Magic Carpet

Once sitting in heavy traffic, Baron von Münchausen thought of a new kind of “magic carpet” type aircraft (see Figure 1.1). The upper surface of the large flat panel is held at a constant temperature T₁ and the lower surface at a temperature T₂ > T₁. He reasoned that, during collision with the hot surface, air molecules acquire additional momentum and therefore will transfer an equal momentum to the panel. The back of the handkerchief estimates he was able to make quickly for 1 m² of such a panel showed that if T₁ = 273 K and T₂ = 373 K (air temperature 293 K) this panel would be able to levitate itself and a payload (the Baron) of about 10³ kg. How did he arrive at this? Is it really possible?

Figure 1.1

SOLUTION

First let us try to reproduce the line of reasoning the Baron was likely to follow. He must have argued that in the z direction the average velocity of a molecule of mass m is

If we consider that during the collision the molecules thermalize, then the average velocities after reflection from the upper and lower surfaces become

Figure 1.2

The forces due to the striking of the molecules on the upper and lower surfaces are, respectively, |F₁| and |F₂| (see Figure 1.2):

Where n is the concentration of the air molecules, and we have used the fact that the number of molecules colliding with 1 m² of the surface per second is approximately n(v) (the exact number is n (v)/4). The net resulting force ∆F is

Substituting for n = P/τ, we have

Unfortunately, this estimate is totally wrong since it assumes that the concentration of molecules is the same above and below the panel, whereas it would be higher near the cold surface and lower near the hot surface  to ensure the same pressure above and below. That’s why irons don’t fly.