Cooling Clara's Car
Clara's car has been sitting in the sun on a hot day. She wants to drive to a restaurant a few minutes away, but she finds that the air inside the car is excruciatingly hot. However, the air outside the car is sufficiently comfortable. Clara would like to cool the car's air down as fast as she possibly can. She realizes there are two effects that can lower the temperature:
- cooling down the air that is already present in the car
- replacing the hot inside air with cooler air
The air conditioning system in Clara's car, like in most cars, gives the option between recirculating the inside air and taking air from outside. Regardless of what she does, the total rate at which air leaves the car will equal the rate of air intake.
The air in Clara's car is also well-mixed at all times. That is, whenever air enters the cabin, it mixes with the inside air quickly enough that, for all intents and purposes, the air temperature is the same at all points in the cabin. This is the temperature of any air that exits the car.
How should Clara cool her car?
| Outside air setting | Recirculation setting | |
| Windows closed | A | B |
| Windows partially open | C | D |
| Windows fully open | E | F |
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1 solution
Any cooling abilities of the car will be best used on air that is already as cold as possible beforehand. Since the outside air is cooler than the inside air, she should therefore use outside air instead of recirculating the inside air.
She should also open the windows as much as possible because some combination of the following two effects will be achieved:
All this applies only if the insider air is hotter than the outside air . Once the inside air reaches the outside temperature, any further cooling will be best achieved using recirculation with windows closed because from that point on the coolest source of air is the inside air itself. However, in this problem, the air is already comfortable by the time this is achieved.
As an additional side note, in real life, the inside air will not be extremely well mixed. This means it is not so clear whether the windows should be open as you might lose cooled air before it has time to mix. On the other hand, you do still get more exchange with outside air and a greater rate of purging the hot inside air. In fact, if the cold air falls to the bottom--away from the windows--then opening the windows will help even more than in Clara's case!