So with your design, if you would put the tubes filled with room temperature water, directly on the processor fins, the heat would transfer to the tubes much faster than with a fan. Ultimately you still need to air cool though. But with liquid cooling you can move the heat outside the laptop and then disperse the heat in as big of a radiator as you want. Like in your case, you could put tubes on the 4 square inch processor and then pump the water outside the laptop into a 10 inch square radiator with a big cooling fan on it. This is almost the same as having a 10 inch cooling fan directly inside the laptop, but of course you can't do that.
The liquid cooling gives you the ability to move the cooling to another area that is easier and larger surface area to disperse heat.
Just like a car engine. The liquid flows past the cylinders and only flows with an area of about 1 square foot. But then the water is moved to the radiator where the heat is dispersed in a 3 square foot radiator, so it removes much more heat than if you had to put cooling fins directly on the cylinders