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Effect of Stagnation Temperature on Supersonic Flow Parameters with Application for Air in Nozzles

Written By

Toufik Zebbiche

Submitted: November 4th, 2010 Published: November 2nd, 2011

DOI: 10.5772/19633

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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 CPis 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 CPand 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 CPand γversus 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.

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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 (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).

CPγdTRTρdρ=0E1

Using relationship between CPand 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 sound velocity is (Ryhming, 1984),

a2=(dPdρ)entropy=constantE3

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 V2and 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.

The momentum equation in differential form can be written as (Moran, 2007, Peterson & Hill1, 1965, & Oosthuisen & Carscallen, 1997):

VdV+dPρ=0E11

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 T0can 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:

PP0=(ρρ0)(TT0)E15

The mass conservation equation is written as (Anderson, 1988& Moran, 2007)

ρVA=constantE16

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 ρand 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.

The critical mass flow rate (Moran, 2007, Zebbiche & Youbi, 2005a, 2005b) can be written in non-dimensional form:

mA*ρ0a0=.A(ρρ0)(aa0)Mcos(θ)dAA*E20

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=0and 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, ρand Afor 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 Mand T, where the reverse analytical expression does not exist.

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3. Calculation procedure

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)
55.5381001.1041.402833.3161107.1921.3502111.0941256.8131.296
...888.8721119.0781.3452222.2051263.4101.294
222.2051001.1011.402944.4271131.3141.3402333.3161270.0971.292
277.7611002.8851.401999.9831141.3651.3362444.4271273.4761.291
305.5381004.6751.4001055.5381151.6581.3322555.5381276.8771.290
333.3161006.4731.3991111.0941162.2021.3282666.6501283.7511.288
361.0941008.2811.3981166.6501170.2801.3252777.7611287.2241.287
388.8721011.9231.3961222.2051178.5091.3222888.8721290.7211.286
416.6501015.6031.3941277.7611186.8931.3192999.9831294.2421.285
444.4271019.3201.3921333.3161192.5701.3173111.0941297.7891.284
499.9831028.7811.3871444.4271204.1421.3133222.2051301.3601.283
555.5381054.5631.3741555.5381216.0141.3093333.3161304.9571.282
611.0941054.5631.3701666.6501225.1211.3063444.4271304.9571.282
666.6501067.0771.3681777.7611234.4091.3033555.5381308.5801.281
722.2051080.0051.3621888.8721243.8831.300
777.7611093.3701.3561999.9831250.3051.298

Table 1.

Variation of CP(T)and γ(T)versus the temperature for air.

For a perfect gas, the γand CPvalues are equal to γ=1.402 and CP=1001.28932 J/(kgK) (Oosthuisen & Carscallen, 1997, Moran, 2007& Zuker & Bilbarz, 2002).. The interpolation of the CPvalues according to the temperature is presented by relation (23) in the form of Horner scheme to minimize the mathematical operations number (Zebbiche, 2010a, 2010b):

CP(T)=a1+T(a2+T(a3+T(a4+T(a5+T(a6+T(a7+T(a8+T(a9+T(a10)))))))))E23

The interpolation (aii=1, 2, …, 10) of constants are illustrated in table 2.

IaiIai
11001.105863.069773 10-12
20.040661287-1.350935 10-15
3-0.00063376983.472262 10-19
42.747475 10-69-4.846753 10-23
5-4.033845 10-9102.841187 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:

forTT¯, we have CP(T)=C¯PforT>T¯, relation (23) is used.

The selected interpolation gives an error less than ε=10-3between 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

H(T)=H0-[c1+T(c2+T(c3+T(c4+T(c5+Tc6+T(c7+T(c8+T(c9+T(c10)))))))))]E25

Where

H0=T0(c1+T0(c2+T0(c3+T0(c4+T0(c5+T0(c6+T0(c7+T0(c8+T0(c9+T0(c10))))))))))E26

and

ci=aii(i=1,2,3,...,10)E27

Figure 1.

Variation of functionFρ(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(T0T)E28
ForT0>T¯,we have two cases:

ifT>T¯:H(T)=relation(24)E29
ifTT¯: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 functionFA(T)in the interval [TS,T*] versusT0

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 TSfor 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+(1b1)[1tanh[b2(1zi)]tanh(b2)]E31

Where

zi=i1N11iNE32

Obtained sivalues, enable to find the value of Tiin nodes i:

Ti=si(TDTG)+TGE33

The temperature TDis equal to T0for Fρ(T), and equal to T*for FA(T).The temperature TGis equal to T*for the critical parameter, and equal to TSfor the supersonic parameter. Taking a value b 1near zero (b 1=0.1, for example) and b 2=2.0, it can condense the nodes towards left edge TSof 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 Kand T2=T0. The value T*can be given with a precision εif the interval of subdivision number Kis satisfied by the following condition:

K=1.4426Log(T0ε)+1E35

If ε=10-8 is taken, the number Kcannot exceed 39. Consequently, the temperature ratio T*/T0can 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=TScan be determined according to the Mach number M=MS. Replacing T=TSand M=MSin relation (10) gives

2H(TS)MS2a2(TS)=0E36

The determination of TSof 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=TSand ρ=ρSin 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=MScan be calculated respectively by using the relations (15), (22) and (19) by replacing T=TS, ρ=ρS, P=PS, a=aSand A=AS.

The integration results of the ratios ρ*/ ρ0, ρS0and AS/A*primarily depend on the values of N, b1and 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 CPand ratio γas 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 PGmodel compared to our HTmodel can be calculated for each parameter. Then, for each value (T0, M), the εerror can be evaluated by the following relationship:

εy(T0,M)=|1yPG(T0,M)yHT(T0,M)|×100E40

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

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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 PGmodel given a supersonic uniform Mach number MSat the exit section of rockets, degrades the desired performances (exit Mach number, pressure force), especially if the temperature T0of 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)=eTS(HT)T*FA(T)dT>ASA*(PG)E43

The relation (36) indicates that the Mach number of the PGmodel 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 PGmodel as present equation (38). Then the shape of the nozzle obtained by PGmodel is included in the nozzle obtained by our model. The temperature T0presented in equation (38) is that correspond to the temperature T0for 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 HTmodel.

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 HTmodel is initially made by the determination of the temperature TSas 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.

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5. Results and comments

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 Kfor HTand PGmodels. 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 T0increases, 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 temperatureT0.

Figure 5.

Variation of the specific heats ratio versusT0.

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 T0unlike to the PGmodel, where they do not depend on T0.. For example, the value of the temperature ratio given by the HTmodel is always higher than the value given by the PGmodel. 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 ofT*/T0versusT0.

Figure 7.

Variation ofρ*0versusT0.

Figure 8.

Variation ofP*/P0versusT0.

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 HTmodel, especially for values of T0.

Figure 9.

Variation of the non-dimensional critical mass flow rate withT0.

Figure 10 presents the variation of the critical sound velocity ratio versus T0. The influence of the T0on this parameter can be found.

Figure 10.

Effect ofT0on the velocity sound ratio.

T*/T0P*/P0ρ*/ρ 0a*/a 0m/A* ρ 0 a 0
PG (γ=1.402)0.83260.52790.63400.91240.5785
T0=298.15 K0.83280.52790.63390.91310.5788
T0=500 K0.83660.52930.63260.91710.5802
T0=1000 K0.85350.53690.62910.92800.5838
T0=2000 K0.86890.54480.62700.93430.5858
T0=2500 K0.87220.54660.62660.93550.5862
T0=3000 K0.87430.54750.62630.93650.5865
T0=3500 K0.87580.54840.62620.93660.5865

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 T0influences on the value of the thermodynamic and geometrical parameters of flow unlike the PGmodel.

The curve 4 of figure 11 is under the curves of the HTmodel, 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/T0M=2.00M=3.00M=4.00M=5.00M=6.00
PG (γ=1.402)0.55430.35600.23710.16590.1214
T0=298.15 K0.55440.35600.23720.16590.1214
T0=500 K0.55770.35810.23860.16690.1221
T0=1000 K0.58100.37310.24810.17360.1269
T0=1500 K0.60310.39110.25940.18100.1323
T0=2000 K0.61630.40580.26940.18730.1366
T0=2500 K0.62450.41620.27780.19280.1403
T0=3000 K0.63010.42330.28480.19770.1473
T0=3500 K0.63400.42850.29010.20180.1462

Table 4.

Numerical values of the temperature ratio at high temperature

Figure 11.

Variation ofT/T0versus Mach number.

ρ/ρ0M=2.00M=3.00M=4.00M=5.00M=6.00
PG (γ=1.402)0.23040.07650.02780.01140.0052
T0=298.15 K0.23040.07650.02780.01140.0052
T0=500 K0.22830.07580.02760.01130.0052
T0=1000 K0.21810.06960.02500.01030.0047
T0=1500 K0.21160.06360.02200.00890.0041
T0=2000 K0.20870.06010.01970.00770.0035
T0=2500 K0.20690.05810.01820.00690.0030
T0=3000 K0.20570.05690.01730.00630.0027
T0=3500 K0.20490.05600.01660.00580.0024

Table 5.

Numerical values of the density ratio at high temperature

Figure 12.

Variation ofρ/ρ0versus Mach number.

P/P0M=2.00M=3.00M=4.00M=5.00M=6.00
PG (γ=1.402)0.12770.02720.00660.00190.0006
T0=298.15 K0.12770.02720.00660.00190.0006
T0=500 K0.12730.02710.00650.00180.0006
T0=1000 K0.12670.02590.00620.00170.0006
T0=1500 K0.12760.02480.00570.00160.0005
T0=2000 K0.12860.02440.00530.00140.0004
T0=2500 K0.12920.02420.00500.00130.0004
T0=3000 K0.12960.02400.00490.00040.0003
T0=3500 K0.12990.02400.00480.00110.0003

Table 6.

Numerical values of the Pressure ratio at high temperature.

Figure 13.

Variation ofP/P0versus Mach number.

A/A*M=2.00M=3.00M=4.00M=5.00M=6.00
PG (γ=1.402)1.68594.220010.647024.749152.4769
T0=298.15 K1.68594.219510.644424.740152.4516
T0=500 K1.69164.237310.689524.844752.6735
T0=1000 K1.72954.473911.399626.501956.1887
T0=1500 K1.75824.782212.639729.776963.2133
T0=2000 K1.77114.993013.861733.586072.0795
T0=2500 K1.77955.121714.822737.210481.2941
T0=3000 K1.78515.209115.504040.384490.4168
T0=3500 K1.78895.272716.009843.000198.7953

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 PGmodel can be used for T0<1000 K.

Figure 15 presents the variation of the sound velocity ratio versus Mach number at high temperature. T0value 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 T0on this parameter. We can found that all the four curves are almost confounded when ME<2.00approximately. 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/a0M=2.00M=3.00M=4.00M=5.00M=6.00
PG (γ=1.402)0.74450.59660.48700.40740.3484
T0=298.15 K0.74500.59700.48730.40760.3486
T0=500 K0.75100.60190.49130.41100.3515
T0=1000 K0.77390.62450.51030.42680.3651
T0=1500 K0.78620.64080.52540.43980.3762
T0=2000 K0.79230.65010.53540.44890.3841
T0=2500 K0.79590.65560.54200.45530.3898
T0=3000 K0.79850.65950.54650.46000.3942
T0=3500 K0.79980.66180.54950.46320.3973

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.

CFM=2.00M=3.00M=4.00M=5.00M=6.00
PG (γ=1.402)1.20781.45191.58021.65231.6959
T0=298.15 K1.20781.45181.58001.65211.6957
T0=500 K1.20761.45191.58021.65231.6958
T0=1000 K1.20721.46131.59191.66461.7085
T0=1500 K1.20621.47481.61231.68711.7317
T0=2000 K1.20481.48321.62881.70691.7527
T0=2500 K1.20421.48791.64011.72211.7694
T0=3000 K1.20381.49121.64791.73371.7828
T0=3500 K1.20331.49361.65331.74221.7932

Table 9.

Numerical values of the thrust coefficient at high temperature

Figure 16.

Variation ofCFversus 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 PGand the HTmodels for several T0values.

It can be seen that the error depends on the values of T0and M. For example, if T0=2000 K and M=3.00, the use of the PGmodel 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 Mand T0, the error εis 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 PGmodel 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 ofPGversus 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 PGmodel. If T0is taken into account, we will see a fall in Mach number of the dimensioned nozzle in comparison with the PGmodel. 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 PGmodel. It should be noticed that the difference becomes considerable if the value T0exceeds 1000 K.

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

One can see that the curves confound until Mach number MS=2.0for 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 T0is lower than 1000 K. For example, if the nozzle delivers a Mach number MS=3.00at the exit section, on the assumption of the PGmodel, the HTmodel 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)1.50002.00003.00004.00005.00006.0000
MS (T0=298.15 K)1.49951.99952.99953.99934.99895.9985
MS (T0=500 K)1.49771.99592.99563.99554.99515.9947
MS (T0=1000 K)1.48791.97052.93983.92374.91455.9040
MS (T0=1500 K)1.48301.95342.87773.81474.77275.7411
MS (T0=2000 K)1.48071.94632.84323.72934.63725.5675
MS (T0=2500 K)1.47921.94172.82453.67654.53605.4209
MS (T0=3000 K)1.47851.93882.81213.64544.46765.3066
MS (T0=3500 K)1.47781.93682.80353.62414.42165.2237

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 HTmodel 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 PGmodel use. The nozzle that is calculated according to the PGmodel provides less cross-section area in comparison with the HTmodel.

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 givenMS=3.00 at the exit.

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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 T0is lower than 1000 K for any value of Mach number, or when the Mach number is lower than 2.0for any value of T0up to approximately 3000 K.

The PGmodel 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 PGmodel is the Mach number because of a nonlinear implicit equation connecting the parameters Tand 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.

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Acknowledgments

The author acknowledges Djamel, Khaoula, Abdelghani Amine, Ritadj Zebbiche and Fettoum Mebrek for granting time to prepare this manuscript.

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Written By

Toufik Zebbiche

Submitted: November 4th, 2010 Published: November 2nd, 2011