What is an intuitive explanation for the concept of heat pumps? I know that it is basically a reversed Carnot process. We can for example take an amount of heat $Q_1$ out of a warmer system and transform part of it into work W. The rest goes to the colder system. If we now reverse that process we need to take heat out of the colder system. For this to be done we need the same amount of energy W we got out of the process previously. But here my problem starts: How do you force the energy to come out of the colder reservoir? How can you explain that without just saying that it is an inversed Carnot process?
You describe the process. For an example, you might think of how a gas refrigerator works. You take a gas and expose it to the cold area, which cools it to the cold temperature. Then you isolate it from the cold area, which costs no energy. You compress it, raising the temperature above that of the hot reservoir, at the cost of physical work. You then connect it to the hot reservoir and let some heat escape. Disconnect it again, then expand it to a temperature below the cold reservoir, recovering some energy in the process. Now connect it to the cold reservoir and you have a cycle with work in and heat moving from cold to hot. This is really just adding details to saying an inverse of the Carnot process.
You use a pump on a separate fluid (gas) system which is connected to the freezer (for example). The pump lowers pressure while the pipe system controls the volume of the fluid gas, and then this fluid can reach a smaller temperate than the internal of the freezer.
The freezer thus passes heat to the fluid system and cools down. The fluid cycles through the pipes to a part with higher pressure etc, which let it raise its temperature, so it can send away the heat to the surroundings.
The issue here is, that this pump performs an amount of work, and this is where $W$ enters the equation.