Goal of Problem Set #9:
This assignment is meant to help you understand:
- the relationships between ordered
and disordered energies,
- the statistical and probabilistic
nature of thermodynamics,
- the role of entropy in determining
what is likely to happen,
- the interplay of work, heat, and
entropy.
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| Your art gallery in New York's Soho district
has only been opened for two months, yet it is already attracting a steady
clientele of the city's moneyed elite. You are exhibiting works by up
and coming artists as well as old masters and having a wonderful time
of it.
Unfortunately, with all the changes in weather and
humidity outside, you are finding it hard to keep the artwork in tiptop
condition. To remedy the situation, you design a climate control system
that keeps the gallery at a fixed temperature and humidity that's optimal
for preservation. Whether the outside air is hot or cold, dry or moist,
your new system will fix it.
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1. When the outside air is hot
and moist, your system's first goal is to extract the excess moisture.
It does this by passing the air over a cold surface so that some of
the moisture condenses and runs down a drain as liquid water. Why
is this process inevitably going to (A) require electricity
(or an equivalent source of ordered energy) and (B) going to
require something to become hotter than the original outside air?
[Note: ignore the possibility of tapping into the neighbor's air conditioning
system or any other trick solution to your predicament. It's just
your gallery and the outside air.]
Answer: (A)
To create the colder surface, you'll need to pump heat against its
natural direction of flow (i.e. toward a hotter region) and that
pumping requires an input of ordered energy. (B) The pumped
heat must go somewhere and the only reasonable destination for much
of it is the outside air, which will become hotter than it was originally.
Why:
Creating a cold region in the midst of a hotter one is going to
require a heat pump: heat must move from the colder region that
you're creating to the hotter one. To keep entropy from decreasing
and to satisfy the second law of thermodynamics, that pumping process
requires a source of ordered energy. The pumped heat, along with
the ordered energy, are dumped into the hotter region as thermal
energy and the hotter region must become hotter still.
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2. Having chilled the air until
it contains only the perfect amount of moisture, you want to reheat
it to the ideal gallery temperature. You could use an electric heater
or a woodstove, but you have another source of heat: the same device
that was chilling the air in the first place. Instead of sending its
heat outdoors, you can use some of that heat to warm the chilled air
back to gallery temperature. But you have to be careful: if you use
all of the heat that's released by the chiller, the gallery air will
become hotter than the outdoor air! Why?
Answer:
Pumping the heat from cold to hot required an input of ordered energy.
That ordered energy becomes thermal energy in the pumping process
and this thermal energy ends up in the hot region. If you make the
gallery air the hot region, the gallery air will be hotter than
the outdoor air.
Why:
If you pump heat from one part of the gallery air to another part
of the gallery air, you are essentially operating an air conditioner
in the middle of the gallery. Sure, heat will move from one side
to the other, but as the air drifts around and mixes, any pumped
heat will end up where it started. But the pumping process itself
creates thermal energy out of ordered energy and this added thermal
energy will serve to raise the overall temperature of the air.
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3. You arrange your climate control
system so that as it pumps heat out of the new gallery air to chill
it and dry it, part of that heat goes to the outdoor air and part
to the indoor air. It turns out that sending some of this heat to
the cooler, indoor air (rather than the hotter, outdoor air) makes
the system more energy efficient--it uses less electricity! Why?
Answer:
Heat pumps require the least ordered energy when the temperature
difference between the cold region (from which they draw heat) and
the hot region (in which they deposit heat) is smallest.
Why:
This system is moving part of its heat from the air that it's chilling
to the air it is trying to heat back up to gallery temperature.
That's great because the temperature difference between these two
airs is small and pumping heat between them requires relatively
little ordered energy.
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4. On one particular day, the outside
air is exactly at gallery-air temperature. However, the outside air
is too moist to use directly. So your climate control system chills
it to extract some moisture and then reheats it back up to gallery
temperature. When all this is done, you have drier air in the gallery
and water running down the drain. Which system has more entropy: (1)
a gallery full of outside air (moist but at the right temperature)
or (2) a gallery full of dried air (drier and at the right temperature)
plus the extracted water in the drain?
Answer:
(1) has more entropy.
Why:
Entropy measures disorder and that disorder includes any type of
disorder. Separating the water from the moist air creates order,
so turning situation (1) into situation (2) involves ordering the
system and lowering its entropy. Another way of looking at this
is observing that situation (2) will become situation (1) naturally
if you let the water evaporate back into the dried gallery air,
but situation (1) won't spontaneously turn into situation (2).
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5. When the weather gets colder,
the climate control system must begin warming the gallery air. Amazingly
enough, it can do this relatively efficiently by transferring heat
from the outside air into the gallery. However, your system must use
electricity to carry out this heat transfer. Why won't the heat transfer
occur naturally?
Answer:
Heat flows naturally from hotter to colder. If the outside air is
colder than the gallery air, heat will not naturally flow from colder
outside air to warmer inside air.
Why:
The temperature part of climate control systems generally involves
pumping heat against its natural direction of flow. It's easy to
let the gallery air's temperature drift toward that of the outside
air, but it takes ordered energy to shift the gallery air's temperature
away from that of the outside. In this case, the outside air is
too cold and you must shift the gallery air's temperature higher.
You can do this by pumping heat in from outdoors, but only through
the use of ordered energy.
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6. Instead of pumping heat into
the gallery air from the outside air, you could simply put an electric
space heater inside the gallery. A space heater turns electric energy
directly into thermal energy. Why would operating a space heater consume
more electric power than operating a heat pump?
Answer:
The heat pump would deliver not only all of the electric energy
it consumed as heat into the gallery, it would also transfer additional
heat from outside into the gallery. The space heater would deliver
no more heat to the gallery than it received as electric energy.
Why:
The space heater does nothing clever with the wonderful, orderly
electric energy it consumes. It merely turns that energy into thermal
energy and dumps it into the gallery air. But the heat pump uses
the electric energy to move thermal energy around. Thermal energy
from outdoors joins the energy consumed by the pump and both are
deposited in the gallery air.
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7. When the air outside is very
cold and dry, your climate control system must humidify the gallery
air. It lets pure water evaporate into the air and raises the moisture
content of that air. As this evaporation occurs, the temperature of
the air drops somewhat because thermal energy is used to turn the
liquid water into water vapor. But lowering the temperature of something
usually means lowering its entropy. Why doesn't this process of mixing
water and air to create slightly colder, moister air violate the second
law of thermodynamics (the one requiring that overall entropy never
decrease)?
Answer:
Letting the two separate materials (water and air) mix creates disorder
and raises the overall entropy.
Why:
Even though the entropy associated with thermal energy drops, the
entropy associated with mixing separated materials increases. Overall,
entropy increases during evaporation!
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8. Keeping the gallery door closed
is important, so you install a sophisticated closing system that keeps
the air exchange between inside and outside to a minimum. Why will
such air exchanges always make your gallery consume extra electricity
(and money)?
Answer:
Such exchanges will always shift the gallery air's temperature and
humidity away from its target and require that your climate control
system pump heat and moisture in or out of the gallery air. That
pumping consumes electricity.
Why:
Heat and moisture flows between the controlled gallery air and uncontrolled
outside air always creates disorder and entropy. Your climate control
system has to undo that entropy creation by "exporting" that entropy
somewhere else. Exporting entropy takes ordered energy.
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