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My concern is will this shorten the life of the motor? And, is it ok to
leave the fan on 24/7 for 2, 3, 4, 5 days? My thinking is to leave it on
for whatever the duration of the heatwave. Correct?

I was told that it wouldn't, but I've replaced my motor twice in 8 years. I
leave mine run 24/h a day 7 days a week.