**3.3 Supersonic nozzle conception**

For supersonic nozzle application, it is necessary to determine the thrust coefficient. For nozzles giving a uniform and parallel flow at the exit section, the thrust coefficient is (Peterson & Hill, 1965 & Zebbiche, Youbi, 2005b)

$$C\_F = \frac{F}{P\_0 A\_\*} \tag{32}$$

Where

428 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

The temperature *TD* is equal to *T0* for *Fρ(T)*, and equal to *T\** for *FA(T).* The temperature *TG* is equal to *T\** for the critical parameter, and equal to *TS* for the supersonic parameter. Taking a value b*<sup>1</sup>* near zero (b*1*=0.1*, for example*) and b*2*=2.0, it can condense the nodes towards left

The stagnation state is given by *M*=0. Then, the critical parameters correspond to *M*=1.00,

The resolution of equation (29) is made by the use of the dichotomy algorithm (Démidovitch & Maron, 1987 & Zebbiche & Youbi, 2006), with *T\*<T0*. It can choose the interval *[T1,T2]*  containing *T\** by *T1=0 K* and *T2=T0*. The value *T\** can be given with a precision *ε* if the interval

0 1 4426 1 *<sup>T</sup> K . Log*

If *ε*=10-8 is taken, the number *K* cannot exceed 39. Consequently, the temperature ratio *T\*/T0*

Taking *T=T\** and *ρ=ρ\** in the relation (14) and integrating the function *Fρ(T)* by using the Simpson's formula with condensation of nodes towards the left end, the critical density ratio

The critical ratios of the pressures and the sound velocity can be calculated by using the

For a given supersonic cross-section, the parameters *ρ=ρS, P=PS, A=AS,* and *T=TS* can be determined according to the Mach number *M=MS*. Replacing *T=TS* and *M=MS* in relation

The determination of *TS* of equation (31) is done always by the dichotomy algorithm, excepting *TS<T\*.* We can take the interval *[T1,T2]* containing *TS*, by (*T1*=0 *K,* and *T2=T\*.*  Replacing *T=TS* and *ρ=ρ<sup>S</sup>* in relation (14) and integrating the function *Fρ(T)* by using the Simpson's method with condensation of nodes towards the left end, the density ratio can be

relations (15) and (22) respectively, by replacing *T=T\*, ρ=ρ\*, P=P\** and *a=a\*,* 

**3.2 Parameters for a supersonic Mach number**

 

edge *TS* of the interval, see figure 3.

Fig. 3. Presentation of the condensation of nodes

for example at the throat of a supersonic nozzle, summarize by:

of subdivision number *K* is satisfied by the following condition:

When *M*=1.00 we have *T=T\**. These conditions in the relation (10), we obtain:

b1=0.1 , b2=2.0 b1=1.0 , b2=2.0 b1=1.9 , b2=2.0

**3.1 Critical parameters**

can be calculated.

is obtained.

(10) gives

obtained.

( ) *T sT T T i iD G G* (28)

<sup>2</sup> 2 () ()0 \* \* *HT a T* (29)

(30)

2 2 2 ( ) ( ) 0 *S SS HT M a T* (31)

$$F = \text{ } \text{ } \text{ } V\_E = \text{ } \text{ } \text{ } M\_E \text{ } a\_E \text{ } \tag{33}$$

The introduction of relations (21), (22) into (32) gives as the following relation:

$$\mathbf{C}\_{F} = \mathcal{Y}(T\_{0}) \,\, M\_{E} \left(\frac{a\_{E}}{a\_{0}}\right) \left(\frac{\rho\_{\*}}{\rho\_{\*}}\right) \left(\frac{a\_{\*}}{a\_{0}}\right) \tag{34}$$

The design of the nozzle is made on the basis of its application. For rockets and missiles applications, the design is made to obtain nozzles having largest possible exit Mach number, which gives largest thrust coefficient, and smallest possible length, which give smallest possible mass of structure.

For the application of blowers, we make the design on the basis to obtain the smallest possible temperature at the exit section, to not to destroy the measuring instruments, and to save the ambient conditions. Another condition requested is to have possible largest ray of the exit section for the site of instruments. Between the two possibilities of construction, we prefer the first one.

### **3.4 Error of perfect gas model**

The mathematical perfect gas model is developed on the basis to regarding the specific heat *CP* and ratio *γ* as constants, which gives acceptable results for low temperature. According to this study, we can notice a difference on the given results between the perfect gas model and developed here model.The error given by the *PG* model compared to our *HT* model can be calculated for each parameter. Then, for each value *(T0, M)*, the *ε* error can be evaluated by the following relationship:

$$\left| \varepsilon\_y(T\_{0'}, M) = \left| 1 - \frac{y\_{PG}(T\_{0'}, M)}{y\_{HT}(T\_{0'}, M)} \right| \times 100\tag{35}$$

The letter *y* in the expression (35) can represent all above-mentioned parameters. As a rule for the aerodynamic applications, the error should be lower than *5%*.

Effect of Stagnation Temperature on Supersonic

950

1.24

Fig. 5. Variation of the specific heats ratio versus *T0*.

1.28

1.32

1.36

1.40

1.44

1000

1050

1100

1150

1200

1250

1300

1350

**5. Results and comments**

of the flow.

Flow Parameters with Application for Air in Nozzles 431

Figures 4 and 5 respectively represent the variation of specific heat *CP(T)* and the ratio *γ(T)*  of the air versus the temperature up to *3550 K* for *HT* and *PG* models. The graphs at high temperature are presented by using the polynomial interpolation (23). We can say that at low temperature until approximately 240 K, the gas can be regarded as calorically perfect, because of the invariance of specific heat *CP(T)* and the ratio *γ(T).* But if *T0* increases, we can see the difference between these values and it influences on the thermodynamic parameters

> 0 1000 2000 3000 4000 *Stagnation Temperature (K)*

> 0 1000 2000 3000 4000 *Stagnation Temperature (K)*

Fig. 4. Variation of the specific heat for constant pressure versus stagnation temperature *T0*.
