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The miner will have free solar + very cheap grid electricity but regular power cuts, so I was wondering if anyone had experience with the miner switching off at least once per day for a few hours. The environment will also be quite high humidity. Will this make my project completely unfeasible, or can Antminers handle this?
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It is not that simple:
For the longevity of the miner, it is better to have it shut down when it is cooled a bit. For that, the fans should run while the hashboards stop mining. You can use the "Sleep" or "Standby" mode on some miners. Although everyone will tell you, it does not matter - and they are right in general. Physically you will do your miners good, I can not tell you how big of a factor it is.
I have experience in regular blackouts. You need to make sure that the miners do not get overcurrent. In my area sometimes the power goes on and off several times in seconds before it comes back. That fried several miner until I installed a switch, that keeps them off until you physically turn them on again, once the energy is safe again. So the message for you here should be, that you make sure your miners only get "good" power.
Humidity is a problem, but it is more of a problem, when your ambient is cold and it condensates. I often have high humidity, but if the miners run and are hot and the ambient inside the room or chamber or whereever they are is hot, it should not be a problem. In your case, if you have them run with solar and they turn off in the evening, it might get cold over night and there might be condesate in your miners in the morning, if your ambient is too low. You can prevent this by having e.g. a miner run permanently to heat the ambient a bit. I would install a humidity sensor to check, so you can find a good solution for this.
Regarding the heatsinks: I am courious, too. I have S19XP incoming.