Just be careful with condensing when you build these systems.
My tap water is well below +10C, pipes run in bathroom and at summer those pipes are often rather wet when I run water, air temperature is higher and can contain more humidity than surface of pipes where temperature is lot less, so water condenses to that cold surface from the indoor air.
This might sound a little, but when pipes constantly are cool it can amount a lot water during 24hours and water is not good at wrong places in the house. Some insulation that is not letting humidity enter to insulation and pipe surface, would solve that and naturally improve efficiency of the system.
Old radiators have had coolant in them probably, here we use Glyckol/water mix of 50/50, your area may wary, but I would be cautious of running water through radiator, some closed loop heat exhanger made from simple bucket and copper spiral might be already better to stay safe side.
Just some random thoughts that came up while reading this stuff.
Table fan, carbdboard + duct tape might be also nice way to get air moving trough radiator with low cost and ok efficiency, also put fan to such air from radiator for lower noise and greater efficiency, that is what I learned when playing with PC water cooling, there I had car heater matrix that I used to cool cooling liquid for pc, system is really similar to this, only that it used metal block to cool cpu and gpu, with house cooling scale is just bigger and block is replaced with cool water source, it does not matter if it is heat or cold that is radiated, only with cold one need to be careful with humidity and condencing, which also comes to play with computer cooling when you go extreme.
Anyway, guys doing water cooling systems by themselves know quite a bit about air and water flow and some even know how to deal with condensing issues too, one just need to scale that up a bit.
4-5C increase in house temperatures when I use my current air cooled pc, boy have I thought about setting up water cooling and to went that heat outside to keep house temps down, at winter of course another radiator to warm the house, cooling is not necessary here, temps are still below +20C daytime, even there was three days with temps bit higher, even up to +24C one day, but less than 10 days this summer when temperature has been above +20C.
Anyway, copper coil to fireplace and transferring heat with such setup to upstairs is something I have been thinking, again lot of similar things applies as with cooling, radiator with fan is so much better transferring heat compared to those normal no fan models that are used in houses.
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