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It would be called a heat pump.
And yes, they’re used all the time. Modern heat pumps are more or less radiators with fans through which liquid is pumped. Air blown over is either heated (hot liquid,) or cooled (chilled liquid.)
In the us, you’ll see them more in comercial buildings rather than residential homes- they’re quite a bit more effecient, with heating or cooling compared with Moving the air the same distance
I think OP is talk specifically about hot water radiators.
He is asking why those rads specifically can't have cold water piped through them in the summer.
I think the answer for that is: they weren't designed to manage the condensation that would occur if the radiator (or pipe) temperature is lower than the dew point. Also, hot water can be a lot more above room temperature than cold water can be below. I think a lot of radiators are actually supplied with steam, not hot water which also lets them potentially use the latent heat of condensation of the steam for even more heat transfer.
From what I've gathered, water based heating systems pump around hot water, not steam. Otherwise, the radiators, which are easily touchable by anyone, would be insanely hot, and thus a significant fire and injury risk.
In order to use them for cooling, I suppose you're going to need a different transport medium, i.e. something like glycol instead of water. This would make the system harder and potentially a bit more dangerous to maintain, limits power when heating (water has higher specific heat than glycol), and you're still stuck dealing with condensation at the radiators.
Using an air system for centralised cooling works. Using a water system is much more problematic.
There are both water and steam radiators.
https://www.thespruce.com/hot-water-vs-steam-radiators-1821916
Most domestic radiators are water radiators, from what I've gathered in a country where most houses are fitted with water based heating.
Modern heat pumps are more or less radiators with fans through which liquid is pumped. Air blown over is either heated (hot liquid,) or cooled (chilled liquid.)
Heat pumps are more like air conditioners run in reverse. Air conditioners and refrigerators are heat pumps. They operate on the same thermodynamic cycle.
Technology Connections? *checks link* yep, Technology Connections.
I think they're becoming more common in residential in the US, at least that's the impression I get from the number of articles and videos that get pushed my way. They're not as good in colder climates, evidently, but hybrid systems where there is still a furnace but a heat pump for cooling seem to be a good alternative in those situations.