That's an overclocked PC... at least 4.0 GHz. I recommend getting an extremely powerful fan. Your budget should be around $300 buddy, because to chill 4 gigs you're going to need some equipment
Bullcrap. 4.0 GHz is the stock speed for that CPU. Cooling the stock 4.0 GHz doesn't take all that much, any of the decent, modern air coolers will do the job quite well, for less than $100. Only when he starts cranking up the clock speed (and voltage) will he need to think about better cooling and there's way more to cooling effectively than just a powerful fan, unless you like noise.
There are three ways to increase cooling capacity: Increase surface area of the heatsink, increase air flow or increase both.
Since he wants to overclock, and as he describes it "extreme" overclocking, and considering he has space limitations inside the case he is going to be relegated to something other than air cooling. Liquid (or water cooling) is typically the next step up, but water cooling isn't "magic", there is nothing special about water over air in terms of its ability to cool as both air and water are actually poor conductors of heat. They are actually classified as insulators, not conductors.
Most liquid cooling systems also use an additive, like anti-freeze in a car radiator to not only prevent freezing, but to increase heat transfer. Even with that, the purpose of going to liquid cooling is to provide a transport mechanism for the heat so you can move the heat to larger radiators (i.e. more surface area) outside the case to dissipate the heat. No matter which cooling system you use, in the end, you are always dissipating the heat to the surrounding air. Liquid allows you to increase your heatsink (radiator) surface area because outside the case, you aren't constrained for size and larger heatsinks can hold larger fans (more volume flow). If you do it right, there's no need for stupidly loud, high speed fans.
Clock speed alone is not a determining factor for how hot a CPU will get. You must consider the thermal design power (TDP) as well. To give you an example, the i7-6700K has a TDP of 95 watts. My i7-5820K has a TDP of 140 watts, yet it runs at a slower clock speed of 3.3 GHz.
Another factor that will affect cooling is the physical size of the CPU. Smaller surfaces don't give up heat as fast as a larger surface (hence, increasing the surface area of the heatsink), but in the case of the CPU, its surface area is set. Theoretically, at some point increasing the size of the heatsink will have no further cooling ability as the heat will not be able to transfer from the CPU to the heatsink any faster.
Using the word "extreme" relative to overclocking is subjective. However, we can probably assume he intends to push his CPU to the absolute limits he is able to attain. For that, it will depend largely on how much cooling he can afford and there are many ways it can be done. Air, classic liquid cooling, combinations of peltiers with other heat transport mechanisms, or even phase change. Each method getting more and more expensive. The absolute most "extreme" would be something like a liquid nitrogen system.
Regardless, overlcocking a CPU is akin to building a high HP, fast car and the old saying holds true in both cases: "Speed is a matter of money, how fast can you afford to go?"
Edited by Ram4x4, 21 May 2016 - 09:43 PM.