_and_storage [9 June 2010].[/fn], no plant yet disposes of its CO2 anywhere other than in the atmosphere.
Footnotes
↑1, ↑12, ↑13 | Aubrecht GJ. Solar Energy: Wind, Photovoltaics, and Large-Scale Installatons. In: Energy – Physical, Environmental, and Social Impact (3), edited by Erik Fahlgren. Upper Saddle River, NJ: Pearson Education Inc., 2006, chapt. 21, 461-465. |
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↑2, ↑3, ↑4 | Kump, L.R., Kasting, J.F., and Crane, R.G. The Atmospheric Circulation System. In: The Earth System (2), edited by Patrick Lynch. Upper Saddle River, New Jersey, USA: 2004, chapt. 4, pp. 55-82. |
↑5 | Environment Canada. Canadian Atlas Level 0 (online). https://collaboration.cmc.ec.gc.ca/science/rpn/modcom/eole/CanadianAtlas0.html [20 May 2009]. |
↑6, ↑7, ↑8 | Gustavson MR. Limits to Wind Power Utilization. Science 204: 13 – 17, 1979. |
↑9 | MacKay DJC. Sustainable Energy – Without the Hot Air (Online). UIT Cambridge. http://www.inference.phy.cam.ac.uk/sustainable/book/tex/ps/253.326.pdf [4 May 2009]. |
↑10 | Betz’ Law http://en.wikipedia.org/wiki/Betz’_law [2012.09.27]. |
↑11 | Learning (online). Solacity Inc. https://www.solacity.com/SiteSelection.htm [20 May 2009]. |
↑14 | Clarke S. Electricity Generation Using Small Wind Turbines At Your Home Or Farm (Online). Ontario Ministry of Agriculture, Foods and Rural Affairs. https://www.omafra.gov.on.ca/english/engineer/facts/03-047.htm#noise [25 May 2009]. |
↑15 | Marris E and Fairless D. Wind Farms’ Deadly Reputation Hard to Shift. Nature 447: 126, 2007. |
↑16 | Keith D. Wind Power and Climate Change (online). University of Calgary. https://www.ucalgary.ca/~keith/WindAndClimateNote.html [20 May 2009]. |
↑17 | Accio Energy. About Accio Energy (online). https://accioenergy.com/about.html [12 June 2009]. |