Heat exchangers are basically a chunk of conductive material, often aluminum, that takes heat from one object and allows it to be released into another. One thing to know about heat, or what is often called thermal energy, is that it wants to equilibrate. Heat will move from the hot place to the colder place, not the other way around, until it is nice and even.
Now, to understand the design of a heat exchanger, we need a bit of theory on the ways heat likes to move around. You have your three main ways for heat to move: conduction, convection, and radiation. The two main culprits at play in a heat exchanger are conduction and convection. Lets focus on those and ignore radiation.
Conduction is when two objects are in direct contact with each other and the heat moves from hot to cold. So you are holding a snowball, your hand conducts heat into it and the snowball feels like it is stealing your heat energy, which it is. Conduction does not work nice and linearly, it is much faster the larger the differences in temperature, which is why cold things can sometimes hurt. The materials matter too, some materials conduct heat much better than others, we call these materials conductors and their counterparts insulators.
Convection is another important piece of a heat exchanger. You can think of convection as a fluid stealing away heat from an object. That fluid can be most anything, and some are better than others. Think of standing outside on a nice fall day and the wind hits you hard, you feel rather cold rather quickly, say it is 55 degrees F ( 13 degree C) outside. Now, think of the last time you were in the water, and the water was 55 degrees F. VERY different story, you are not just a little cold you are freezing! This effect is much quicker if the water or air are moving faster, like a river. That is because water steals your heat much better than air does. So now we know that moving fluids (air/water) can steal you heat away rather well.
Now, why do they have fins.
Lets combine what we know now and add one more little piece to our new heat transfer knowledge. The efficiency of a heat exchanger is tied to its surface area and the amount of contact it has with the hot thing you are trying to cool. You want lots of good direct contact with your hot thing to allow conduction to do its job, you also want lots of contact with your fluid so it can steal away the heat faster via convection. The fins that you see on many heat exchangers are doing exactly this, they are increasing the surface area to allow convection to work much faster. More fins, faster cooling. Then you add in something to move your fluid across the fins and you are really cooling!
In conclusion:
Hot thing => conducts to metal chunk => convection steals away the heat from the metal=> heat is transferred away via the cooling fluid => the fins make the surface area larger so heat is transferred faster.
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