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I noticed while reading about compressors that they have various types and are generally made for a specific function. However, are there types of compressors (or pumps) that can compress both water and water vapor?

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Water at "normal" pressures is assumed to be incompressible.

However, if you subject water to about 200 atmospheres of pressure then its volume will reduce by about 1%.

So, water vapor will behave differently, depending on the ratio of gas to liquid. There are tables which you can consult to work out the density or specific volume for given temperatures.

Those tables are often called "steam tables".

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Just a remark to your terminology: if the main function is the TRANSPORT of gas, the device is called VENTILATOR. The pressure change is "small" (relative to an arbitrary reference). If the main function is the COMPRESSION of the gas, the device is a COMPRESSOR (high pressure change, eventually small mass flow). The label PUMP is mostly used for liquids. The TRANSPORT of liquids often need high pressure changes because of GRAVITATION. However as SolarMike says, that is not a compression in the sense that the rise in density is neglible in technical applications.

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