what's this and what does it do?

inline shutoff valve with a small somthing on the outflow side with a small plastic tube running into the overflow pan for water heater.
This small tube has a steady stream pooring out.
What is is and what does it do?

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Would it not be better to install a expansion tank?

Only if you don’t want to waste water.

Sounds, by the description of it being on the “outward flow” side, that it is improperly installed.

Sounds like it needs replacing…and Jim’s right, it should be installed on the cold water inlet side, downsteam of the shut off.