Drinking fountains are such simple devices, yet they're incredibly likely to misbehave. About half the fountains you encounter deliver so little water that it barely dribbles out of their nozzles and the other fountains typically spray water up into your eyes like fire hoses.
1. Water enters a typical drinking fountain through a large pipe at almost zero velocity and a pressure that's about 6 times atmospheric pressure (normal atmospheric pressure plus an additional 5 atmospheres). The water then passes through a nozzle and sprays up into the air. If the drinking fountain did nothing to alter the water's total energy as it flows, how high would the water jet rise above the nozzle before coming to a stop and then descending? (Assume that the nozzle and water jet are vertical, and neglect air resistance.)
Answer:
The jet of water would rise 50 meters.
Why: The pressure of water in
a vertical column rises by 1 atmosphere for every 10 meters of depth. Thus the
pressure at the bottom of a 50-meter tall column of water is 6 atmospheres (1
atmosphere of pressure to support the atmosphere overhead and 5 more
atmospheres of pressure to support the weight of the water above). If you
attach the drinking fountain's inlet to the bottom of this column and let the
water flow, it will soon reach steady state. At that point, water throughout
the system will have a constant total energy per volume. The water at the top
of the 50-meter column has all of its available energy (we are neglecting the
pressure energy associated with atmospheric pressure itself, which isn’t really
available here) in the form of gravitational potential energy. Therefore, you
can expect that when water in the jet has converted all of its available energy
into gravitational potential energy, it must be at this same height: 50 meters
above the nozzle.
2. Once the water in this full-energy jet has left the nozzle, it's surrounded only by air at atmospheric pressure. What is the water pressure in this freely moving jet at a height of 10 centimeters above the nozzle?
Answer:
The water pressure in the jet is atmospheric pressure.
Why: Since the water jet is
exposed to atmospheric pressure all around and it isn't accelerating inward or
outward (spreading wider or becoming narrower), its pressure must also be
atmospheric.
3. You lean your head over into this full-energy water jet and place your mouth 10 centimeters above the nozzle. What is the approximate pressure of the water as it hits your teeth and comes abruptly to a complete stop?
Answer: The pressure against your teeth is just a
little less than 6 atmospheres of pressure (a little less than the pressure
inside the drinking fountain before the nozzle).
Why: When the water comes to a
stop, virtually all of its total energy returns to the form of pressure (a
little remains as gravitational potential energy). Since the water had a
pressure of 6 atmospheres before spraying out the nozzle, it will again have almost
6 atmospheres when it stops on your teeth. The slight rise of 10 centimeters
will reduce the energy available to pressure ever so slightly.
4. To prevent you from losing teeth or fillings, the people who built the drinking fountain included a pressure regulator that causes water in the free jet to stop rising only 10 cm above the nozzle. What fraction of its original energy does the water retain after passing through this pressure regulator and what probably becomes of that missing energy?
Answer: The water retains only about 1 part in 500
of its original energy (10 centimeters divided by 50 meters). The rest of that
energy is turned into thermal energy.
Why: Since the full-pressure
jet will rise to 50 meters, something must reduce the rise by a factor of 500
in order for it to rise only 10 centimeters. Since gravitational potential
energy is proportional to height, this reduction in height by a factor of 500
is equivalent to a reduction in total energy by a factor of 500. The regulator
converts 99.8% of the water's total energy into thermal energy, leaving it with
just enough energy to rise 10 centimeters above the nozzle before coming to a
stop.
Your internship at the Supreme Court is on hold after the justice you were planning to work for broke a leg hang-gliding. So you decide to use your intellect working at another job: night chef at the local Krispy Kreme donut shop. You are in charge of the whole donut production system. This system begins with the machine that squeezes narrow rings of dough out onto a little conveyor belt. The dough rings rise as they travel along the conveyor for roughly half an hour. The now large rings of dough drop into a river of hot oil and cook. And after being flipped and then lifted up onto a drying belt, a river of liquid glaze covers each donut with a sugary coating. When no one is looking, it's all you can do not to sit at the end of the glazing station with tongs, a bib, and a smile on your face.
5. During the rising process, yeast in the dough converts some of the dough's sugar molecules into simpler molecules that include carbon dioxide gas. These gas molecules accumulate in the dough. In terms of density, pressure, and force, explain why these gas molecules cause the dough to rise, even against the force of gravity and the inward elastic forces of the dough itself.
Answer: The dough was originally in equilibrium,
with all the forces balancing everywhere to zero. But as the density of gas
molecules increases inside the dough, the pressure inside the dough also
increases and upsets the equilibrium. The internal pressure increase causes
outward acceleration of the dough. The dough moves outward until it reaches a
new, larger equilibrium size.
Why: There are many ways to
answer this question but they all should address the fact that the dough was
originally in equilibrium (zero net force throughout) and that the increasing
density of gas inside the dough upsets this equilibrium. As dense gas bubbles
appear inside the dough, pressure imbalances across the walls of these bubbles
cause the bubbles to swell up and the dough becomes larger.
6. The moment a ring of dough falls into the hot oil, it begins to swell up significantly. Explain this sudden swelling of the donut.
Answer: Increasing the temperature of the trapped
gas increases that gas's pressure. The bubbles expand as the result of the
pressure imbalance and the dough expands overall.
Why: Heating a trapped bubble
of gas causes that gas to increase in pressure. With more pressure pushing
outward on the bubble's walls, the bubble is no longer in equilibrium. Instead,
its walls accelerate outward and it soon reaches a new, larger equilibrium
size.
7. The glazing station uses a pump to recirculate the thick, hot, liquid glaze. Glaze accumulates in a pan below the donuts and is pumped up to a spigot above the donuts. It pours from that spigot as a smooth curtainlike sheet of glaze through which the donuts must pass on their way to the sales counter. The pipes carrying the glaze from pan to pump to spigot are surprisingly wide, despite the fact that they don't carry all that much glaze each second. In terms of viscosity and pressure, why does the glazing station need such large-diameter pipes for handling the glaze?
Answer: The rate at which fluid flows through a
pipe, given a particular pressure imbalance across the pipe, decreases as the
fluid's viscosity increases and increases with the pipe diameter. To compensate
for the slow flow of high viscosity glaze, they need large diameter pipes.
Why: The thick glaze
experiences large internal friction-like forces, which tend to slow all the
glaze down to the speed of the pipe walls (i.e. stationary). To overcome these
slowing effects of viscosity, they use very wide pipes--ones that get the
central core of the flowing glaze far away from the walls of the pipes.
8. The thick glaze pours as a smooth curtain over the donuts and leaves each of them with a smooth shiny layer of glaze. If watery glaze were used instead of thick glaze, it would splatter all over and leave the donuts looking like they'd been sprinkled with droplets. Why does the glaze's thick character cause it to flow so smoothly and form such a smooth layer on the donuts?
Answer: The thick glaze's high viscosity keeps it
flowing orderly (laminar flow) so that it forms smooth coatings. In contrast,
watery glaze's low viscosity can't overcome inertia and the glaze tears itself
apart and flows disorderly (turbulent flow) so that it forms blotchy coatings.
Why: Thicker fluids tend to
experience laminar (orderly) flow in situations where thinner fluids would
become turbulent (disorderly flow). In the glazing station, the tendency for the
glaze to remain orderly allows it to form smooth coatings on the donuts.