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All industries can use air compressors.
I have not see them in Hospitals since most air used their is purified oxygen. (extra clean)
Factories are the most common as the use air pressure to operate tools and machines.
This could be anything from hand tools to presses or arms that lift or move parts.
They often use air operated cylinders to open valves or gates to control the flow of product to the manufacturing process.
I have seen sizes from a small three to five gallon tanks for home or shop use to as big as a truck to supply an aircraft hanger.