Your thinking may be unduly influenced by your experience with home Air Conditioning. That is ALWAYS done with a powered module outside the house that uses a fan to exhaust heated air. But the overall mechanism is much more complex. That unit outside contains a compressor that accepts warm refrigerant gas from the Evaporator Coil in the inside unit (whether that also contains a furnace or not) and compresses it to a much higher pressure. Under these higher pressure conditions that refrigerant gas is at even higher temperatures - much hotter that the outside air. THEN the hot gas is passed through a Condenser (heat exchanger) that has a fan blowing warm outside air over it. However the warm air is still MUCH cooler than the hot compressed gas inside, so heat IS removed from the gas, reducing its energy content so low that it condenses into a liquid form. The heat energy removed is the Heat of Vaporization of the refrigerant - that is, the difference in heat energy between the liquid and gaseous forms of that material. The resulting liquid has a temperature just a bit higher than the outside air that cooled it. So the cooled liquid then flows through a high-pressure pipe back to the interior cooling system. It is fed into the Evaporator (another heat exchanger) inside that through an Expansion Valve that restricts the rate of flow of the liquid into the Evaporator, so that the Pressure inside the Evaporator is MUCH lower than in the high-pressure liquid line. This allows the liquid to evaporate into a gas again. To do that it requres heat, so the gas gets very cold, and the in-the-house air blowing (by fan) through the inside unit then is chilled a lot by the cold Evaporation fins. This move heat from the room air into the refrigerant gas. The final result in here is that the gas is now re-warmed and heads back to the outside Compressor unit.
The key point here is that a refrigeration unit uses devices to cause CHANGES of STATE (between liquid and gas forms) of the refrigerant material at TWO places. In each location the heat exchanger has a hot side and a cold side so that the AIR passing over it is COOLED inside the house and HEATED outside the house. In both locations there is a big temperture difference created by the system to maxmize the rate of heat transfer.
A computer liquid-cooling system is a MUCH simpler system. It uses a liquid that never changes its form to absorb heat from the CPU chip with a modest temperture difference - the CPU interior temp can get up to 100 C at worst - and the heated liquid leaving the contact block on the way to the cooling rad is just a bit above room temperature. That's a fair temp difference. But when that fluid passes though the rad and is cooled by room air, cooling can happen only to the point that the cooled liquid coming BACK from the rad is only a couple degrees warmer than the AIR used to cool the rad. This system cannot move a LOT of heat rapidly, and is limited by how cool the room air is.
Bear this in mind also. A CPU may be the hottest chip in your machine (the GPU chip wull be close to that, too), but the highest-power CPU chips now in common use convert into heat at a maximum rate of 150 W, and AVERAGE heat generation is lower. That is the same as the heat generated by ONE traditional incandescent lamp of 100 W to 150 W. Few people would go to a lot of effort to design a system to move that small amount of heat outside.
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