**8. References**

442 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

123456 *Mach number for Perfect Gas*

Fig. 19. Correction of the Mach number at High Temperature of a nozzle dimensioned on

0 2 4 6 8 10 12 14 16 *Non-dimensional X-coordinates*

If we accept an error lower than 5%, we can study a supersonic flow using a perfect gas relations, if the stagnation temperature *T0* is lower than 1000 K for any value of Mach number, or when the Mach number is lower than *2.0* for any value of *T0* up to

The *PG* model is represented by an explicit and simple relations, and do not request a high time to make calculation, unlike the proposed model, which requires the resolution of a nonlinear algebraic equations, and integration of two complex analytical functions. It takes

The basic variable for our model is the temperature and for the *PG* model is the Mach number because of a nonlinear implicit equation connecting the parameters *T* and *M*.

Fig. 20. Shapes of nozzles at high temperature corresponding to same Mach number

variation througout the nozzle and given *MS*=3.00 at the exit.

From this study, we can quote the following points:

more time for calculation and for data processing.

0

the perfect gas model.

0.0 1.0 2.0 3.0 4.0 5.0

**6. Conclusion**

approximately 3000 K.

1

2

3

4

5

<sup>6</sup> *M (HT)*


**17** 

Illia Dubrovskyi

*Ukraine* 

(1)

**R** - the characteristic

**R R** . Here the

**Statistical Mechanics That Takes into** 

**Law - Theory and Application** 

*Institute for Metal Physics National Academy of Science* 

**Account Angular Momentum Conservation** 

The fundamental problem of statistical mechanics is obtaining an ensemble average of physical quantities that are described by phase functions (classical physics) or operators (quantum physics). In classical statistical mechanics the ensemble density of distribution is defined in the phase space of the system. In quantum statistical mechanics the space of functions that describe microscopic states of the system play a role similar to the classical phase space. The probability density of the system detection in the phase space must be normalized. It depends on external parameters that determine the macroscopic state of the

An in-depth study of the statistical mechanics foundations was presented in the works of A.Y. Khinchin (Khinchin, 1949, 1960). For classical statistical mechanics an invariant set was introduced. It would be mapped into itself by transforming with the Hamilton equations. The phase point of the isolated system remains during the process of the motion at the invariant set at all times. If the system is in the stationary equilibrium state, this invariant set has a finite measure. The Ergodic hypothesis asserts that in this case the probability d*P* **R**

function of the invariant set, which is equal to one if the point **R** belongs to this set, and is

element. The number of distinguishable states in a phase space volume element d is

. The system that will be under consideration is a collection of *<sup>N</sup>* structureless

integral goes over all phase space . This is microcanonical distribution. A characteristic

A hypersurface in a hyperspace is a set with zero measure. Therefore the invariant set is determined as a thin layer that nearly envelops the hypersurface in the phase space. The

 **R**

<sup>d</sup> <sup>d</sup> 2 ! *<sup>N</sup> <sup>P</sup>*

**R**

particles. The averaged value of a phase function *F***R** is *FF P* d

where - the measure (phase volume) of the invariant set ;

equal to zero in all other points of the phase space; =1 d= d d *<sup>N</sup>*

<sup>3</sup>

*N*

*f z* **R** , where *f* **R** is a phase function and *z*

*i i <sup>i</sup>* **p r** - the phase space volume

to detect this system at any point **R** of the phase space is:

**1. Introduction** 

system.

 <sup>1</sup> <sup>3</sup> 2 ! *<sup>N</sup> <sup>N</sup>*

is it's fixed value.

function often would be presented as

