There are many different kinds of heat pumps, but they all operate on the same basic principle -- heat transfer. This means that rather than burning fuel to create heat, the device moves heat from one place to another. There's a key to making this all happen -- heat naturally flows downhill. This means that it tends to move from a location with a high temperature to a location with a lower temperature. Pretty simple. What a heat pump does is use a small amount of energy to switch that process into reverse, pulling heat out of a relatively low-temperature area, and pumping it into a higher temperature area. So heat is transferred from a "heat source," like the ground or air, into a "heat sink," like your home.
One of the most common types of heat pumps is the air-source heat pump. This marvel of modern technology takes heat from the air outside your home and pumps it inside through refrigerant-filled coils, not too different from what's on the back of your fridge. The air source variety is pretty basic, and you'll find two fans, the refrigerator coils, a reversing valve and a compressor inside to make it work.
And since you asked, here's how this kind of heat pump works:
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This system is more commonly known as an air-air heat pump, because it takes heat from outdoor air and transfers it to indoor air ducts. With the right kind of modifications, air-source systems can also work with other types of indoor heating systems.
The key to allowing the air-air heat pump to also cool is the reversing valve. This versatile part reverses the flow of the refrigerant, so that the system begins to operate in the opposite direction. So instead of pumping heat inside your home, the heat pump releases it, just like your air conditioner does. When the refrigerant is reversed it absorbs heat on the indoor side of the unit and flows to the outside. It's here that the heat is released, allowing the refrigerant to cool down again and flow back inside to pick up more heat. This process repeats itself until you're nice and cool.