Variation of
1. Introduction
The obtained results of a supersonic perfect gas flow presented in (Anderson, 1982, 1988& Ryhming, 1984), are valid under some assumptions. One of the assumptions is that the gas is regarded as a calorically perfect, i. e., the specific heats
The presented mathematical relations are valid in the general case independently of the interpolation form and the substance, but the results are illustrated by a polynomial interpolation of the 9^{th} degree. The obtained mathematical relations are in the form of nonlinear algebraic equations, and so analytical integration was impossible. Thus, our interest is directed towards to the determination of numerical solutions. The dichotomy method for the solution of the nonlinear algebraic equations is used; the Simpson’s algorithm (Démidovitch & Maron, 1987& Zebbiche & Youbi, 2006, Zebbiche, 2010a, 2010b) for numerical integration of the found functions is applied. The integrated functions have high gradients of the interval extremity, where the Simpson’s algorithm requires a very high discretization to have a suitable precision. The solution of this problem is made by introduction of a condensation procedure in order to refine the points at the place where there is high gradient. The Robert’s condensation formula presented in (Fletcher, 1988) was chosen. The application for the air in the supersonic field is limited by the threshold of the molecules dissociation. The comparison is made with the calorically perfect gas model.
The problem encounters in the aeronautical experiments where the use of the nozzle designed on the basis of the perfect gas assumption, degrades the performances. If during the experiment measurements are carried out it will be found that measured parameters are differed from the calculated, especially for the high stagnation temperature. Several reasons are responsible for this deviation. Our flow is regarded as perfect, permanent and non-rotational. The gas is regarded as calorically imperfect and thermally perfect
To determine the application limits of the perfect gas model, the error given by this model is compared with our results.
2. Mathematical formulation
The development is based on the use of the conservation equations in differential form. We assume that the state equation of perfect gas (
Using relationship between
The integration of the relation (2) gives the adiabatic equation of a perfect gas at high temperature.
The sound velocity is (Ryhming, 1984),
The differentiation of the state equation of a perfect gas gives:
Substituting the relationship (2) in the equation (4), we obtain after transformation:
Equation (5) proves that the relation of speed of sound of perfect gas remains always valid for the model at high temperature, but it is necessary to take into account the variation of the ratio
The equation of the energy conservation in differential form (Anderson, 1988& Moran, 2007) is written as:
The integration between the stagnation state (
Where
Dividing the equation (6) by
Dividing the relation (7) by the sound velocity, we obtain an expression connecting the Mach number with the enthalpy and the temperature:
The relation (10) shows the variation of the Mach number with the temperature for calorically imperfect gas.
The momentum equation in differential form can be written as (Moran, 2007, Peterson & Hill1, 1965, & Oosthuisen & Carscallen, 1997):
Using the expression (3), the relationship (10), can be written as:
Where
The density ratio relative to the temperature
The pressure ratio is obtained by using the relation of the perfect gas state:
The mass conservation equation is written as (Anderson, 1988& Moran, 2007)
The taking logarithm and then differentiating of relation (16), and also using of the relations (9) and (12), one can receive the following equation:
Where
The integration of equation (17) between the critical state (
To find parameters
The critical mass flow rate (Moran, 2007, Zebbiche & Youbi, 2005a, 2005b) can be written in non-dimensional form:
As the mass flow rate through the throat is constant, we can calculate it at the throat. In this section, we have
The determination of the velocity sound ratio is done by the relation (5). Thus,
The parameters
3. Calculation procedure
In the first case, one presents the table of variation of C_{P} and γ versus the temperature for air (Peterson & Hill, 1965, Zebbiche 2010a, 2010b). The values are presented in the table 1.
T (K) | C_{P}(J/(KgK) | γ(T) | T (K) | C_{P }(J/(Kg K) | γ(T) | T (K) | C_{P }J/(Kg K) | γ(T) | ||
55.538 | 1001.104 | 1.402 | 833.316 | 1107.192 | 1.350 | 2111.094 | 1256.813 | 1.296 | ||
. | . | . | 888.872 | 1119.078 | 1.345 | 2222.205 | 1263.410 | 1.294 | ||
222.205 | 1001.101 | 1.402 | 944.427 | 1131.314 | 1.340 | 2333.316 | 1270.097 | 1.292 | ||
277.761 | 1002.885 | 1.401 | 999.983 | 1141.365 | 1.336 | 2444.427 | 1273.476 | 1.291 | ||
305.538 | 1004.675 | 1.400 | 1055.538 | 1151.658 | 1.332 | 2555.538 | 1276.877 | 1.290 | ||
333.316 | 1006.473 | 1.399 | 1111.094 | 1162.202 | 1.328 | 2666.650 | 1283.751 | 1.288 | ||
361.094 | 1008.281 | 1.398 | 1166.650 | 1170.280 | 1.325 | 2777.761 | 1287.224 | 1.287 | ||
388.872 | 1011.923 | 1.396 | 1222.205 | 1178.509 | 1.322 | 2888.872 | 1290.721 | 1.286 | ||
416.650 | 1015.603 | 1.394 | 1277.761 | 1186.893 | 1.319 | 2999.983 | 1294.242 | 1.285 | ||
444.427 | 1019.320 | 1.392 | 1333.316 | 1192.570 | 1.317 | 3111.094 | 1297.789 | 1.284 | ||
499.983 | 1028.781 | 1.387 | 1444.427 | 1204.142 | 1.313 | 3222.205 | 1301.360 | 1.283 | ||
555.538 | 1054.563 | 1.374 | 1555.538 | 1216.014 | 1.309 | 3333.316 | 1304.957 | 1.282 | ||
611.094 | 1054.563 | 1.370 | 1666.650 | 1225.121 | 1.306 | 3444.427 | 1304.957 | 1.282 | ||
666.650 | 1067.077 | 1.368 | 1777.761 | 1234.409 | 1.303 | 3555.538 | 1308.580 | 1.281 | ||
722.205 | 1080.005 | 1.362 | 1888.872 | 1243.883 | 1.300 | |||||
777.761 | 1093.370 | 1.356 | 1999.983 | 1250.305 | 1.298 |
For a perfect gas, the
The interpolation (
I | ai | I | ai | |
1 | 1001.1058 | 6 | 3.069773 10-12 | |
2 | 0.04066128 | 7 | -1.350935 10-15 | |
3 | -0.000633769 | 8 | 3.472262 10-19 | |
4 | 2.747475 10-6 | 9 | -4.846753 10-23 | |
5 | -4.033845 10-9 | 10 | 2.841187 10-27 |
A relationship (23) gives undulated dependence for temperature approximately low than
Thus:
for
The selected interpolation gives an error less than
Once the interpolation is made, we determine the function
Substituting the relation (23) in (8) and writing the integration results in the form of Horner scheme, the following expression for enthalpy is obtained
Where
and
Taking into account the correction made to the function
For
The determination of the ratios (14) and (19) require the numerical integration of
Due to high gradient at the left extremity of the interval, the integration with a constant step requires a very small step. The tracing of the functions is selected for
A Condensation of nodes is then necessary in the vicinity of
Where
Obtained
The temperature
The stagnation state is given by
When
The resolution of equation (29) is made by the use of the dichotomy algorithm (Démidovitch & Maron, 1987& Zebbiche & Youbi, 2006), with
If
Taking
The critical ratios of the pressures and the sound velocity can be calculated by using the relations (15) and (22) respectively, by replacing
3.2. Parameters for a supersonic Mach number
For a given supersonic cross-section, the parameters
The determination of
Replacing
The ratios of pressures, speed of sound and the sections corresponding to
The integration results of the ratios
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)
Where
The introduction of relations (21), (22) into (32) gives as the following relation:
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
The letter
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
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
The relation (36) indicates that the Mach number of the
The second situation consists to preserving the shape of the nozzle dimensioned on the basis of PG model for the aeronautical applications considered the
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
5. Results and comments
Figures 4 and 5 respectively represent the variation of specific heat
5.1. Results for the critical parameters
Figures 6, 7 and 8 represent the variation of the critical thermodynamic ratios versus
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
Figure 10 presents the variation of the critical sound velocity ratio versus
T*/T0 | P*/P0 | ρ*/ρ 0 | a*/a 0 | m/A* ρ 0 a 0 | |
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 |
5.2. Results for the supersonic parameters
Figures 11, 12 and 13 presents the variation of the supersonic flow parameters in a cross-section versus Mach number for
The curve 4 of figure 11 is under the curves of the
T/T0 | 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 |
T0=298.15 K | 0.5544 | 0.3560 | 0.2372 | 0.1659 | 0.1214 |
T0=500 K | 0.5577 | 0.3581 | 0.2386 | 0.1669 | 0.1221 |
T0=1000 K | 0.5810 | 0.3731 | 0.2481 | 0.1736 | 0.1269 |
T0=1500 K | 0.6031 | 0.3911 | 0.2594 | 0.1810 | 0.1323 |
T0=2000 K | 0.6163 | 0.4058 | 0.2694 | 0.1873 | 0.1366 |
T0=2500 K | 0.6245 | 0.4162 | 0.2778 | 0.1928 | 0.1403 |
T0=3000 K | 0.6301 | 0.4233 | 0.2848 | 0.1977 | 0.1473 |
T0=3500 K | 0.6340 | 0.4285 | 0.2901 | 0.2018 | 0.1462 |
ρ/ρ0 | 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 |
T0=298.15 K | 0.2304 | 0.0765 | 0.0278 | 0.0114 | 0.0052 |
T0=500 K | 0.2283 | 0.0758 | 0.0276 | 0.0113 | 0.0052 |
T0=1000 K | 0.2181 | 0.0696 | 0.0250 | 0.0103 | 0.0047 |
T0=1500 K | 0.2116 | 0.0636 | 0.0220 | 0.0089 | 0.0041 |
T0=2000 K | 0.2087 | 0.0601 | 0.0197 | 0.0077 | 0.0035 |
T0=2500 K | 0.2069 | 0.0581 | 0.0182 | 0.0069 | 0.0030 |
T0=3000 K | 0.2057 | 0.0569 | 0.0173 | 0.0063 | 0.0027 |
T0=3500 K | 0.2049 | 0.0560 | 0.0166 | 0.0058 | 0.0024 |
P/P0 | 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 |
T0=298.15 K | 0.1277 | 0.0272 | 0.0066 | 0.0019 | 0.0006 |
T0=500 K | 0.1273 | 0.0271 | 0.0065 | 0.0018 | 0.0006 |
T0=1000 K | 0.1267 | 0.0259 | 0.0062 | 0.0017 | 0.0006 |
T0=1500 K | 0.1276 | 0.0248 | 0.0057 | 0.0016 | 0.0005 |
T0=2000 K | 0.1286 | 0.0244 | 0.0053 | 0.0014 | 0.0004 |
T0=2500 K | 0.1292 | 0.0242 | 0.0050 | 0.0013 | 0.0004 |
T0=3000 K | 0.1296 | 0.0240 | 0.0049 | 0.0004 | 0.0003 |
T0=3500 K | 0.1299 | 0.0240 | 0.0048 | 0.0011 | 0.0003 |
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 |
T0=298.15 K | 1.6859 | 4.2195 | 10.6444 | 24.7401 | 52.4516 |
T0=500 K | 1.6916 | 4.2373 | 10.6895 | 24.8447 | 52.6735 |
T0=1000 K | 1.7295 | 4.4739 | 11.3996 | 26.5019 | 56.1887 |
T0=1500 K | 1.7582 | 4.7822 | 12.6397 | 29.7769 | 63.2133 |
T0=2000 K | 1.7711 | 4.9930 | 13.8617 | 33.5860 | 72.0795 |
T0=2500 K | 1.7795 | 5.1217 | 14.8227 | 37.2104 | 81.2941 |
T0=3000 K | 1.7851 | 5.2091 | 15.5040 | 40.3844 | 90.4168 |
T0=3500 K | 1.7889 | 5.2727 | 16.0098 | 43.0001 | 98.7953 |
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
Figure 15 presents the variation of the sound velocity ratio versus Mach number at high temperature.
Figure 16 shows the variation of the thrust coefficient versus exit Mach number for various values of
a/a0 | M=2.00 | M=3.00 | M=4.00 | M=5.00 | M=6.00 |
PG (γ=1.402) | 0.7445 | 0.5966 | 0.4870 | 0.4074 | 0.3484 |
T0=298.15 K | 0.7450 | 0.5970 | 0.4873 | 0.4076 | 0.3486 |
T0=500 K | 0.7510 | 0.6019 | 0.4913 | 0.4110 | 0.3515 |
T0=1000 K | 0.7739 | 0.6245 | 0.5103 | 0.4268 | 0.3651 |
T0=1500 K | 0.7862 | 0.6408 | 0.5254 | 0.4398 | 0.3762 |
T0=2000 K | 0.7923 | 0.6501 | 0.5354 | 0.4489 | 0.3841 |
T0=2500 K | 0.7959 | 0.6556 | 0.5420 | 0.4553 | 0.3898 |
T0=3000 K | 0.7985 | 0.6595 | 0.5465 | 0.4600 | 0.3942 |
T0=3500 K | 0.7998 | 0.6618 | 0.5495 | 0.4632 | 0.3973 |
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 |
T0=298.15 K | 1.2078 | 1.4518 | 1.5800 | 1.6521 | 1.6957 |
T0=500 K | 1.2076 | 1.4519 | 1.5802 | 1.6523 | 1.6958 |
T0=1000 K | 1.2072 | 1.4613 | 1.5919 | 1.6646 | 1.7085 |
T0=1500 K | 1.2062 | 1.4748 | 1.6123 | 1.6871 | 1.7317 |
T0=2000 K | 1.2048 | 1.4832 | 1.6288 | 1.7069 | 1.7527 |
T0=2500 K | 1.2042 | 1.4879 | 1.6401 | 1.7221 | 1.7694 |
T0=3000 K | 1.2038 | 1.4912 | 1.6479 | 1.7337 | 1.7828 |
T0=3500 K | 1.2033 | 1.4936 | 1.6533 | 1.7422 | 1.7932 |
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
It can be seen that the error depends on the values of
We can deduce for the error given by the thrust coefficient that it is equal to
5.4. Results for the supersonic nozzle application
Figure 18 presents the variation of the Mach number through the nozzle for
Figure 19 present the correction of the Mach number of nozzle giving exit Mach number
One can see that the curves confound until Mach number
MS (PG γ=1.402) | 1.5000 | 2.0000 | 3.0000 | 4.0000 | 5.0000 | 6.0000 |
MS (T0=298.15 K) | 1.4995 | 1.9995 | 2.9995 | 3.9993 | 4.9989 | 5.9985 |
MS (T0=500 K) | 1.4977 | 1.9959 | 2.9956 | 3.9955 | 4.9951 | 5.9947 |
MS (T0=1000 K) | 1.4879 | 1.9705 | 2.9398 | 3.9237 | 4.9145 | 5.9040 |
MS (T0=1500 K) | 1.4830 | 1.9534 | 2.8777 | 3.8147 | 4.7727 | 5.7411 |
MS (T0=2000 K) | 1.4807 | 1.9463 | 2.8432 | 3.7293 | 4.6372 | 5.5675 |
MS (T0=2500 K) | 1.4792 | 1.9417 | 2.8245 | 3.6765 | 4.5360 | 5.4209 |
MS (T0=3000 K) | 1.4785 | 1.9388 | 2.8121 | 3.6454 | 4.4676 | 5.3066 |
MS (T0=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
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
The
The basic variable for our model is the temperature and for the
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 C_{P} 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.
Acknowledgments
The author acknowledges Djamel, Khaoula, Abdelghani Amine, Ritadj Zebbiche and Fettoum Mebrek for granting time to prepare this manuscript.
References
- 1.
Anderson J. D. Jr. 1982 Modern Compressible Flow. With Historical Perspective, (2^{nd} edition), Mc Graw-Hill Book Company,0-07001-673-9 York, USA. - 2.
Anderson J. D. Jr. 1988 Fundamentals of Aerodynamics , (2^{nd} edition), Mc Graw-Hill Book Company,0-07001-656-9 York, USA. - 3.
Démidovitch B. Maron I. 1987 - 4.
Fletcher C. A. J. 1988 Computational Techniques for Fluid Dynamics: Specific Techniques for Different Flow Categories, Vol. II, Springer Verlag,0-38718-759-6 Heidelberg. - 5.
Moran M. J. 2007 Fundamentals of Engineering Thermodynamics , John Wiley & Sons Inc., 6th Edition,978-8-0471787358 USA - 6.
Oosthuisen P. H. Carscallen W. E. 1997 Compressible Fluid Flow. Mc Grw-Hill,0-07-0158752-9 York, USA. - 7.
Peterson C. R. Hill P. G. 1965 Mechanics and Thermodynamics of Propulsion , Addition-Wesley Publishing Company Inc.,0-20102-838-7 York, USA. - 8.
Ralston A. Rabinowitz P. A. 1985 A First Course in Numerical Analysis. (2^{nd} Edition), McGraw-Hill Book Company,0-07051-158-6 York, USA. - 9.
Ryhming I. L. 1984 - 10.
Zebbiche T. 2007 Stagnation Temperature Effect on the Prandtl Meyer Function . - 11.
Zebbiche T. Youbi Z. 2005a Supersonic Flow Parameters at High Temperature. Application for Air in nozzles. German Aerospace Congress 2005, DGLR-2005-0256, 26-29 Sep. 2005,978-3-83227-492-4 Friendrichshafen, Germany. - 12.
Zebbiche T. Youbi Z. 2005b Supersonic Two-Dimensional Minimum Length Nozzle Conception. Application for Air. German Aerospace Congress 2005, DGLR-2005-0257, 26-29 Sep. 2005,978-3-83227-492-4 Friendrichshafen, Germany. - 13.
Zebbiche T. Youbi Z. 2006 Supersonic Plug Nozzle Design at High Temperature. Application for Air, AIAA Paper2006 0592 ^{th} AIAA Aerospace Sciences Meeting and Exhibit, 9-12 Jan. 2006,978-1-56347-893-2 Reno Nevada, Hilton, USA. - 14.
Zebbiche T. 2010a Supersonic Axisymetric Minimum Length Conception at High Temperature with Application for Air. - 15.
Zebbiche T. 2010b - 16.
Zuker R. D. Bilbarz O. 2002