Here is an interesting story about iPads. The short version is that American Airlines has obtained FAA approval to replace their huge flight manuals with iPads. As far as I understand, the flight manual contains everything you might possibly want to know about that aircraft and weighs 35 pounds.
American Airlines claims that by switching from paper to iPads, they will save 1.2 million dollars in fuel over the course of a year. You know this blog post won't end here.
Why Do You Use More Fuel With More Weight?
First, why do airplanes fly? Just a quick note - this is a much debated topic. So, that is your warning. Let me give a super simple explanation of flying. This will be so simple that it will probably make you angry. Here is a side view of a wing that is moving through the air.
Here I am representing the air as a bunch of small balls. When one of these balls collides with the wing, it changes its momentum. With a change in momentum, there must be a net force. The net force for this air-ball's change in momentum comes from the wing. Since forces are an interaction between objects, the air must also exert a force on the wing with the same magnitude. When a whole bunch of these air balls collide with the wing, the net result is a "lift" as well as a "drag".
Of course there has to be another force. If you want your plane to go at a constant speed, you would need a thrust force to balance the drag force. But what about the weight of the plane? If you increase the weight, you would also have to increase the lift. How can you do this? Really there are two ways. First you could have the plane go faster. This means you get more collisions with air-balls as well as a greater change in momentum for the air-balls. Both of these would lead to greater lift.
The other option would be to change the angle of the wing. If the wing is tilted more towards a vertical orientation, air balls will be deflected down more and result in a greater upward force. Of course this would also produce more drag. So either way, more lift means more drag. There is no escaping this effect.
BUT WHAT ABOUT BERNOULLI'S PRINCIPLE!!!!!? Yes. You are correct. I didn't say anything about pressure and lift and stuff like that. Why? Well, if you treat the air as particles you don't need to deal with pressure and Bernoulli's principle. You would use that if you wanted to treat the air as a continuous fluid. Of course even the air-ball collision model isn't perfect. However, it is useful enough to show how increased mass in a plane would lead to more drag and thus require more fuel.
How Much Fuel?
Let me first look at the money. The claim by American Airlines is that replacing a 35 pound flight manual with an iPad saves 1.2 million dollars per year. Now, I am not sure how many flight manuals this would replace, but I would suspect that it would have to be replaced by more than 1 iPad for redundancy. If one iPad weighs about 1.5 pounds, then the net weight savings would be 32 pounds (14.5 kg). Let me call the costs savings ΔCyf for a certain change in mass (the yf stands for yearly fuel). Then I can write:
So, this says that if you decrease the mass by 14.5 kg (I will use kilograms now) then the yearly fuel cost will decrease by 1.2 million dollars. But what if I look at other changes in mass? I can put other values in for the change in mass and calculate the change in cost if I assume that the relationship between fuel cost and mass is linear in this region. This isn't a bad assumption. Why? If the change in mass is tiny compared to the total mass of the plane, then it will at least be approximately true.
What about 1 bag of peanuts? Suppose that each American Airlines flight removed just one bag of peanuts (that they no longer give you) from the flight. How much in yearly fuel savings would this be? Well, first I need the mass of a bag of peanuts. Instead of searching for the exact mass for a bag of peanuts, let me just get a ballpark mass of 25 grams. That seems close. So if each American Airlines reduced the payload mass by 25 grams, they would save
That is 2,000 dollars a year for 1 bag of peanuts per flight. What if they took away all the peanuts? Here I will need to make quick guess. How many bags of peanuts would they bring on board each flight? This is tough since I really want an average. How about an average of 300 seats per flight with an average 400 bags of peanuts? Sounds good enough for me. This would lead to a yearly cost savings of right around 800 THOUSAND dollars. Quite close to the savings from the paper-based flight manual.
A Luggage Problem
American Airlines charges a fee for you to check a bag on the plane - like most airlines. According to this AA page, the bag can have a mass up to 23 kg without an extra extra fee. So, how does the fee they charge the passenger compare to the price of fuel needed to accommodate this extra mass? Ok, this will be little more difficult - but of course that won't stop me.
Here are the fees according to American Airlines:
- Flight within the US: $25 for the first bag.
- Flight in the US: $35 for the second bag.
- Flight in the US: $150 for more bags. Wow. That's seems harsh.
- Flying from the US and going through Europe makes the second bag $60. What if you go around Europe instead?
I am going to stop there. It seems like $25 for the first bag is the price for all "first" checked bag. But now I need to make some estimates. What is the average number of checked bags per flight? What is the average checked bag mass? Let me give a range of values.
- Average checked bag mass: 9 - 14 kg. Remember, this is just a guess.
- Average number of checked bags per flight? If I stick with the average of 300 seats per flight, then maybe there would be an average of 250 passengers per flight. Maybe half of these check a bag. Let's go with 120 - 180 first checked bags. Yes, this is a total guess.
With this, what would be the range of "first checked bag" masses? Oh, and by "first checked bag" I am indicating that I am ignoring the other bags checked since they don't charge the $25 fee for these.
Using both the high-end and low end mass of bags (average per flight), this would cost American Airlines a yearly fuel cost of 89 to 208 Million Dollars. Yup.
But is the $25 fee excessive? This is a little bit tougher to answer. The above calculation just looks at the yearly averages. It doesn't look at the cost per bag. So, I need to know how many people are paying for checked bags a year. According to this American Airlines info page, an average day has 275,000 passengers with an average 3,400 flights and more than 300,000 pieces of luggage.
From this, I get an average of just 80 passengers per flight. Arrg my first passenger estimate was a bit off. Oh well. Also this says that there would be 1.09 bags per passenger. Now is that checked bags or carry on? If you read their statement, it says the "handle more than 300,000 pieces of luggage". So, if a passenger has a connecting flight, their one bag could be handled more than once. I guess it depends on the definition of "handle".
If I go with 80 passengers per flight with 0.5 first checked bags per person, this would be an estimated fuel cost of 30 - 46 million dollars per year. Remember this is just for the first bags. Now what about the $25 baggage fee? If there are an average of 275,000 passengers per day, this would mean 100 million passengers per year. If half of these pay the $25 baggage fee, American Airlines would collect 1.25 BILLION dollars in fees. This is way higher than fuel costs for these first bags.
Maybe the fee is there to cover all the bags, including the carry on bags. Let's say these 100 million passengers per year bring an average total luggage mass of 30 kilograms. The yearly fuel cost for these bags would be 198 million dollars. This is still much less than the $25 baggage fee.
What would a fair baggage fee be? Well, first you have to choose what this fee pays for. The airline has lots of expenses related to luggage. There are the people that handle them there is the space on the plane and stuff like that. Let's go back to the first estimate for the yearly cost of checked bags at 46 million per year. If I stick with the 50% of passengers checking a bag on a flight - this would be 50 million passengers. This makes the calculation pretty easy. ONE dollar per checked bag. OK, change it to 2 dollars per checked bag. I wouldn't mind paying 2 dollars per bag.
One important point. The baggage fee calculation might not be valid. It is a slight stretch to assume that change in fuel costs for a mass difference of 14.5 kg is the same linear function for a mass change of 2000 kg. Even though the fuel-mass function isn't likely to be linear that whole way, this probably still gives a fair estimate. There, I said it.
Another Way to Save Money
If American Airlines can save 1.2 million per year with the lower mass of the iPads for flight manuals. Are there any other ways to shave off some weight during the flight? Yup. I have an idea. What if all passengers had to relieve themselves of excess bodily fluids before the flight? Just to be clear, I am talking about urination.
Lets go back to the 80 passengers per flight. Suppose each one stops by the restroom before getting on the plane. How much mass would this account for? I will just guess that on average, a person could produce 300 ml of fluid. Oh sure, some people could fill a whole liter - but there could be some passengers that have stage fright as well. I think 300 ml is a fair (even low end) estimate. It seems plausible that urine would have a density similar to the density of water (although I have not experimentally verified this).
Using a pee density of 1000 kg/m3 I get an average urine mass of 0.3 kg. With an average of 80 passengers, this is total mass savings of 24 kilograms. Using the same model for the savings from the iPad, this would have a yearly fuel savings of 1.98 million dollars. It would have the added benefit of not having to get up in the middle of the flight to visit the restroom. So, it makes sense and saves money. This should be a law.
