It's basically just a compressor and heat exchanger, same as a fridge or a freezer. So you'll always have a hot side and a cold side, and unsurprisingly there's a definite proportional relationship between surface areas and temperature. What sort of cold side temperature and deltaT are you aiming at, as that could widely vary the design of the heat pump.
You can try to find an old compressor from a freezer that is complete with a pretuned pressure reduction valve. You can make a simple condenser and evaporator out of copper pipe and solder all peaces together. The biggest problem might be to fill the system with liquid. If you have not already looked at this: http://en.wikipedia.org/wiki/Heat_pump
Well, that is the tricky part. It's been a while since I did science math but, I'm going out on a limb here and guessing that by deltaT you mean the change in Time or the amount of time it takes to change a variable? If so, I'm trying to use it to keep the PC cool when gaming. Basically, the idea is to use my counter rotating fans in the design to pump in and out the air and heat some how if possible. It is kinda like a water cooling system minus the liquid within the case.
deltaT is the difference between the cold side and the hot side temp. The deltaT is controlled by the two pressures in the system. You should remember that sub zero temps cause watercondensate, that would damage your computer parts. So one option is to go above zero and use an aircooler, or sub zero thats cooling a watersystem with anti freeze coolant in. The condeser can be placed on the roof.
^^ Actually whenever you go below ambient temperature, the moisture in the air starts to condensate on the cold surfaces. It doesn't need to be sub zero.
I think you should exclude the method of cooling the surrounding air because, as said above, there will most likely be moist in the air. The other choices I know of is cooling a secondary system of water, which you can cool down to around -40 degrees or whatever temp the water allows before freezing. Or cooling the parts directly down to even lower temps. Both these two options require special parts that can handle the low temps. Unless someone else have other input, my best choice is a water based system.
