Characteristic temperature Compressibility factor

Here / is the universal gas constant (=8314 J/kgmol K), vv is the molecular weight of the fluid, R = R/w is the characteristic gas constant and Z is the compressibility factor, which will be a function of temperature in conditions of boiling or condensing. 

Since there is no inlet velocity, the stagnation values of temperature, pressure etc. are unchanged from the basic values. Treating air as a perfect gas with a compressibility factor of unity, we may use the characteristic gas equation to calculate the specific volume at nozzle inlet as... 

Here the expansion is done in powers of inverse volume, which guarantees that in the ideal-gas state (V= 00) the compressibility factor is Z = 1. This series expansion is known as the virial equation and its coefficients as the virial coefficients B is the second virial coefficient, C is the third, and so on. These are characteristic of the fluid and depend on temperature only. 

The equation for the compressibility factor is a cubic polynomial in Z. This is the reason that the van der Waals equation is referred to as a cubic equation of state. The parameters A and B depend on pressure and temperature. If T and P are given, the compressibility factor can be calculated from eg. r2. 7l As a cubic polynomial, this equation may have one or three real roots. Multiplicity of roots is a characteristic property of equations of state that are capable of describing phase transitions. 

These forms of a generalized equation of state only require the critical temperature and the critical pressure as substance-specific parameters. Therefore, these correlations are an example for the so-called tsvo-parameter corresponding-states principle, which means that the compressibility factor and thus the related thermodynamic properties for all substances should be equal at the same values of their reduced properties. As an example, the reduced vapor pressure as a function of the reduced temperature should have the same value for all substances, provided that the regarded equation of state can reproduce the PvT behavior of the substance on the basis of the critical data. In reality, the two-parameter corresponding-states principle is only well-suited to reflect the properties of simple, almost spherical, nonpolar molecules (noble gases as Ar, Kr, Xe). For all other molecules, the correlations based on the two-parameter corresponding-states principle reveal considerable deviations. To overcome these limitations, a third parameter was introduced, which is characteristic for a particular substance. The most popular third parameter is the so-called acentric factor, which was introduced by Pitzer ... 

The principle of corresponding states can be used to express the pressure, tan-perature, and specific volume in terms of reduced variables. Experimental observations reveal that the compressibility factor, Z Equation (2.24), for different fluids exhibits similar behavior when correlated as a function of reduced temperature, T, and reduced pressure, The reduced variables may be defined with respect to some characteristic quantity. For example, they can be defined as follows with respect to critical temperature and critical pressure ... 

EOV EOS theory was developed by formulation of canonical function of the Boltzmann distribution of energies and derivation of thermodynamic pressure. The theorem of corresponding states says that the same compressibility factor can be expected for all fluids when compared at reduced temperature and pressure. A two-parameter correlation for compressibility factor, Z, can be derived using the theorem of corresponding states. EOV EOS obeys the corresponding state principle. Characteristic temperature, pressure, and specific volume used in EOV EOS are tabulated for 16 commonly used polymers. 

Chueh s method for calculating partial molar volumes is readily generalized to liquid mixtures containing more than two components. Required parameters are and flb (see Table II), the acentric factor, the critical temperature and critical pressure for each component, and a characteristic binary constant ktj (see Table I) for each possible unlike pair in the mixture. At present, this method is restricted to saturated liquid solutions for very precise work in high-pressure thermodynamics, it is also necessary to know how partial molar volumes vary with pressure at constant temperature and composition. An extension of Chueh s treatment may eventually provide estimates of partial compressibilities, but in view of the many uncertainties in our present knowledge of high-pressure phase equilibria, such an extension is not likely to be of major importance for some time. 

From the thermodynamic standpoint, the basic components of stars can be considered as photons, ions and electrons. The material particle gas (fermions) and the photon gas (bosons) react differently under compression and expansion. Put n photons and n material particles into a box. Let R be the size of the box (i.e. a characteristic dimension or scale factor). The relation between temperature and size is TR = constant for the photons and TR = constant for the particles. This difference of behaviour is very important in the Big Bang theory, for these equations show quite unmistakably that matter cools more quickly than radiation under the effects of expansion. Hence, a universe whose energy density is dominated by radiation cannot remain this way for long, in fact, no longer than 1 million years. 

The dependence of heating on pressure itself may be due to a decrease, with increasing pressure, of the characteristic size of dispersion grains. There is also an alternative explanation, connected with increasing devitrification temperature of a compressed matrix this factor causes a displacement of the temperature region of the intense recombination of active centers and creates the conditions for a higher degree of conversion at the front. 

---