I was reading in the papers how some-airline-or-the-other increased their prices for extra luggage, citing increased fuel costs.
Now I'm a bit skeptical. Using the (wrong) Bernoulli-effect explanation of lift, I get this:
More luggage$\implies$more lift needed $\implies$ more speed needed$\:\:\:\not \!\!\!\! \implies$more fuel needed.
At this point, I'm only analysing the cruise situation. When the plane is accelerating, this will come into effect, but more on that later.
Now, I know that the correct description of lift involves the Coanda effect and conservation of momentum, but I don't know it well enough to analyse this. Also, there will be drag forces which I haven't (and don't know how to) factored in. I can see that viscosity must be making a change (otherwise planes wouldn't need engines once they're up there), but I don't know how significant a 1kg increase of weight would be.
So, my question is: Are airlines justified in equating extra baggage to fuel?
Bonus questions:
- If more baggage means more fuel, approximately what should the price be for each extra kilo of baggage?
- What happens when we consider takeoff and landing? Does a heavier plane have to use a significantly large amount of fuel?
Best Answer
In your own question you recognize that the Bernoulli equation is the wrong thing to apply to this situation, because obviously there are dissipative losses involved.
My preferred way of looking at this is recognizing there is a lift to drag ratio that exists as a metric for aircraft. This can be 4:1 or 25:1 depending on the plane. Regardless, provided that we accept the existence of this ratio in the first place, then the airlines are justified in the claim that more weight $\rightarrow$ more fuel. Limiting the discussion to cruising, it then becomes a simple multiplication of weight times lift to drag ratio to find fuel use.
The other flaw in your argument is, of course, the assumption that speed can be increased to compensate for more weight. A cursory reading into the flow path of turbo-machinery will disprove this. The jet engines will be most efficient at the designed cruise speed and rotation speed, and any deviation from that will alter the angles at which the air hits the rows in the turbine, causing efficiency to decrease. In the real world, drag also tends to increase as some power of velocity, which in itself will probably predict some marked decrease in the lift to drag ratio, again, making the plane consume more fuel. If the plane uses different altitudes to compensate for different weights with the same velocity, then more dense air will obviously cause more drag. It's true that these are ultimately viscous losses, but this flow is turbulent, and its likely that drag will scale as something close to $\propto \rho v^2$ (density times velocity squared) as a result of that fact. As the density increases fuel consumption will too.