**4. Application**

The design of a supersonic propulsion nozzle can be considered as example. The use of the obtained dimensioned nozzle shape based on the application of the *PG* model given a supersonic uniform Mach number *MS* at the exit section of rockets, degrades the desired performances (exit Mach number, pressure force), especially if the temperature *T0* of the combustion chamber is higher. We recall here that the form of the nozzle structure does not change, except the thermodynamic behaviour of the air which changes with *T0*. Two situations can be presented.

The first situation presented is that, if we wants to preserve the same variation of the Mach number throughout the nozzle, and consequently, the same exit Mach number *ME*, is necessary to determine by the application of our model, the ray of each section and in particular the ray of the exit section, which will give the same variation of the Mach number, and consequently another shape of the nozzle will be obtained.

$$M\_S \text{(HT)} = M\_S \text{ (PG)} \tag{36}$$

$$M\_S \text{ (PG)} = \frac{\sqrt{2 \, H \left[ T\_{S \text{ (HT)}} \right]}}{a \left[ T\_{S \text{ (HT)}} \right]} \tag{37}$$

\* \* ( ) ( ) ( ) *A S S T\* TS HT F T dT A A HT e PG A A* (38)

The relation (36) indicates that the Mach number of the *PG* model is preserved for each section in our calculation. Initially, we determine the temperature at each section; witch presents the solution of equation (37). To determine the ratio of the sections, we use the relation (38). The ratio of the section obtained by our model will be superior that that determined by the *PG* model as present equation (38). Then the shape of the nozzle obtained by *PG* model is included in the nozzle obtained by our model. The temperature *T0* presented in equation (38) is that correspond to the temperature *T0* for our model.

The second situation consists to preserving the shape of the nozzle dimensioned on the basis of PG model for the aeronautical applications considered the *HT* model.

$$\frac{A\_S}{A\_\*} \text{(HT) } = \frac{A\_S}{A\_\*} \text{(PG)}\tag{39}$$

$$M\_{\mathbb{S}}\left(HT\right) < M\_{\mathbb{S}}\left(PG\right) \tag{40}$$

The relation (39) presents this situation. In this case, the nozzle will deliver a Mach number lower than desired, as shows the relation (40). The correction of the Mach number for *HT* model is initially made by the determination of the temperature *TS* as solution of equation (38), then determine the exit Mach number as solution of relation (37). The resolution of equation (38) is done by combining the dichotomy method with Simpson's algorithm.
