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Given the overall pressure ratio of a multi-stage compressor, how can I calculate the increase in pressure at each stage?

Does pressure increase linearly at every stage, or does it increase logarithmically or exponentially?

Intuition tells me that pressure increases logarithmically since the latter stages of a compressor can't compress the air as much as the first stages, but intuition hasn't served me so well in thermodynamics...

Thanks!

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  • $\begingroup$ you can increase the later stages' efficiency by decreasing the area through which the air goes most turbine engines will do this $\endgroup$
    – ratchet freak
    Dec 3, 2014 at 21:33
  • $\begingroup$ That makes sense, but how does that affect the increase in pressure? $\endgroup$
    – techSultan
    Dec 3, 2014 at 22:26

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Firstly, your intuition doesn't really apply in terms of basic thermodynamics (but, as I'll explain, you're kind of right):

Boyle's Law: "For a fixed mass of gas, at a constant temperature, the product (pressure x volume) is a constant."

Pressure does increase linearly [assuming the temperature stays the same]: if you squash an amount of a gas into half the volume, the pressure will double (assuming you keep the temperature the same). If you squash it into half the volume again, the pressure will double again (4x the original)

However you're also right in that we don't simply increase pressure by the same amount as we go along: Each stage compresses the air by a different amount (otherwise an engine would look like a weird triangle). and perhaps more importantly we can't maintain a constant temperature in an engine... as we compress the air into a smaller volume the temperature increases unless the air is being cooled in some way.

Essentially, though, you can calculate the pressure change at each stage if you know how much the volume reduces by (ie how much smaller each chamber is than the last) and the temperature of the air in each.

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  • $\begingroup$ Exactly. it depends on how each compression stage is designed. $\endgroup$
    – rbp
    Dec 4, 2014 at 14:18
  • $\begingroup$ I forgot about Boyle's Law! Unfortunately, I don't know the change in volume or the design of the compressor stage, so I can't use it. $\endgroup$
    – techSultan
    Dec 6, 2014 at 2:17
  • $\begingroup$ For a fixed mass of gas The mass of gas is not fixed though, is it ? You can augment the pressure of a fluid in a conduit while maintaining a constant section by "slowing it down". Bernoulli's_principle $\endgroup$
    – YaJBoy
    Oct 26, 2019 at 10:29
  • $\begingroup$ @YaJBoy, the mass is always fixed, because mass is (in and off itself in a non-relativistic case) a conserved quantity. You use conservation of mass to relate the velocity with cross-section. Then you use conservation of energy (a.k.a Bernoulli's equation) to calculate the pressure. Assuming the density changes are negligible, you can solve the equations in that order, but if the pressure changes are large, you need to throw in the adiabatic compression equation and then you get a set of three equations. Of course an engine then adds some non-adiabatic expansion. $\endgroup$
    – Jan Hudec
    Oct 28, 2019 at 10:12

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