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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 CP is constant and does not depend on the temperature, which is not valid in the real case when the temperature increases (Zebbiche & Youbi, 2005b, 2006, Zebbiche, 2010a, 2010b). The aim of this research is to develop a mathematical model of the gas flow by adding the variation effect of CP and γ with the temperature. In this case, the gas is named by calorically imperfect gas or gas at high temperature. There are tables for air (Peterson & Hill, 1965) for example) that contain the values of CP and γ versus the temperature in interval 55 K to 3550 K. We carried out a polynomial interpolation of these values in order to find an analytical form for the function CP(T).
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 9th 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. The theory of perfect gas does not take account of this temperature.
To determine the application limits of the perfect gas model, the error given by this model is compared with our results.
The development is based on the use of the conservation equations in differential form. We assume that the state equation of perfect gas (P=ρRT) remains valid, with R=287.102 J/(kg K). For the adiabatic flow, the temperature and the density of a perfect gas are related by the following differential equation (Moran, 2007& Oosthuisen & Carscallen, 1997& Zuker & Bilbarz, 2002, Zebbiche, 2010a, 2010b).
CPγdT−RTρdρ=0E1
Using relationship between CP and γ [CP=γR/(γ-1)], the equation (1) can be written at the following form:
dρρ=dTT[γ(T)−1]E2
The integration of the relation (2) gives the adiabatic equation of a perfect gas at high temperature.
The differentiation of the state equation of a perfect gas gives:
dPdρ=ρRdTdρ+RTE4
Substituting the relationship (2) in the equation (4), we obtain after transformation:
a2(T)=γ(T)RTE5
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 γ(T).
The equation of the energy conservation in differential form (Anderson, 1988& Moran, 2007) is written as:
CPdT+VdV=0E6
The integration between the stagnation state (V0≈ 0, T0) and supersonic state (V, T) gives:
V2=2H(T)E7
Where
H(T)=∫TT0CP(T)dTE8
Dividing the equation (6) by V2 and substituting the relation (7) in the obtained result, we obtain:
dVV=−CP(T)2H(T)dTE9
Dividing the relation (7) by the sound velocity, we obtain an expression connecting the Mach number with the enthalpy and the temperature:
M(T)=2H(T)a(T)E10
The relation (10) shows the variation of the Mach number with the temperature for calorically imperfect gas.
Using the expression (3), the relationship (10), can be written as:
dρρ=Fρ(T)dTE12
Where
Fρ(T)=CP(T)a2(T)E13
The density ratio relative to the temperature T0 can be obtained by integration of the function (13) between the stagnation state (ρ0,T0) and the concerned supersonic state (ρ,T):
ρρ0=Exp(−∫TT0Fρ(T)dT)E14
The pressure ratio is obtained by using the relation of the perfect gas state:
The taking logarithm and then differentiating of relation (16), and also using of the relations (9) and (12), one can receive the following equation:
dAA=FA(T)dTE17
Where
FA(T)=CP(T)[1a2(T)−12H(T)]E18
The integration of equation (17) between the critical state (A*, T*) and the supersonic state (A, T) gives the cross-section areas ratio: *
AA*=Exp(∫TT*FA(T)dT)E19
To find parameters ρ and A, the integrals of functions Fρ(T) and FA(T) should be found. As the analytical procedure is impossible, our interest is directed towards the numerical calculation. All parameters M, ρ and A depend on the temperature.
As the mass flow rate through the throat is constant, we can calculate it at the throat. In this section, we have ρ=ρ*, a=a*, M=1, θ=0 and A=A*. Therefore, the relation (20) is reduced to:
m˙A*ρ0a0=(ρ*ρ0)(a*a0)E21
The determination of the velocity sound ratio is done by the relation (5). Thus,
aa0=[γ(T)γ(T0)]1/2[TT0]1/2E22
The parameters T, P, ρ and A for the perfect gas are connected explicitly with the Mach number, which is the basic variable for that model. For our model, the basic variable is the temperature because of the implicit equation (10) connecting M and T, where the reverse analytical expression does not exist.
In the first case, one presents the table of variation of CP and γ versus the temperature for air (Peterson & Hill, 1965, Zebbiche 2010a, 2010b). The values are presented in the table 1.
T (K)
CP(J/(KgK)
γ(T)
T (K)
CP (J/(Kg K)
γ(T)
T (K)
CP 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
Table 1.
Variation of CP(T) and γ(T) versus the temperature for air.
For a perfect gas, the γ and CP values are equal to γ=1.402 and CP=1001.28932 J/(kgK) (Oosthuisen & Carscallen, 1997, Moran, 2007& Zuker & Bilbarz, 2002).. The interpolation of the CP values according to the temperature is presented by relation (23) in the form of Horner scheme to minimize the mathematical operations number (Zebbiche, 2010a, 2010b):
The interpolation (aii=1, 2, …, 10) of constants are illustrated in table 2.
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
Table 2.
Coefficients of the polynomial CP(T).
A relationship (23) gives undulated dependence for temperature approximately low thanT¯=240K. So for this field, the table value (Peterson & Hill, 1965), was taken
C¯P=Cp(T¯)=1001.15868J/(kgK)E24
Thus:
forT≤T¯, we have CP(T)=C¯PforT>T¯, relation (23) is used.
The selected interpolation gives an error less than ε=10-3 between the table and interpolated values.
Once the interpolation is made, we determine the function H(T) of the relation (8), by integrating the function CP(T) in the interval [T, T0]. Then, H(T) is a function with a parameter T0and it is defined when T≤T0.
Substituting the relation (23) in (8) and writing the integration results in the form of Horner scheme, the following expression for enthalpy is obtained
Variation of function Fρ(T) in the interval [TS,T0] versusT0.
Taking into account the correction made to the function CP(T), the function H(T) has the following form:
ForT0<T¯,
H(T)=C¯P(T0−T)E28
ForT0>T¯,we have two cases:
ifT>T¯:H(T)=relation(24)E29
ifT≤T¯:H(T)=C¯P(T¯−T)+H(T¯)E30
The determination of the ratios (14) and (19) require the numerical integration of Fρ(T) and FA(T) in the intervals [T, T0] and [T, T*] respectively. We carried out preliminary calculation of these functions (Figs. 1, 2) to see their variations and to choice the integration method.
Figure 2.
Variation of the function FA(T) in the interval [TS,T*] versus T0
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 T0=500 K (low temperature) and MS=6.00 (extreme supersonic) for a good representation in these ends. In this case, we obtain T*=418.34 K and TS=61.07 K. the two functions presents a very large derivative at temperature TS.
A Condensation of nodes is then necessary in the vicinity of TS for the two functions. The goal of this condensation is to calculate the value of integral with a high precision in a reduced time by minimizing the nodes number. The Simpson’s integration method (Démidovitch & Maron, 1987& Zebbiche & Youbi, 2006) was chosen. The chosen condensation function has the following form (Zebbiche & Youbi, 2005a):
si=b1zi+(1−b1)[1−tanh[b2⋅(1−zi)]tanh(b2)]E31
Where
zi=i−1N−11≤i≤NE32
Obtained si values, enable to find the value of Ti in nodes i:
Ti=si(TD−TG)+TGE33
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 b1 near zero (b1=0.1, for example) and b2=2.0, it can condense the nodes towards left edge TS of the interval, see figure 3.
Figure 3.
Presentation of the condensation of nodes
3.1. Critical parameters
The stagnation state is given by M=0. Then, the critical parameters correspond to M=1.00, for example at the throat of a supersonic nozzle, summarize by:
When M=1.00 we have T=T*. These conditions in the relation (10), we obtain:
2H(T*)−a2(T*)=0E34
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 of subdivision number K is satisfied by the following condition:
K=1.4426Log(T0ε)+1E35
If ε=10-8 is taken, the number K cannot exceed 39. Consequently, the temperature ratio T*/T0 can be calculated.
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 is obtained.
The critical ratios of the pressures and the sound velocity can be calculated by using the relations (15) and (22) respectively, by replacing T=T*, ρ=ρ*, P=P* and a=a*,
3.2. Parameters for a supersonic Mach number
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 (10) gives
2H(TS)−MS2a2(TS)=0E36
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=ρS 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 obtained.
The ratios of pressures, speed of sound and the sections corresponding to M=MS can be calculated respectively by using the relations (15), (22) and (19) by replacing T=TS, ρ=ρS, P=PS, a=aS and A=AS.
The integration results of the ratios ρ*/ ρ0, ρS/ρ0 and AS/A* primarily depend on the values of N, b1 and b2.
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)
CF=FP0A*E37
Where
F=mVE=mMEaEE38
The introduction of relations (21), (22) into (32) gives as the following relation:
CF=γ(T0)ME(aEa0)(ρ*ρ*)(a*a0)E39
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:
εy(T0,M)=|1−yPG(T0,M)yHT(T0,M)|×100E40
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%.
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.
MS(HT)=MS(PG)E41
MS(PG)=2H[TS(HT)]a[TS(HT)]E42
ASA*(HT)=e∫TS(HT)T*FA(T)dT>ASA*(PG)E43
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.
ASA*(HT)=ASA*(PG)E44
MS(HT)<MS(PG)E45
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.
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.
Figure 4.
Variation of the specific heat for constant pressure versus stagnation temperature T0.
Figure 5.
Variation of the specific heats ratio versus 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.
Figure 6.
Variation of T*/T0 versus T0.
Figure 7.
Variation of ρ*/ρ0 versus T0.
Figure 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.
Figure 9.
Variation of the non-dimensional critical mass flow rate with T0.
Figure 10 presents the variation of the critical sound velocity ratio versus T0. The influence of the T0 on this parameter can be found.
Figure 10.
Effect of T0 on the velocity sound ratio.
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
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 cross-section versus Mach number for T0 =1000 K, 2000 K and 3000 K, including the case of perfect gas for γ=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.
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
Table 4.
Numerical values of the temperature ratio at high temperature
Figure 11.
Variation of T/T0 versus Mach number.
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
Table 5.
Numerical values of the density ratio at high temperature
Figure 12.
Variation of ρ/ρ0 versus Mach number.
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
Table 6.
Numerical values of the Pressure ratio at high temperature.
Figure 13.
Variation of P/P0 versus Mach number.
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
Table 7.
Numerical Values of the cross section area 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.
Figure 14.
Variation of the critical cross-section area ratio versus Mach number.
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
Table 8.
Numerical values of the sound velocity ratio at high temperature.
Figure 15.
Variation of the ratio of the velocity sound versus Mach number.
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
Table 9.
Numerical values of the thrust coefficient at high temperature
Figure 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.
Figure 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.
Figure 18.
Effect of stagnation temperature on the variation of the Mach number through the nozzle.
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
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.
Figure 19.
Correction of the Mach number at High Temperature of a nozzle dimensioned on the perfect gas model.
Figure 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:
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.
The author acknowledges Djamel, Khaoula, Abdelghani Amine, Ritadj Zebbiche and Fettoum Mebrek for granting time to prepare this manuscript.
References
1.AndersonJ. D.Jr.1982Modern Compressible Flow. With Historical Perspective, (2nd edition), Mc Graw-Hill Book Company, 0-07001-673-9 York, USA.
2.AndersonJ. D.Jr.1988Fundamentals of Aerodynamics, (2nd edition), Mc Graw-Hill Book Company, 0-07001-656-9 York, USA.
3.DémidovitchB.MaronI.1987Eléments de calcul numérique, Editions MIR, 978-2-72989-461-0 Moscou, USSR.
4.FletcherC. A. J.1988 Computational Techniques for Fluid Dynamics: Specific Techniques for Different Flow Categories, Vol. II, Springer Verlag, 0-38718-759-6 Heidelberg.
5.MoranM. J.2007Fundamentals of Engineering Thermodynamics, John Wiley & Sons Inc., 6th Edition, 978-8-0471787358 USA
6.OosthuisenP. H.CarscallenW. E.1997 Compressible Fluid Flow. Mc Grw-Hill, 0-07-0158752-9 York, USA.
7.PetersonC. R.HillP. G.1965Mechanics and Thermodynamics of Propulsion, Addition-Wesley Publishing Company Inc., 0-20102-838-7 York, USA.
8.RalstonA.RabinowitzP. A.1985A First Course in Numerical Analysis. (2nd Edition), McGraw-Hill Book Company, 0-07051-158-6 York, USA.
9.RyhmingI. L.1984Dynamique des fluides, Presses Polytechniques Romandes, Lausanne, 2-88074-224-2
10.ZebbicheT.2007Stagnation Temperature Effect on the Prandtl Meyer Function. AIAA Journal, 45 N 04, 952954 , April 2007, 0001-1452 USA
11.ZebbicheT.YoubiZ.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.ZebbicheT.YoubiZ.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.ZebbicheT.YoubiZ.2006 Supersonic Plug Nozzle Design at High Temperature. Application for Air, AIAA Paper 20060592th AIAA Aerospace Sciences Meeting and Exhibit, 9-12 Jan. 2006, 978-1-56347-893-2 Reno Nevada, Hilton, USA.
14.ZebbicheT.2010a Supersonic Axisymetric Minimum Length Conception at High Temperature with Application for Air. Journal of British Interplanetary Society (JBIS), 63 N 04-05, 171192 , May-June 2010, 0007