An equation of state (for gases) is a mathematical model used to roughly describe or predict the state properties of a gas. At present, there is no single equation of state that accurately predicts the properties of all gases under all conditions. Therefore, a number of much more accurate equations of state have been developed for gases in specific temperature and pressure ranges. The "gas models" that are most widely discussed are "Perfect Gas", "Ideal Gas" and "Real Gas". Each of these models has its own set of assumptions to facilitate the analysis of a given thermodynamic system[13]. Each successive model expands the temperature range of coverage to which it applies. The image of first powered flight at Kitty Hawk, North Carolina illustrates one example on the successful application of these relationships in 1903. More recent examples include the 2009 maiden flights of the first solar powered aircraft, the Solar Impulse, and the first commercial airliner to be constructed primarily from composite materials, the Dreamliner.
Perfect gas
By definition, a perfect gas is one in which intermolecular forces are negligible due to the separation of the molecules and any particle collisions are elastic.
Perfect gas equation of state
The symbol n represents the number of particles grouped by moles of a substance. All other symbols in these equations use notation described earlier in the Macroscopic Section. These relationships are valid only when used with absolute temperatures and pressures.
- Chemist's version – PV = nRT
The gas constant, R, in this expression has different units than the Gas Dynamicist's version. The Chemist's version emphasizes numbers of particles (n), while the latter emphasizes the particle mass in the density term ρ.
- Gas Dynamicist's version- P = ρRT
There are two subclassifications to a perfect gas although various textbooks either omit or combine the following simplifications into a general "perfect gas" definition. For sake of clarity, these simplifications are defined separately in the following two subsections.
Calorically perfect
The Calorically perfect gas model is the most restrictive from a temperature perspective[14], as it adds the following condition:
- Constant specific heats (valid for most gases below 1000 K)
- u = CvT, h = CpT
Here u represents internal energy, h represents enthalpy, and the C terms represent the specific heat capacity at either constant volume or constant pressure, respectively.
Although this may be the most restrictive model from a temperature perspective, it is accurate enough to make reasonable predictions within the limits specified. A comparison of calculations for one compression stage of an axial compressor (one with variable Cp, and one with constant Cp) produces a deviation small enough to support this approach. As it turns out, other factors come into play and dominate during this compression cycle. These other effects would have a greater impact on the final calculated result than whether or not Cp was held constant. (examples of these real gas effects include compressor tip-clearance, separation, and boundary layer/frictional losses, etc.)
Thermally perfect
A thermally perfect gas is:
- in thermodynamic equilibrium
- not chemically reacting (chemical equilibrium)
- cp - cv = R (still valid even though specific heats vary with temperature)
- Internal energy, enthalpy, and specific heats are functions of temperature only.
- u = u(T), h = h(T), du = CvdT, dh = CpdT
This type of approximation is useful for modeling, for example, a turbine where temperature fluctuations are usually not large enough to cause any significant deviations from the thermally perfect gas model. Heat capacity is still allowed to vary, though only with temperature and the molecules are not permitted to dissociate.[15]
Ideal gas
An "ideal gas" is a simplified "real gas" with the assumption that the compressibility factor Z is set to 1 meaning that this pneumatic ratio remains constant. A compressibility factor of one also requires the four state variables to follow the ideal gas law.
This approximation is more suitable for applications in engineering although simpler models can be used to produce a "ball-park" range as to where the real solution should lie. An example where the "ideal gas approximation" would be suitable would be inside a combustion chamber of a jet engine[16]. It may also be useful to keep the elementary reactions and chemical dissociations for calculating emissions.
Real gas
Each one of the assumptions listed below adds to the complexity of the problem's solution. As the density of a gas increases with pressure rises, the intermolecular forces play a more substantial role in gas behavior which results in the ideal gas law no longer providing "reasonable" results. At the upper end of the engine temperature ranges (e.g. combustor sections - 1300 K), the complex fuel particles absorb internal energy by means of rotations and vibrations that cause their specific heats to vary from those of diatomic molecules and noble gases. At more than double that temperature, electronic excitation and dissociation of the gas particles begins to occur causing the pressure to adjust to a greater number of particles (transition from gas to plasma).[17] Finally, all of the thermodynamic processes were presumed to describe uniform gases whose velocities varied according to a fixed distribution. Using a non-equilibrium situation implies the flow field must be characterized in some manner to enable a solution. One of the first attempts to expand the boundaries of the ideal gas law was to include coverage for different thermodynamic processes by adjusting the equation to read pVn = constant and then varying the n through different values such as the specific heat ratio, γ.
Real gas effects include those adjustments made to account for a greater range of gas behavior:
- Compressibility effects (Z allowed to vary from 1.0)
- Variable heat capacity (specific heats vary with temperature)
- Van der Waals forces (related to compressibility, can substitute other equations of state)
- Non-equilibrium thermodynamic effects
- Issues with molecular dissociation and elementary reactions with variable composition.
For most applications, such a detailed analysis is excessive. Examples where "Real Gas effects" would have a significant impact would be on the Space Shuttle re-entry where extremely high temperatures and pressures are present or the gases produced during geological events as in the image of the 1990 eruption of Mount Redoubt.
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