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Air Pressure and Fluid Flow
Name: Chris H.
Status: student
Age: 17
Location: N/A
Country: N/A
Date: Thursday, November 28, 2002
Question:
I have a bottle with two holes (one in the top one in the
bottom), I am performing an experiment showing how when the hole in the
top is covered (air hole) the water will not flow out the small hole in
the bottom. As there becomes less and less water in the bottle the time
delay between covering the hole at the top of the bottle and water coming
out the bottom increases - why?
Replies:
The underlying reason is in the way pressure and volume are related in a gas.
Let us start with the top hole uncovered, with air flowing into the bottle and
water flowing out the bottom hole. Now cover the top hole and let us look at
the forces on the cylinder of water that extends from the bottom hole up
to the
top surface of the water inside the bottle.
1) Atmospheric pressure below the bottom hole, acting over the area of the
bottom hole; this is your only upward force and it is equal to P_o * A, where
P_o is the pressure outside the bottle, and A is the area of the hole.
2) The gravitational force W of the cylinder of water; this is a downward
force, obviously.
3) Air trapped in the bottle, with a pressure that will vary as water
leaves the
bottle (with the top plugged). This pressure also acts over the area of the
hole, and it produces a downward force equal to P_i * A, where P_i is the
pressure in the bottle. (Do not get sucked in by the notion of some sort of
vacuum inside the bottle sucking the water in. The air inside the bottle
pushes
water down. Period.)
All sideways forces on the cylinder cancel out, and are therefore ignorable.
Initially, the air in the bottle is at atmospheric pressure, and its force
cancels the upward force from the air under the bottle. The remaining force
is the weight of the water, and this causes water to move down through the
hole.
As water leaves the bottle, the air inside increases in volume, and this
causes
its pressure to decrease. If the amount of air stays constant, and its
temperature does not change, the equation that relates pressure and volume is
Pressure * Volume = constant,
so, as the volume increases the pressure decreases. As the pressure
decreases,
the total downward force on the cylinder decreases. Here is an expression for
that force:
Force = P_o * A + W - P_i * A.
If P_i decreases to the point that the total downward force is zero, then
water
will stop flowing out the bottom hole.
How long will that take? Let us assume for simplicity that we can ignore the
changing height of the water cylinder. Let us assume for concreteness
that the
water stops flowing when the pressure inside the bottle is just half of the
pressure outside. If we start with a small volume of air, it does not
take much
to double that volume (halve the pressure) -- only a small quantity of
water has
to leave. But if we have a large air volume in the bottle, then a lot of water
must exit in order for that air volume to double. This is why it takes longer
for the water to stop flowing as the amount of water in the bottle
decreases.
Tim Mooney
Chris,
I have not performed this experiment myself but I am not surprised
that there is a relationship between the amount of water in the jar and
how long it takes for the flow to stop. Actually, it is probably related
to the amount of air in the bottle. In order for the flow to stop, the
atmospheric pressure outside the bottle must equal the force due to the
weight of the water in the bottle plus the force due to the air inside
the bottle. When you cover the bottle the water continues to pour out
until the air pressure inside the bottle drops sufficiently that the
external air pressure can hold the water in. If there is very little
air space it does not take long for enough water to pour out, increasing
the air volume in the bottle, to achieve this balance. When there is
lots of air to begin with much more water must flow out before the air
pressure inside drops enough to hold the water in.
Greg Bradburn
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