Module 4, Lesson 2

Flight

Objective: By the end of this lesson you will be able to make estimates of the energy required for flight. This will involve balancing forces and determining the lift-to-drag ratios of a number of planes. You will also appreciate that the physics of flight is universal and that the same laws govern large and small bodies.

Big Ideas

• To fly, you have to move in a fluid, and move that fluid around, which takes energy.

• By looking at how well a plane glides (uses gravitation potential energy to move forward) we can determine how much energy it takes fly.

Introduction

In the last lesson we reviewed the energy costs of many types of transportation, but we didn’t really discuss why these modes of transportation cost energy. Our first lesson will discuss the energy cost of flight. The actual aspects of flight are quite complicated and, as we will see, involve a balance between lift and drag and trust and gravity.

Physics of Flight

The basic physics of flying has been well understood since about 19171. In this lesson we do not intend to go into these details here; we are going to concentrate on the energy and environmental cost of air transport. The energy cost of transport is how much energy it takes to move a given mass a given distance2. For an unconventional approach to the pedagogy of flight, see Ref.3 For a beautiful video debunking the standard textbook explanation of how airfoils work, see Ref.4. The wikipedia page also has a good explanation http://en.wikipedia.org/wiki/Lift_(force).

Why do we need energy to fly?

In order to fly, aircrafts or birds need to continually push air downwards so that the deflected air pushes back and provides a lift force. This continual addition of momentum to the air requires a continual input of mechanical energy. For straight and level flight, this energy comes from the burning of fossil fuels in the engines of an aircraft, or the conversion of food energy in the muscles of a bird. For gliding flight, the energy comes from the loss of potential energy as the glider descends. 

1. Measuring the energy cost of flying.

Consider the forces acting on an aircraft in level flight at a constant velocity. All forces have to sum to zero, so we can be sure that the lift L equals the weight W, and that the thrust from the engines T equals the total drag D. The best measure of the quality of an aircraft, from the point of view of minimizing the energy cost, is the ratio of lift to drag, L/D. Plainly, the less drag you have for a given weight, the less thrust, and therefore energy, you need to get the aircraft from one place to another. The newest airliners like the Boeing 787 have lift to drag ratios of about 20. High performance sailplanes (i.e. gliders) have much higher ratios, but they are not configured to carry useful loads.

This video gives an overview of the topics discussed later in the lesson.

Glider_final9.avi (right-click and choose "Save Link As..." to download to your computer) http://www.youtube.com/watch?feature=player_embedded&v=0UpPjAst5mw#!

 Now consider what happens if we turn off the engines and let the aircraft glide. For the sum of the forces to be zero, the aircraft pitches down and assumes a glide slope of angle θ. You can see from the diagram below that tan θ = D/L.         

So an aircraft with L/D = 20 will glide at an angle tan θ = 1/20, i.e. θ ≈ 3°. It is no coincidence that all airports require airliners to approach the runway for landing at an angle 3°, for at this angle airliners are almost gliding, and the engines are therefore fairly quiet.

Question: Given L/D it is a simple step to calculate the energy cost of transport. If an airliner has a glide slope of 1 in 20, that means it loses potential energy mgh for every distance of d = 20h travelled. Calculate the energy cost of transport (use units of MJ per tonne per km).

Answer: The energy cost of transport, energy divided by (mass times distance) = mgh/(md) = gh/(20h) = g/20 ≈ 0.5 m/s2 = 0.5 MJ/(tonne km). 

Now look up the fuel and range statistics for a Boeing 747-300 (the long-haul version, which we choose because short-haul versions spend a larger proportion of their time and energy taxiing around airports, accelerating and braking):5

Maximum mass6: 378 tonnes

Fuel capacity: 199,000 L

Range: 12,400 km

If we reckon on the aircraft carrying, on average, half its fuel capacity, and fuel has a density of 0.7 kg/L, then at the mid-point of a long flight, the mass of aircraft is about 300 tonnes. Jet fuel has a heat of combustion of around 34 MJ/L.

Energy cost = (127,000 L)(34 MJ/L)/((200 tonnes)(15,700 km)) = 1.8 MJ/(tonne.km)

Hmm. This nothing like the 0.5 MJ/(tonne.km) we estimated from the glide slope. We need to take account of the fact that the engines are not 100% efficient, in fact they are only about 35% efficient7, so for every 100 J of fuel burnt, only 35 J goes into pushing the aircraft forward; the rest just heats the environment (directly).

Question: Using a glide slope of 1 in 18 (which is good for the 747 8) and and engine efficiency of 35%, recalculate the energy cost.

Energy cost = (g/18)/(0.35) ≈ 1.6 MJ/(tonne.km)

This is about 10% less than the in-service data - not bad for a simple calculation. No wading through tables of data. Just two numbers: the glide slope and the engine efficiency. This instantly tells us the two things that have to be improved to reduce the energy cost in terms of MJ/(tonne.km): the glide slope and the engine efficiency. However, each is a long slog. The

glide slope is determined by the aerodynamic cleanliness of the aircraft, particularly the slenderness of the wings. However, the overall shape of a jet airliner has changed little since the Boeing 707 prototype first flew in 1954, and so there have not been enormous improvements in L/D. 

Table 1. Lift-to-drag ratios for aircraft and birds (years shown are first-flight dates)

Type (L/D)max

$5 balsa glider9 4

Sparrow10 4

Gull8 11

Albatross8 20

Piper Warrior (shown in movie) 10

Boeing 707 (1954)7 18.5

Boeing 747 (1969, shown in movie)7 18

Boeing 787 (2009)11 21.5

Note how small things like sparrows and balsa gliders tend not to fly very well. Big birds like the Albatross fly much better than small ones. Ditto for aircraft. How to set up a simple experiment to measure the glide slope and L/D ratio for a balsa or paper glider can be found in our "Balsa Gliders and 747s" article12.

Engine efficiency is determined in part by how hot one can run the combustion chamber, and this is determined by the quality of materials used. A factor of two has been gained in the last 50 years; efficiencies have risen from about 17% to 35%6.

 

Energy per passenger-km or per tonne of freight

So far we have only considered the energy cost per tonne of aircraft, as that is the number basic physics tells us. Plainly a more telling quantity is the energy per passenger-km. Given that the energy per tonne-km is closely constrained by physics, the next most consideration in airliner design is the mass of the aircraft divided by the number of passengers. The more passengers you can get into a lighter aircraft the better. Here, materials science is having an effect. The Boeing 787 is largely built of composite materials while the 747 and earlier models were entirely aluminum. Comparing the long-haul 787-9 with the long-haul 747-300 we see that the former has a mass of 0.7 tonnes per seat, and the latter 0.8 tonnes per seat - a 14% improvement13.

 

The Jevons Paradox

Will these improvements in efficiency help the environment? Unlikely. As W. Stanley Jevons noticed when studying the British coal industry in the 1860s14, the more efficiently a resource is used, the more that resource gets used. In other words, improvements in the energy efficiency of flying are unlikely to reduce the carbon footprint of the global aviation business. The more efficiently planes fly, the cheaper flying will become, and the more people will fly, and the greater will be the GHG emissions. Too bad.

  

150 tonnes fuel x 44/14 = 470 tonnes CO2

  They still do it better than we do

2. Solar Powered Airplanes

If we were to coat a Boeing 747 jumbo jet with solar panels, is it possible to fly it using only the energy generated by these solar panels?

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)1.  Its typical cruising speed is Mach 0.8, or about 800 km/h, which is 0.22 km/s2. The specific energy of jet fuel is 36 MJ/L3. This means:

Total energy used in flight:

Total time over which the energy is used:

Average Power:

Part 2: Solar Panels

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”)4. 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 biofuel5.

Solar panels are slabs of photovoltaic cells, and use the photoelectric effect to generate electricity from sunlight5. 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”6. Due to the material properties of photovoltaic cells (they’re semiconductors) excited electrons are free to travel through the cell, but only along one direction6. 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%5. Common photovoltaics have efficiencies of around 10%, but we can assume 20%, which can be reached by the most expensive photovoltaics available today5.

Powering a 747 With Solar Panels

Question: We will cover the top surface of the wings of a 747 with solar panels. Using the diagram below estimate the total power that solar panels could generate. Give your answer in megawatts. Can the solar panels power the plane?

Figure 1.  The schematics of at Boeing 747-8 7.

Answer: Anything between 6.3 MW and 0.5 MW is acceptable.

The wing area is 525 m2 8. This gives a power of:

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:

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 alone9.

Figure 2. A to-scale comparison of a Boeing 747-87 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 photovoltaics10.

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.

Resources

References for Section 1 on Measuring the energy cost of flying:

1. J. D. Anderson, History of Aerodynamics (Cambridge 1997)2. Energy cost of transport /article/energy-cost-transport3. C. E. Waltham "Flight without Bernoulli" http://www.phas.ubc.ca/~waltham/pubs/

flight_without_bernoulli_34.pdf 4. Holger Babinsky, "How airplane wings really work" http://gizmodo.com/5878773/

clever-1+minute-video-shows-how-airplane-wings-really-work5. Boeing 747 http://en.wikipedia.org/wiki/Boeing_7476. a. b. I cannot bring myself to call it "weight".7. a. b. c. IPCC report on aviation http://www.grida.no/publications/other/ipcc_sr/?src=/climate/

ipcc/aviation/097.htm8. a. b. c. Peter Wegener, What makes airplanes fly?, (Springer, NY, 1991) p.169.9. C. E. Waltham "The flight of a balsa glider"http://www.phas.ubc.ca/~waltham/pubs/

balsa_glider_35.pdf 10. Henk Tennekes, The simple science of flight, (MIT, Cambridge 1997) p.83. Uses the French

term finesse for L/D.11. Piano-X analysis, http://www.lissys.demon.co.uk/samp1/index.html12. Balsa gliders and 747s /article/balsa-gliders-and-747s13. We compare long-haul aircraft because short-haul ones carry much less fuel and

consequently have much lower mass per seat; the short haul 787 is only 0.5 tonnes per seat14. Jevons paradox  http://www.eoearth.org/article/Jevons_paradox

References for Section 2 of covering the plane with solar panels.

1. Boeing.  747-8 Technical Characteristics (online).  http://www.boeing.com/commercial/747family/747-8_fact_sheet.html [27 May 2010].

2. Wikipedia.  Speed of Sound (online).  http://en.wikipedia.org/wiki/Speed_of_sound [27 May 2010].

3. /article/useful-numbers

4. Ostlie, Dale A & Caroll, Bradley W.  An Introduction to Modern Stellar Astrophysics. 

5. a. b. c. d. MacKay DJC.  Sustainable Energy Without the Hot Air. Solar (online).  http://www.withouthotair.com/ [27 May 2010].

6. a. b. Wikipedia.  Solar Cell (online).   http://en.wikipedia.org/wiki/Solar_cell [27 May 2010].

7. a. b. Boeing.  747 Schematic (online).  http://www.boeing.com/commercial/747family/pf/pf_exterior.html [27 May 2010].

8. Top Speed. 2011 Boeing 747-8 (online).http://www.topspeed.com/aviation/aviation-reviews/boeing/2011-boeing-747-8-ar86257.html[1 June 2010].

9. Google Earth’s ruler function

10. McKay, DJC.  Sustainable Energy Without the Hot Air. Planes (online).  http://www.withouthotair.com/ [27 May 2010].

Part 1: © Physics and Astronomy Outreach Program at the University of British Columbia (Oliver Millar, Christian Villar (movie) and Chris Waltham (text) 2010/07/23)

Part 2: © Physics and Astronomy Outreach Program at the University of British Columbia (Mo Chen, Charles Zhu 2010-05-27)