I am happy someone wants to discuss
the h in 1/(hA) is related to fluid conductivity (~10-20 times better for liquid metal vs. water) and heatsink geometry & conducivity (~same in either case). In my thermo textbook they actually derive values of h from this. So liquid metal cooling wins hands down here.
You are right! But this part isn't important in total thermal resistance
The second equation, which should be 1/(m'Cp), depends on the mass flow rate (m' = density times flow rate) and specific heat (Cp) of the fluid (~3 times better for water).
Water wins hands down here.
For a standard 180gph
(A pump for liquid metal with this number has not been developed yet and it is expected no this product in the future) pump: m' = 12.5g/s, Cp = 4J/g,
(for liquid metal, cp=0.365 J/g) so R = 1 / (12.5 * 4) = 0.02C / W. So an 80W cpu would lose only 1.6C from water or ~5C from liquid metal. Not much. Also keep in mind that heat transfer temperature.
Most cooling methods do nearly all of their cooling at the hottest parts near the tiny CPU, and not much everywhere else.
(Wrong, if you check carefully for any liquid cooling system, an ambient heat exchanger is always there. You must guarantee heat can be taken away by air finally. Otherwise, the system will die immediately. Therefore, total thermal resistance must includes this ambient heat exchanger. Unfortunately, liquid metal performs badly there.)
Using liquid metal (or a copper base plate, for example), helps spread out that heat a great deal, dramatically reducing the 1/(hA) term. So unless the flow rate is below, let's say ~60gph (which might be the case), liquid metal should win overall.
(Check the 1st page of this post and see the performance of Nanocooler's devices. Then think carefully and ask why its performance doesn't match your estimation. Another easy way is to call nanocoolers and tell them you have a better way to save liquid metal cooling technique. )
COP 3.0, huh? Well a TEC could be advertised with a COP of 2.0 assuming low load (which is how they normally rate COP).
(NOT COP, It is ZT) I think TECs are still by far the best solution for cooling CPUs below room temperature, because refrigeration is just too big and expensive. I'm working on a setup to overclock my next 2 video cards at between -57.5C (3 stage) and -20C (2 stage), depending on the card's power consumption (40W-80W). Over 8 years I'll supposedly get my money back plus ~$500 from not buying more expensive video cards. These estimates come from a close look at TEC performance curves, the thermal resistivities of my heatsinks, the contact resistance of thermal compound, etc. The supposed money saved is still up in the air, and I'll have to take a close look at some actual video cards to be sure.
Here's the secret: Run your TECs at 6V instead of 12V to get the higher efficiency. To get 6V, just wire 2 in series. That will make it easier to stack multiple stages and easier to cool the final stage. For a more conservative setup, I'd recommend running 4 72W TECs in parallel to cool an 80W load, or 2 in parallel to cool a 40W load. Don't supercool the CPU directly. Use a water block on the CPU and cool the fluid. This keeps your heat spread out more. 2-4 TECs may seem like a lot, but remember at 6V you don't get as much performance - your "72W" TECs aren't 72W anymore. They're roughly equivalent to 40W TECs, except for their higher efficiency. But it's worth it because power hungry 12V TECs are next to impossible to cool. Why use multiple 72W TECs instead of 1 powerful TEC? Again, it's to spread out the heat more. 4 times the area (and 4 times the heatsink size) means 1/4 the losses. By my numbers (which could be wrong), this setup should get your chip down to 17C with cheap (but decent) heatsinks & fans cooling the TECs, or as low as 12C with expensive heatsinks & fans cooling the TECs. Power consumption is only 20W per TEC, so you don't need a seperate power supply like most setups.
The common way to do this is to get a 212W TEC, a $48 heatsink, air cool it with a massive 60dB fan (!), get a very powerful case fan, get an auxillary power supply to run the TEC, get a relay to automatically turn on your auxillary power supply and then hopefully you'll be down to 20C. If that's not enough to spend by itself, your electricity bills will make you wonder why you didn't just buy a better PC. Not to mention the difficulty you'll have hearing your games with your fan running, unless you replace the fan and heatsink with a very expensive water cooling setup (your ordinary water cooling setup won't cut it). You want 2 stages to run it even colder? Okay, let's get 3-4 more 212W TECs...
There is some comparison among TEC solution and Non-TEC solution. I am sure you can find somewhere. The baseline is, you must have a heat sink with very low thermal resistance so that you don't need too much increase in temperature to dissipate the extra heat from TEC itself (electricity). Otherwise, it will get worse.