**5. Results and comments**

430 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

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

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,

2[ ] ( ) [ ]

( ) ( ) ( ) *A*

*A A HT e PG* 

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

The second situation consists to preserving the shape of the nozzle dimensioned on the basis

\* \* ( ) ( ) *A A S S HT PG*

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

\* \*

*A A*

*S S*

() () *MS S HT M PG* (36)

*A A* (39)

( ) () *MS S HT M PG* (40)

 *a T* (37)

(38)

( ) ()

*S HT*

*S HT*

*F T dT*

 

 *H T* 

and consequently another shape of the nozzle will be obtained.

*S* 

*M PG* 

*T\**

in equation (38) is that correspond to the temperature *T0* for our model.

of PG model for the aeronautical applications considered the *HT* model.

*TS HT* 

**4. Application**

algorithm.

situations can be presented.

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 of the flow.

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

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

Effect of Stagnation Temperature on Supersonic

0.52

Fig. 8. Variation of *P\* /P0* versus *T0*.

0.576

0.578

0.580

0.582

0.584

0.586

0.588

0.53

0.53

0.54

0.54

0.55

0.55

Flow Parameters with Application for Air in Nozzles 433

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

Figure 9 shows that mass flow rate through the critical cross section given by the perfect gas

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

theory is lower than it is at the *HT* model, especially for values of *T0*.

Fig. 9. Variation of the non-dimensional critical mass flow rate with *T0*.

### **5.1 Results for the critical parameters**

Figures 6, 7 and 8 represent the variation of the critical thermodynamic ratios versus *T0*. It can be seen that with enhancement *T0,* the critical parameters vary, and this variation becomes considerable for high values of *T0* unlike to the *PG* model, where they do not depend on *T0*.. For example, the value of the temperature ratio given by the *HT* model is always higher than the value given by the *PG* model. The ratios are determined by the choice of *N*=300000, *b1*=0.1 and *b2*=2.0 to have a precision better than *ε*=10 -5. The obtained numerical values of the critical parameters are presented in the table 3.

Fig. 6. Variation of *T\* /T0* versus *T0*.

Fig. 7. Variation of *ρ\* /ρ0* versus *T0*.

Figures 6, 7 and 8 represent the variation of the critical thermodynamic ratios versus *T0*. It can be seen that with enhancement *T0,* the critical parameters vary, and this variation becomes considerable for high values of *T0* unlike to the *PG* model, where they do not depend on *T0*.. For example, the value of the temperature ratio given by the *HT* model is always higher than the value given by the *PG* model. The ratios are determined by the choice of *N*=300000, *b1*=0.1 and *b2*=2.0 to have a precision better than *ε*=10 -5. The obtained

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

0 1000 2000 3000 4000 *Stagnation temperature (K)*

numerical values of the critical parameters are presented in the table 3.

**5.1 Results for the critical parameters**

0.82

0.620

Fig. 7. Variation of *ρ\* /ρ0* versus *T0*.

0.624

0.628

0.632

0.636

0.640

Fig. 6. Variation of *T\* /T0* versus *T0*.

0.83

0.84

0.85

0.86

0.87

0.88

0.89

Fig. 8. Variation of *P\* /P0* versus *T0*.

Figure 9 shows that mass flow rate through the critical cross section given by the perfect gas theory is lower than it is at the *HT* model, especially for values of *T0*.

Fig. 9. Variation of the non-dimensional critical mass flow rate with *T0*.

Effect of Stagnation Temperature on Supersonic

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 11. Variation of *T/T0* versus Mach number.

Table 5. Numerical values of the density ratio at high temperature

Flow Parameters with Application for Air in Nozzles 435

Table 4. Numerical values of the temperature ratio at high temperature

*T*/*T*<sup>0</sup> *M*=2.00 *M*=3.00 *M*=4.00 *M*=5.00 *M*=6.00 *PG* (*γ*=1.402) 0.5543 0.3560 0.2371 0.1659 0.1214 *T*0=298.15 K 0.5544 0.3560 0.2372 0.1659 0.1214 *T*0=500 K 0.5577 0.3581 0.2386 0.1669 0.1221 *T*0=1000 K 0.5810 0.3731 0.2481 0.1736 0.1269 *T*0=1500 K 0.6031 0.3911 0.2594 0.1810 0.1323 *T*0=2000 K 0.6163 0.4058 0.2694 0.1873 0.1366 *T*0=2500 K 0.6245 0.4162 0.2778 0.1928 0.1403 *T*0=3000 K 0.6301 0.4233 0.2848 0.1977 0.1473 *T*0=3500 K 0.6340 0.4285 0.2901 0.2018 0.1462

> 123456 *Mach number*

*ρ*/*ρ*<sup>0</sup> *M*=2.00 *M*=3.00 *M*=4.00 *M*=5.00 *M*=6.00 *PG* (*γ*=1.402) 0.2304 0.0765 0.0278 0.0114 0.0052 *T*0=298.15 K 0.2304 0.0765 0.0278 0.0114 0.0052 *T*0=500 K 0.2283 0.0758 0.0276 0.0113 0.0052 *T*0=1000 K 0.2181 0.0696 0.0250 0.0103 0.0047 *T*0=1500 K 0.2116 0.0636 0.0220 0.0089 0.0041 *T*0=2000 K 0.2087 0.0601 0.0197 0.0077 0.0035 *T*0=2500 K 0.2069 0.0581 0.0182 0.0069 0.0030 *T*0=3000 K 0.2057 0.0569 0.0173 0.0063 0.0027 *T*0=3500 K 0.2049 0.0560 0.0166 0.0058 0.0024

Figure 10 presents the variation of the critical sound velocity ratio versus *T0*. The influence of the *T0* on this parameter can be found.

Fig. 10. Effect of *T0* on the velocity sound ratio.


Table 3. Numerical values of the critical parameters at high temperature.

#### **5.2 Results for the supersonic parameters**

Figures 11, 12 and 13 presents the variation of the supersonic flow parameters in a crosssection versus Mach number for *T0* =1000 K*,* 2000 K and 3000 K*,* including the case of perfect gas for *=1.402*. When *M=1*, we can obtain the values of the critical ratios. If we take into account the variation of *CP(T),* the temperature *T0* influences on the value of the thermodynamic and geometrical parameters of flow unlike the *PG* model.

The curve 4 of figure 11 is under the curves of the *HT* model, which indicates that the perfect gas model cool the flow compared to the real thermodynamic behaviour of the gas, and consequently, it influences on the dimensionless parameters of a nozzle. At low temperature and Mach number, the theory of perfect gas gives acceptable results. The obtained numerical values of the supersonic flow parameters, the cross section area ratio and sound velocity ratio are presented respectively if the tables 4, 5, 6, 7 and 8.

Figure 10 presents the variation of the critical sound velocity ratio versus *T0*. The influence

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

> *\**

0

*\* a*

*a* 0 0 *\**

*m A ρ a*

*P* <sup>0</sup>

PG (γ=1.402) 0.8326 0.5279 0.6340 0.9124 0.5785 T0=298.15 K 0.8328 0.5279 0.6339 0.9131 0.5788 T0=500 K 0.8366 0.5293 0.6326 0.9171 0.5802 T0=1000 K 0.8535 0.5369 0.6291 0.9280 0.5838 T0=2000 K 0.8689 0.5448 0.6270 0.9343 0.5858 T0=2500 K 0.8722 0.5466 0.6266 0.9355 0.5862 T0=3000 K 0.8743 0.5475 0.6263 0.9365 0.5865 T0=3500 K 0.8758 0.5484 0.6262 0.9366 0.5865

Figures 11, 12 and 13 presents the variation of the supersonic flow parameters in a crosssection versus Mach number for *T0* =1000 K*,* 2000 K and 3000 K*,* including the case of perfect

account the variation of *CP(T),* the temperature *T0* influences on the value of the

The curve 4 of figure 11 is under the curves of the *HT* model, which indicates that the perfect gas model cool the flow compared to the real thermodynamic behaviour of the gas, and consequently, it influences on the dimensionless parameters of a nozzle. At low temperature and Mach number, the theory of perfect gas gives acceptable results. The obtained numerical values of the supersonic flow parameters, the cross section area ratio

*=1.402*. When *M=1*, we can obtain the values of the critical ratios. If we take into

of the *T0* on this parameter can be found.

0.910

Fig. 10. Effect of *T0* on the velocity sound ratio.

0 *T\**

**5.2 Results for the supersonic parameters**

gas for  *T* <sup>0</sup>

*P\**

Table 3. Numerical values of the critical parameters at high temperature.

thermodynamic and geometrical parameters of flow unlike the *PG* model.

and sound velocity ratio are presented respectively if the tables 4, 5, 6, 7 and 8.

0.915

0.920

0.925

0.930

0.935

0.940


Table 4. Numerical values of the temperature ratio at high temperature

Fig. 11. Variation of *T/T0* versus Mach number.


Table 5. Numerical values of the density ratio at high temperature

Effect of Stagnation Temperature on Supersonic

*T0*<1000 K.

in the table 9.

Flow Parameters with Application for Air in Nozzles 437

Table 7. Numerical Values of the cross section area ratio at high temperature.

temperature. *T0* value influences on this parameter.

Figure 14 represent the variation of the critical cross-section area section ratio versus Mach number at high temperature. For low values of Mach number and *T0*, the four curves fuses and start to be differs when *M*>2.00. We can see that the curves 3 and 4 are almost superposed for any value of *T0*. This result shows that the *PG* model can be used for

Figure 15 presents the variation of the sound velocity ratio versus Mach number at high

Figure 16 shows the variation of the thrust coefficient versus exit Mach number for various values of *T0*. It can be seen the effect of *T0* on this parameter. We can found that all the four curves are almost confounded when *ME<2.00* approximately. After this value, the curves begin to separates progressively. The numerical values of the thrust coefficient are presented

> 123456 *Mach number*

Fig. 14. Variation of the critical cross-section area ratio versus Mach number.

*1*

*2*

*3 4*

*A*/*A*\* *M*=2.00 *M*=3.00 *M*=4.00 *M*=5.00 *M*=6.00 *PG* (*γ*=1.402) 1.6859 4.2200 10.6470 24.7491 52.4769 *T*0=298.15 K 1.6859 4.2195 10.6444 24.7401 52.4516 *T*0=500 K 1.6916 4.2373 10.6895 24.8447 52.6735 *T*0=1000 K 1.7295 4.4739 11.3996 26.5019 56.1887 *T*0=1500 K 1.7582 4.7822 12.6397 29.7769 63.2133 *T*0=2000 K 1.7711 4.9930 13.8617 33.5860 72.0795 *T*0=2500 K 1.7795 5.1217 14.8227 37.2104 81.2941 *T*0=3000 K 1.7851 5.2091 15.5040 40.3844 90.4168 *T*0=3500 K 1.7889 5.2727 16.0098 43.0001 98.7953

Fig. 12. Variation of *ρ/ρ0* versus Mach number.


Table 6. Numerical values of the Pressure ratio at high temperature.

Fig. 13. Variation of *P/P0* versus Mach number.

123456 *Mach number*

*P*/*P*<sup>0</sup> *M*=2.00 *M*=3.00 *M*=4.00 *M*=5.00 *M*=6.00 *PG* (*γ*=1.402) 0.1277 0.0272 0.0066 0.0019 0.0006 *T*0=298.15 K 0.1277 0.0272 0.0066 0.0019 0.0006 *T*0=500 K 0.1273 0.0271 0.0065 0.0018 0.0006 *T*0=1000 K 0.1267 0.0259 0.0062 0.0017 0.0006 *T*0=1500 K 0.1276 0.0248 0.0057 0.0016 0.0005 *T*0=2000 K 0.1286 0.0244 0.0053 0.0014 0.0004 *T*0=2500 K 0.1292 0.0242 0.0050 0.0013 0.0004 *T*0=3000 K 0.1296 0.0240 0.0049 0.0004 0.0003 *T*0=3500 K 0.1299 0.0240 0.0048 0.0011 0.0003

> 123456 *Mach number*

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Fig. 12. Variation of *ρ/ρ0* versus Mach number.

0.0

Fig. 13. Variation of *P/P0* versus Mach number.

0.1

0.2

0.3

0.4

0.5

0.6

Table 6. Numerical values of the Pressure ratio at high temperature.



Figure 14 represent the variation of the critical cross-section area section ratio versus Mach number at high temperature. For low values of Mach number and *T0*, the four curves fuses and start to be differs when *M*>2.00. We can see that the curves 3 and 4 are almost superposed for any value of *T0*. This result shows that the *PG* model can be used for *T0*<1000 K.

Figure 15 presents the variation of the sound velocity ratio versus Mach number at high temperature. *T0* value influences on this parameter.

Figure 16 shows the variation of the thrust coefficient versus exit Mach number for various values of *T0*. It can be seen the effect of *T0* on this parameter. We can found that all the four curves are almost confounded when *ME<2.00* approximately. After this value, the curves begin to separates progressively. The numerical values of the thrust coefficient are presented in the table 9.

Fig. 14. Variation of the critical cross-section area ratio versus Mach number.

Effect of Stagnation Temperature on Supersonic

0.0

the same time. When *ME*=2.00*.*

Fig. 16. Variation of *CF* versus exit Mach number.

**5.3 Results for the error given by the perfect gas model**

between the *PG* and the *HT* models for several *T0* values.

0.5

1.0

1.5

2.0

Flow Parameters with Application for Air in Nozzles 439

Table 9. Numerical values of the thrust coefficient at high temperature

*CF M*=2.00 *M*=3.00 *M*=4.00 *M*=5.00 *M*=6.00 *PG* (*γ*=1.402) 1.2078 1.4519 1.5802 1.6523 1.6959 *T*0=298.15 K 1.2078 1.4518 1.5800 1.6521 1.6957 *T*0=500 K 1.2076 1.4519 1.5802 1.6523 1.6958 *T*0=1000 K 1.2072 1.4613 1.5919 1.6646 1.7085 *T*0=1500 K 1.2062 1.4748 1.6123 1.6871 1.7317 *T*0=2000 K 1.2048 1.4832 1.6288 1.7069 1.7527 *T*0=2500 K 1.2042 1.4879 1.6401 1.7221 1.7694 *T*0=3000 K 1.2038 1.4912 1.6479 1.7337 1.7828 *T*0=3500 K 1.2033 1.4936 1.6533 1.7422 1.7932

> 123456 *Exit Mach number*

Figure 17 presents the relative error of the thermodynamic and geometrical parameters

It can be seen that the error depends on the values of *T0* and *M*. For example, if *T0*=2000 K and *M*=3.00, the use of the *PG* model will give a relative error equal to *ε*=14.27 % for the temperatures ratio, *ε*=27.30 % for the density ratio, error *ε*=15.48 % for the critical sections ratio and *ε*=2.11 % for the thrust coefficient. For lower values of *M* and *T0*, the error *ε* is weak. The curve 3 in the figure 17 is under the error 5% independently of the Mach number,

We can deduce for the error given by the thrust coefficient that it is equal to *ε*=0.0 %*,* if *ME*=2.00 approximately independently of *T0.* There is no intersection of the three curves in

which is interpreted by the use potential of the *PG* model when *T0<*1000 K*.* 


Table 8. Numerical values of the sound velocity ratio at high temperature.

Fig. 15. Variation of the ratio of the velocity sound versus Mach number.

*a*/*a*<sup>0</sup> *M*=2.00 *M*=3.00 *M*=4.00 *M*=5.00 *M*=6.00

123456 *Mach number*

Fig. 15. Variation of the ratio of the velocity sound versus Mach number.

*PG* (*γ*=1.402) 0.7445 0.5966 0.4870 0.4074 0.3484 *T*0=298.15 K 0.7450 0.5970 0.4873 0.4076 0.3486 *T*0=500 K 0.7510 0.6019 0.4913 0.4110 0.3515 *T*0=1000 K 0.7739 0.6245 0.5103 0.4268 0.3651 *T*0=1500 K 0.7862 0.6408 0.5254 0.4398 0.3762 *T*0=2000 K 0.7923 0.6501 0.5354 0.4489 0.3841 *T*0=2500 K 0.7959 0.6556 0.5420 0.4553 0.3898 *T*0=3000 K 0.7985 0.6595 0.5465 0.4600 0.3942 *T*0=3500 K 0.7998 0.6618 0.5495 0.4632 0.3973

Table 8. Numerical values of the sound velocity ratio at high temperature.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1


Table 9. Numerical values of the thrust coefficient at high temperature

Fig. 16. Variation of *CF* versus exit Mach number.

### **5.3 Results for the error given by the perfect gas model**

Figure 17 presents the relative error of the thermodynamic and geometrical parameters between the *PG* and the *HT* models for several *T0* values.

It can be seen that the error depends on the values of *T0* and *M*. For example, if *T0*=2000 K and *M*=3.00, the use of the *PG* model will give a relative error equal to *ε*=14.27 % for the temperatures ratio, *ε*=27.30 % for the density ratio, error *ε*=15.48 % for the critical sections ratio and *ε*=2.11 % for the thrust coefficient. For lower values of *M* and *T0*, the error *ε* is weak. The curve 3 in the figure 17 is under the error 5% independently of the Mach number, which is interpreted by the use potential of the *PG* model when *T0<*1000 K*.* 

We can deduce for the error given by the thrust coefficient that it is equal to *ε*=0.0 %*,* if *ME*=2.00 approximately independently of *T0.* There is no intersection of the three curves in the same time. When *ME*=2.00*.*

Effect of Stagnation Temperature on Supersonic

0.0 1.0 2.0 3.0 4.0

1.0

nozzle.

model.

1.5

2.0

2.5

3.0 *M*

Flow Parameters with Application for Air in Nozzles 441

0 2 4 6 8 10 12 14 16

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

Fig. 18. Effect of stagnation temperature on the variation of the Mach number through the

*MS* (*PG γ*=1.402) 1.5000 2.0000 3.0000 4.0000 5.0000 6.0000 *MS* (*T*0=298.15 K) 1.4995 1.9995 2.9995 3.9993 4.9989 5.9985 *MS* (*T*0=500 K) 1.4977 1.9959 2.9956 3.9955 4.9951 5.9947 *MS* (*T*0=1000 K) 1.4879 1.9705 2.9398 3.9237 4.9145 5.9040 *MS* (*T*0=1500 K) 1.4830 1.9534 2.8777 3.8147 4.7727 5.7411 *MS* (*T*0=2000 K) 1.4807 1.9463 2.8432 3.7293 4.6372 5.5675 *MS* (*T*0=2500 K) 1.4792 1.9417 2.8245 3.6765 4.5360 5.4209 *MS* (*T*0=3000 K) 1.4785 1.9388 2.8121 3.6454 4.4676 5.3066 *MS* (*T*0=3500 K) 1.4778 1.9368 2.8035 3.6241 4.4216 5.2237

Figure 20 presents the supersonic nozzles shapes delivering a same variation of the Mach number throughout the nozzle and consequently given the same exit Mach number *MS*=3.00. The variation of the Mach number through these 4 nozzles is illustrated on curve 4 of figure 18. The three other curves 1, 2, and, 3 of figure 15 are obtained with the *HT* model use for *T0*=3000 K, 2000 K and 1000 K respectively. The curve 4 of figure 20 is the same as it is in the figure 13a, and it is calculated with the *PG* model use. The nozzle that is calculated according to the *PG* model provides less cross-section area in comparison with the *HT*

*(a): Shape of nozzle, dimensioned on the consideration of the PG model for MS=3.00. (b): Variation of the Mach number at high temperature through the nozzle.* 

Table 10. Correction of the exit Mach number of the nozzle.

*(a)*

*4 3 2*

*(b) <sup>1</sup>*

Curve 3 Error compared to HT model for (T0=1000 K)

(a): Temperature ratio. (b): Density ratio. (c): Critical sections ratio. (d): Thrust coefficient.

Fig. 17. Variation of the relative error given by supersonic parameters of *PG* versus Mach number.

### **5.4 Results for the supersonic nozzle application**

Figure 18 presents the variation of the Mach number through the nozzle for *T0*=1000 K, 2000 K and 3000 K, including the case of perfect gas presented by curve 4. The example is selected for *MS*=3.00 for the *PG* model. If *T0* is taken into account, we will see a fall in Mach number of the dimensioned nozzle in comparison with the *PG* model. The more is the temperature *T0*, the more it is this fall. Consequently, the thermodynamics parameters force to design the nozzle with different dimensions than it is predicted by use the *PG* model. It should be noticed that the difference becomes considerable if the value *T0* exceeds 1000 K.

Figure 19 present the correction of the Mach number of nozzle giving exit Mach number *MS*, dimensioned on the basis of the *PG* model for various values of *T0*.

One can see that the curves confound until Mach number *MS=2.0* for the whole range of *T0*. From this value, the difference between the three curves 1, 2 and 3, start to increase. The curves 3 and 4 are almost confounded whatever the Mach number if the value of *T0* is lower than 1000 K. For example, if the nozzle delivers a Mach number *MS=3.00* at the exit section, on the assumption of the *PG* model, the *HT* model gives Mach number equal to *MS*=2.93, 2.84 and 2.81 for *T0*=1000 K, 2000 K and 3000 K respectively. The numerical values of the correction of the exit Mach number of the nozzle are presented in the table 10.

*1*

*2*

*3*

*1*

*2*

*3*

(a): Temperature ratio. (b): Density ratio. (c): Critical sections ratio. (d): Thrust coefficient.

noticed that the difference becomes considerable if the value *T0* exceeds 1000 K.

correction of the exit Mach number of the nozzle are presented in the table 10.

dimensioned on the basis of the *PG* model for various values of *T0*.

Fig. 17. Variation of the relative error given by supersonic parameters of *PG* versus Mach

Figure 18 presents the variation of the Mach number through the nozzle for *T0*=1000 K, 2000 K and 3000 K, including the case of perfect gas presented by curve 4. The example is selected for *MS*=3.00 for the *PG* model. If *T0* is taken into account, we will see a fall in Mach number of the dimensioned nozzle in comparison with the *PG* model. The more is the temperature *T0*, the more it is this fall. Consequently, the thermodynamics parameters force to design the nozzle with different dimensions than it is predicted by use the *PG* model. It should be

Figure 19 present the correction of the Mach number of nozzle giving exit Mach number *MS*,

One can see that the curves confound until Mach number *MS=2.0* for the whole range of *T0*. From this value, the difference between the three curves 1, 2 and 3, start to increase. The curves 3 and 4 are almost confounded whatever the Mach number if the value of *T0* is lower than 1000 K. For example, if the nozzle delivers a Mach number *MS=3.00* at the exit section, on the assumption of the *PG* model, the *HT* model gives Mach number equal to *MS*=2.93, 2.84 and 2.81 for *T0*=1000 K, 2000 K and 3000 K respectively. The numerical values of the

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0

*1*

*(d)*

*2*

*3*

20

40

60

*(b)*

80

100

123456 *Mach number*

123456 *Exit Mach number*

*1*

*2*

*3*

*3*

*2*

*1*

123456 *Mach number*

123456 *Mach number*

Curve 1 Error compared to HT model for (T0=3000 K) Curve 2 Error compared to HT model for (T0=2000 K) Curve 3 Error compared to HT model for (T0=1000 K)

**5.4 Results for the supersonic nozzle application**

*(a)*

5

0

number.

10

20

30

*(c)*

40

50

10

15

20

*(a): Shape of nozzle, dimensioned on the consideration of the PG model for MS=3.00. (b): Variation of the Mach number at high temperature through the nozzle.* 



Table 10. Correction of the exit Mach number of the nozzle.

Figure 20 presents the supersonic nozzles shapes delivering a same variation of the Mach number throughout the nozzle and consequently given the same exit Mach number *MS*=3.00. The variation of the Mach number through these 4 nozzles is illustrated on curve 4 of figure 18. The three other curves 1, 2, and, 3 of figure 15 are obtained with the *HT* model use for *T0*=3000 K, 2000 K and 1000 K respectively. The curve 4 of figure 20 is the same as it is in the figure 13a, and it is calculated with the *PG* model use. The nozzle that is calculated according to the *PG* model provides less cross-section area in comparison with the *HT* model.

Effect of Stagnation Temperature on Supersonic

make a suitable interpolation.

particular case of our model.

**7. Acknowledgment**

**8. References**

Flow Parameters with Application for Air in Nozzles 443

The relations presented in this study are valid for any interpolation chosen for the function

We can choose another substance instead of the air. The relations remain valid, except that it is necessary to have the table of variation of *CP* and *γ* according to the temperature and to

The cross section area ratio presented by the relation (19) can be used as *a source of comparison for verification of the dimensions calculation of various supersonic nozzles.* It provides a uniform and parallel flow at the exit section by the method of characteristics and the Prandtl Meyer function (Zebbiche & Youbi, 2005a, 2005b, Zebbiche, 2007, Zebbiche, 2010a & Zebbiche, 2010b). The thermodynamic ratios can be used to determine the design

We can obtain the relations of a perfect gas starting from the relations of our model by annulling all constants of interpolation except the first. In this case, the *PG* model becomes a

The author acknowledges Djamel, Khaoula, Abdelghani Amine, Ritadj Zebbiche and

Anderson J. D. Jr.. (1982), *Modern Compressible Flow. With Historical Perspective*, (2nd edition), Mc Graw-Hill Book Company, ISBN 0-07-001673-9. New York, USA. Anderson J. D. Jr. (1988), *Fundamentals of Aerodynamics*, (2nd edition), Mc Graw-Hill Book

Démidovitch B. et Maron I. (1987), *Eléments de calcul numérique*, Editions MIR, ISBN 978-2-

Fletcher C. A. J. (1988), Computational Techniques for Fluid Dynamics: Specific Techniques

Moran M. J., (2007). Fundamentals of Engineering Thermodynamics, John Wiley & Sons Inc.,

Oosthuisen P. H. & Carscallen W. E., (1997), *Compressible Fluid Flow.* Mc Grw-Hill, ISBN 0-

Peterson C.R. & Hill P. G. (1965), *Mechanics and Thermodynamics of Propulsion*, Addition-Wesley Publishing Company Inc., ISBN 0-201-02838-7, New York, USA. Ralston A. & Rabinowitz P. A. (1985). *A First Course in Numerical Analysis*. (2nd Edition), McGraw-Hill Book Company, ISBN 0-07-051158-6, New York, USA. Ryhming I. L. (1984), *Dynamique des fluides*, Presses Polytechniques Romandes, Lausanne,

Zebbiche T. (2007). Stagnation Temperature Effect on the Prandtl Meyer Function. *AIAA Journal,* Vol. 45 N° 04, PP. 952-954, April 2007, ISSN 0001-1452, USA Zebbiche T. & Youbi Z. (2005a). Supersonic Flow Parameters at High Temperature.

*26-29 Sep. 2005, ISBN 978-3-8322-7492-4, Friendrichshafen, Germany.* 

Application for Air in nozzles. *German Aerospace Congress 2005, DGLR-2005-0256,* 

for Different Flow Categories, Vol. II, Springer Verlag, ISBN 0-387-18759-6, Berlin,

*CP(T)*. The essential one is that the selected interpolation gives small error.

parameters of the various shapes of nozzles under the basis of the *HT* model.

Fettoum Mebrek for granting time to prepare this manuscript.

Company, ISBN 0-07-001656-9, New York, USA.

6th Edition, ISBN 978-8-0471787358, USA

7298-9461-0, Moscou, USSR.

07-0158752-9, New York, USA.

ISBN 2-88074-224-2, Suisse.

Heidelberg.

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

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.

### **6. Conclusion**

From this study, we can quote the following points:

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 approximately 3000 K.

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 more time for calculation and for data processing.

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*.

The relations presented in this study are valid for any interpolation chosen for the function *CP(T)*. The essential one is that the selected interpolation gives small error.

We can choose another substance instead of the air. The relations remain valid, except that it is necessary to have the table of variation of *CP* and *γ* according to the temperature and to make a suitable interpolation.

The cross section area ratio presented by the relation (19) can be used as *a source of comparison for verification of the dimensions calculation of various supersonic nozzles.* It provides a uniform and parallel flow at the exit section by the method of characteristics and the Prandtl Meyer function (Zebbiche & Youbi, 2005a, 2005b, Zebbiche, 2007, Zebbiche, 2010a & Zebbiche, 2010b). The thermodynamic ratios can be used to determine the design parameters of the various shapes of nozzles under the basis of the *HT* model.

We can obtain the relations of a perfect gas starting from the relations of our model by annulling all constants of interpolation except the first. In this case, the *PG* model becomes a particular case of our model.
