This module introduces, somewhat incidentally, the concept of estimation, both in the sense of rounding extremely precise numbers off to remove excessive significant figures, and simplifying a complex process (the flight of a 747) to a simpler one.
Part 1: Roughly how much power is necessary to keep a 747 aloft?
Flight is a complex process, and the amount of power needed for a 747 during flight will vary depending on what it’s doing. Whether or not the plane is rising, falling, or cruising at a constant altitude, the density of air at the altitude it’s flying, the mass of its payload, and prevailing winds will all affect the energy necessary to keep the plane going. We don’t want to deal with any of these complications, so we’ll make a very rough estimate of the amount of power needed to keep a 747 flying.
A modern passenger 747 has a maximum range of around 15,000 km, during which it uses 200,000 L of jet fuel (this assumes that travelling the maximum range requires the entire tank of fuel)[fn]Boeing. 747-8 Technical Characteristics (online). http://www.boeing.com/commercial/747family/747-8_fact_sheet.html [27 May 2010].[/fn]. Its typical cruising speed is Mach 0.8, or about 800 km/h, which is 0.22 km/s[fn]Wikipedia. Speed of Sound (online). http://en.wikipedia.org/wiki/Speed_of_sound [27 May 2010].[/fn]. The specific energy of jet fuel is 36 MJ/L[fn]/article/useful-numbers[/fn]. This means:
Total energy used in flight:
$E = 2 \times 10^5 \textnormal{ L} \times 36 \times 10^6 \textnormal{ J/L} = 7.2 \times 10^{12} \textnormal{ J}$
Total time over which the energy is used:
$t = \dfrac{15000 \textnormal{ km}}{0.22 \textnormal{ km/s}} = 6.8 \times 10^4 \textnormal{ s}$
Average Power:
$P_{ave} = \dfrac{E}{t} = \dfrac{7.2 \times 10^{12} \textnormal{ J}}{6.8 \times 10^4 \textnormal{ s}} = 100 \textnormal{ MW}$
Part 2: Solar Panels
How Do Solar Panels Work?
The sun radiates power toward us, in the form of sunlight: this power is what drives everything from the water cycle to the growth of plants (and the creation of fossil fuels). The power from the sun given to an area of 1 m2, assuming the sun is directly above us, is 1365 W/m2 (this is known as a “flux”)[fn] Ostlie, Dale A & Caroll, Bradley W. An Introduction to Modern Stellar Astrophysics. [/fn]. The goal of solar power generation is to turn this incident sunlight into power we can use, such as mechanical or electrical power. A number of methods exist for extracting power from the sun, the two must straightforward being solar thermal power (using sunlight for cooking, or for heating water to run a turbine) and the growing of food and biofuel[fn]MacKay DJC. Sustainable Energy Without the Hot Air. Solar (online). http://www.withouthotair.com/ [27 May 2010].[/fn].
Solar panels are slabs of photovoltaic cells, and use the photoelectric effect to generate electricity from sunlight[fn value=5][/fn]. Here’s how they work: light from the sun hits the photovoltaic cell. The result is a transfer of energy into the electrons of the cell, raising them into an “excited state”[fn]Wikipedia. Solar Cell (online). http://en.wikipedia.org/wiki/Solar_cell [27 May 2010].[/fn]. Due to the material properties of photovoltaic cells (they’re semiconductors) excited electrons are free to travel through the cell, but only along one direction[fn value=6][/fn]. This travel is a (DC) electric current, which can be attached to a load, such as a light bulb or motor.
Of course, we have to worry about efficiency, define to be the ratio between the power output and the power input. The efficiency limit for solar panels is around 30%; use of concentrators may increase this to around 60%[fn value=5][/fn]. Common photovoltaics have efficiencies of around 10%, but we can assume 20%, which can be reached by the most expensive photovoltaics available today[fn value=5][/fn].
Powering a 747 With Solar Panels
Figure 1. The schematics of at Boeing 747-8 [fn]Boeing. 747 Schematic (online). http://www.boeing.com/commercial/747family/pf/pf_exterior.html [27 May 2010].[/fn].
We will cover the top surface of the wings of a 747 with solar panels. The wing area is 525 m2 [fn]Top Speed. 2011 Boeing 747-8 (online).
http://www.topspeed.com/aviation/aviation-reviews/boeing/2011-boeing-747-8-ar86257.html
[1 June 2010]. [/fn]This gives a power of:
$P = 1365 \textnormal{ W/m}^2 \times 525 \textnormal{ m}^2 = 7.1 \times 10^5 \textnormal{ W}$
This is less than 1% of the needed power for the plane to take flight. To power an airplane, we would need solar panels covering an area:
$A = \dfrac{1.0 \times 10^8 \textnormal{ W}}{1.365 \times 10^2 \textnormal{ W/m}^2} = 7.3 \times 10^4 \textnormal{ m}^2 = (270 \textnormal{ m})^2$
This is about the footprint of BC Place in Vancouver, which doesn’t sound too bad until we consider that at any given moment tens of thousands of planes are aloft in North America alone[fn]Google Earth’s ruler function [/fn].
Figure 2. A to-scale comparison of a Boeing 747-8[fn value=7][/fn] to the area (in blue) solar panels would need to cover in order to power a 747’s flight (270 m)2.
Complications
There are lots of complications to this problem, of course. Several were mentioned throughout the document, and a few more are listed below:
- At high latitudes, the sun isn’t directly overhead, even at midday. In fact, the angle of incoming solar radiation depends not only on latitude but also on the time of day and the season. An airplane flying level to the ground at high latitudes will get much less than 1365 W/m2 (simple trigonometry shows that the flux from the Sun at latitude θ should be 1365 x cos θ where θ is the angle between the normal to the plane of the solar panel and the direction of the solar flux).
- Flying any solar-powered airplane at night is impossible, since the flux from the Moon is orders of magnitude smaller than the flux from the sun. Flying below clouds is also problematic, since clouds reduce the power from the Sun by a factor of 106.
- If we decided instead to use ground-based solar power to fuel our airplanes, there are other forms of solar power generation, such as the Stirling engines mentioned earlier, that potentially have higher efficiencies than photovoltaics[fn]McKay, DJC. Sustainable Energy Without the Hot Air. Planes (online). http://www.withouthotair.com/ [27 May 2010].[/fn].
Summary
It isn’t possible to power a 747 by coating the plane in solar panels and using power collected from the panels to fly the plane.