Compression and Expansion of Gases
Isothermal and isentropic gas compression and expansion processes.
The relationship between pressure and density when compressing  or expanding  a gas depends on the nature of the process. The process can be
 isothermal,
 isentropic (adiabatic)
 polytropic
Isothermal Compression/Expansion Processes
If compression or expansion of gas takes place under constant temperature conditions  the process is said to be isothermal. The isothermal process can be expressed with the Ideal Gas Law as
p / ρ = constant (1)
where
p = absolute pressure (Pa, N/m^{2})
ρ = density (kg/m^{3})
The isothermal process can also be expressed as
pV = constant (1a)
or
p_{1}V_{1} = p_{2}V_{2 }(1b)
where
V = gas volume (m^{3}, ft^{3}...)
suffix_{1} denotes initial conditions and suffix_{2} denotes final conditions
Isentropic (or adiabatic) Compression/Expansion Processes
If compression or expansion of gas takes place with no flow of heat energy either into or out of the gas  the process is said to be isentropic or adiabatic. The isentropic (adiabatic) process can be expressed with the Ideal Gas Law as
p / ρ^{k} = constant (2)
where
k = c_{p} / c_{v}  the ratio of specific heats  the ratio of specific heat at constant pressure  c_{p}  to the specific heat at constant volume  c_{v}
The isentropic or adiabatic process can also be expressed as
pV^{k }= constant (2a)
or
p_{1}V_{1}^{k} = p_{2}V_{2}^{k } (2b)
Polytropic Compression/Expansion Process
An ideal isothermal process must occur very slowly to keep the gas temperature constant. An ideal adiabatic process must occur very rapidly without any flow of energy in or out of the system. In practice most expansion and compression processes are somewhere in between, or said to be polytropic.
The polytropic process can be expressed as
pV^{n }= constant (3a)
or
p_{1}V_{1}^{n} = p_{2}V_{2}^{n }(3b)
where
n = polytropic index or exponent (ranging 1 to 1.4)
Related Topics

Fluid Mechanics
The study of fluids  liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time.
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