Can we stop writing essays about what thermodynamics is, and help the OP instead?
Would love to. But posting incorrect information is not helping the OP.
Sure. Lets get away from your general statements and calculate instead, what force is acting on a body of air say 20°C hotter than surrounding air trapped in a volume 40x120x1,5mm. I just measured these dimensions on a traditional cooler. You can then recalculate this force using area of 120x1,5mm to pressure and compare this figure to a pressure created by a fan.
You are not making any sense. You cannot randomly pull arbitrary numbers out of thin air, pretend they represent the whole, then use that to make moot the real point. And who says what the dimensions of a "traditional" cooler are? Coolers come in all shapes and sizes. Even OEM coolers come in all shapes and sizes these days.
But also, it is not even about the dimensions as much as it is about the total
surface area of the fins, not to mention the composition of the metal. Copper, for example, conducts heat better than aluminum. The design of the heatsink matters. The use of heatpipes, for example, to more quickly move the heat from the block to the fins can make a significant difference.
The air is not "trapped". So why you are trying to suggest it is, is simply, and deceptively misleading. There is no reason any air should be trapped inside a computer case. Period. If it is, then it is time clean out the dust, or remove the cover you put over the heatsink or the case's vents.
the cooler needs to heat up anyway to move enough air. Check reviews of Noctus NH-P1 cooler and see, how hot is it getting.
Come on! This is just total nonsense. The cooler does not "NEED" to heat up. That is NOT how it works. If the temperature of the cooler is just 1° warmer than the surrounding air, heat
WILL start to move into that air. The Laws of Physics, dude! Cold is the absence of heat. Heat is energy. Heat moves into cold.
What you are suggesting is if I put a monstrous heat sink on a CPU, a heat sink 10 times bigger and more efficient than it needs to be, because it will not get hot, it will not provide enough cooling for the CPU. Does that make sense? Of course not.
You are suggesting that Noctus cooler is only doing any good when it gets "hot". Total nonsense. It gets hot because the heat from the CPU has moved into it.
Now "hot" is a relative term. What I should have said above (my apologies for not doing so), is
"IF" the heatsink is getting
"TOO" "HOT", then (1) it is too small for the job and/or (2) there is not enough space around the heatsink to allow the heated air to freely move away and/or (3), there is not an adequate supply of cool air coming in.
Now, in light of SL2's reminder, unless the OP has further questions, I agree it is time to move on.