Compression and Expansion of Gases

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/m²)

ρ = density (kg/m³)

The isothermal process can also be expressed as

pV = constant                    (1a)

or

p₁V₁ = p₂V₂                           (1b)

where

V = gas volume (m³, ft³...)

suffix₁ denotes initial conditions and suffix₂ 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

p₁V₁^k  = p₂V₂^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

pVⁿ = constant                      (3a)

or

p₁V₁ⁿ  = p₂V₂ⁿ                       (3b)

where

n = polytropic index or exponent (ranging 1 to 1.4)