Adiabatic expansion of a gas
What is an adiabatic process?
A process is called adiabatic when no heat is exchanged over system boundaries. In particular, this is the case when an expansion or contraction of a gas occurs so quickly that there is not sufficient time during the compression or expansion for the gas to exchange heat across system boundaries, for example to cylinder walls. The change dU in internal energy U is only done by work W.
For example, for an internal combustion engine running at 6000rpm, the compression phase takes one hundred of a second. This is not enough time to transfer heat in a substantial amount to the piston and the cylinder walls. On the other extreme, the rise or fall of air masses is also considered adiabatic, even if it takes hours. Heat is exchanged only at the boundaries where an air mass is in contact with a colder or warmer air mass. This contact surface is neglectable to the spatial extension of an air mass that extends hundreds of kilometers horizontally and some km vertically. It may take days before the air masses have exchanged heat and thus have completely mixed.
The state variables for for pressure P and volume V of a gas submitted to a reversible adiabatic process follow the formula:
or for the transistion from a first state 1 to a second state 2 the formula can be rewritten:
If we are interested to know the pressure P2 when a volume V1 of a pressure P1 is expanded to a volume V2 we can rearrange the formula to:
For a expansion ratio r = V2/V1 we finally arrive at :
Examples for adiabatic indexes
In theory, the so-called adiabatic index γ for ideal monatomic gases (noble gases) is 1.666; for diatomic gases (nitrogen, oxygen …) 1.4 and for triatomic gases, such as superheated steam 1.333. For nonideal gases, the adiabatic index is often renamed to κ and has to be determined experimentally. The adiabatic index is also a dependent on the temperature.
| gas | κ |
|---|---|
| air | 1.4 |
| steam, superheated [10] | 1.300 |
| steam, dry saturated [10] | 1.135 |
| steam, wet [10] | 1.113 |
| from pressure steam table p < 1atm | 1.069 - 1.062 |
| from pressure steam table p > 1atm | 1.069 - 0.856 |
| from temperature steam table T < 100°C | 1.080 - 1.064 |
| from temperature steam table T > 100°C | 1.080 - 1.099 |
The last four values were calculated from the steam tables related to steam of 100°C at atmospheric pressure. We see that steam obviously does not behave like an ideal gas. Especially the calculation of the temperature is very sensitive. For high expansion rates a variation of the third digit of kappa changes the calculated temperature by several degrees Celsius. It therefore seems more reliable to use the steam tables rather than calculations with the adiabatic index.