The Spool and the Card
© by Ed Seykota, 1999
|The spool-and-card levitator is a good illustration for
lift since it is graphic, easy to construct and nicely isolates the
phenomenon. It is a cute and curious device in which air flows down
through a spool and onto a card below. The downward flow of air lifts
the card up against the spool. Yes, that's right, it actually lifts it!
The author constructed a larger laboratory-grade model of the levitator
and used it to make measurements of pressure, airflow and lift. He also
built a mathematical model of it based on the Theory of Radial Momentum.
Result: the Radial Momentum model explains the levitator both
qualitatively and quantitatively.
When you blow down through the spool of thread, the index card adheres
to the bottom of the spool. (A sewing pin up through the center of the
card can help to keep it centered.) Contrary to popular belief,
Bernoulli's Principle does not explain the phenomenon. The Theory of
Radial Momentum does.
Momentum is the tendency of an object in motion to stay in motion.
Radial Momentum is the tendency of a group of objects (or a fluid) that
is radiating out from a center of radiation, to continue to radiate.
Radial Momentum is the sum of the momenta of all the objects with
respect to their center of radiation. An explosion in outer space
radiates mass in all directions. In this case, the linear and angular
momenta are zero while the Radial Momentum is the sum of the momenta of
all the particles from the center. Like linear momentum and angular
momentum, Radial Momentum also remains constant until some force acts
upon it. A group of objects may have linear, angular and Radial Momentum
all in superposition. Classical physics often deals with linear momentum
and angular momentum. Radial Momentum, which also obeys the classical
laws, is generally overlooked. It is a very important piece of physics
that explains lift and other pressure drop phenomena.
When a group of objects with Radial Momentum radiates away from the
center of radiation, it occupies more space and therefore becomes less
dense and, in the case of a fluid, develops less pressure. This is the
fundamental principle behind pressure drop and lift: radiating fluid
becomes less dense and loses pressure. Most instances of pressure
decrease in nature occur as a result of Radial Momentum. Other
mechanisms use angle-of-attack and/or pistons.
The levitator works by Radial Momentum. The small central valve just
under the central well, between the spool and the card is the bottleneck
that controls the flow. It has a cylindrical shape, entrains the air to
flow radially and invests the air with Radial Momentum. This Radial
Momentum carries the air out in expanding rings. As it expands it
becomes less dense and loses pressure. Eventually, counter pressure and
friction absorb the Radial Momentum and limit the radius of the effect.
Detail of Spool and Card
Air enters the spool at the top and flows into the well and out through
the valve. Radial Momentum carries the air out between the spool (green)
and the card (yellow). As the air expands into larger rings, it loses
density and pressure. It is this radial expansion of the air and not the
air velocity (as many textbooks indicate) that accounts for lift.
Your lungs provide a high and consistent flow of air. The gap between
the spool and the card is small. The air comes out of the valve with
high Radial Momentum. As the air expands, its pressure falls. It is this
momentum-induced expansion of the air (and not its velocity) that
motivates the pressure drop. The ambient pressure below the card
elevates the card. Incidentally, this may decrease the gap even further,
and increase the valve escape velocity, drawing the card up, even
tighter to the spool.
Of course, many good things come to an end, including levitating
cards. In this case, no real lungs can maintain a constant flow against
an arbitrarily small valve. Your lungs would have to deliver unlimited
pressure. Also, at velocities near and above the speed of sound, valves
tend to choke on the shock wave and this limits velocity. So as the gap
becomes smaller and smaller, the valve escape velocity and the mass flux
through the valve eventually stop increasing. Your lungs and the valve
both set limits on Radial Momentum. Absent the limits, if you could blow
with an infinite pressure through an infinitely small orifice, you would
create, in the theoretical limit, a suction cup.
Once the air leaves the valve area and radial Momentum begins to
carry it into larger and larger rings, two counter-effects combine to
reduce Radial Momentum and limit the radius of the effect. First, as
low-pressure air in the gap flows toward the circumference of the spool,
it experiences a positive pressure gradient that slows the air; this
reduces the Radial Momentum. Second, as the air flows along between the
face of the spool and the face of the card, it experiences drag or skin
friction, both from the viscosity of the air and from its colliding with
the roughness on the walls. Skin friction is a surface phenomenon,
generally independent of the size of the gap, so for very small gaps it
can become overwhelming and determine the equilibrium position for the
card. Valve and friction effects do not directly motivate lift; they act
against it. Still, a numerical model of the levitator does not come to
equilibrium without their inclusion.
During construction of the levitator model, the author discovers that
while the theory and mathematics of Radial Momentum are both
straightforward and relatively simple, the theories and mathematics for
valves and air friction are rather complex. Most of the work in building
the fluid-mechanics model of the levitator lies in obtaining accurate
equations to describe valve and friction phenomena. Still, a theory had
better be able to predict behavior pretty well, especially if you are
using it to pin the tail on a couple hundred years of fallacious
thinking, so including all the details is essential. All in all, the
levitator model is not perfect fit. In fluid dynamics, almost nothing
fits exactly. The model is pretty close, however, and far better than
the Bernoulli-based models. In addition, it provides an open, detailed
and rigorous look under the hood at the workings of Radial Momentum.
This web site presents a catalog of several other experiments, many
of which use standard textbook devices to demonstrate Bernoulli's
Principle. In each and every case, the author shows how radial Momentum
accounts for the behavior while Bernoulli's Principle does not.