**1. Introduction**

420 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

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А1. Bulletin 2003/15. 09.04.2003.

262.

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

Effect of Stagnation Temperature on Supersonic

Mach number with the enthalpy and the temperature:

Hill1, 1965, & Oosthuisen & Carscallen, 1997):

Using the expression (3), the relationship (10), can be written as:

0

*<sup>ρ</sup> Exp <sup>ρ</sup>*

The pressure ratio is obtained by using the relation of the perfect gas state:

Where

obtain:

Where

(*ρ,T*):

calorically imperfect gas.

Flow Parameters with Application for Air in Nozzles 423

<sup>0</sup>

( ) 2 ( ) *dV C T <sup>P</sup> dT*

( ) *<sup>P</sup> T*

*T*

Dividing the equation (6) by *V2* and substituting the relation (7) in the obtained result, we

Dividing the relation (7) by the sound velocity, we obtain an expression connecting the

2 ( ) ( ) ( ) *H T*

The relation (10) shows the variation of the Mach number with the temperature for

The momentum equation in differential form can be written as (Moran, 2007, Peterson &

<sup>0</sup> *dP V dV ρ*

> ( ) *<sup>ρ</sup> <sup>d</sup><sup>ρ</sup> F T dT*

> > 2 ( ) ( ) ( ) *P*

*F T dT* 

(14)

*ρ*

*C T F T*

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

0

*T*

*T*

*ρ*

*MT* 

<sup>2</sup> *V HT* 2 ( ) (7)

*HT C T dT* (8)

*V HT* (9)

*a T* (10)

(11)

*<sup>ρ</sup>* (12)

*a T* (13)

are responsible for this deviation. Our flow is regarded as perfect, permanent and nonrotational. 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.

## **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).

$$\frac{\mathbf{C}\_P}{\chi} \, dT - \frac{RT}{\rho} \, d\rho = 0 \tag{1}$$

Using relationship between *CP* and *γ* [*CP=γR/(γ-1)]*, the equation (1) can be written at the following form:

$$\frac{d\rho}{\rho} = \frac{dT}{T\left[\mathcal{Y}\left(T\right) - 1\right]}\tag{2}$$

The integration of the relation (2) gives the adiabatic equation of a perfect gas at high temperature.

The sound velocity is (Ryhming, 1984),

$$a^2 = \left(\frac{dP}{d\rho}\right)\_{entropy = \text{cons }\tan t} \tag{3}$$

The differentiation of the state equation of a perfect gas gives:

$$\frac{dP}{d\rho} = \rho \text{ R } \frac{dT}{d\rho} + \text{ R } T \tag{4}$$

Substituting the relationship (2) in the equation (4), we obtain after transformation:

$$a^2(T) = \chi(T) \ R \ T \tag{5}$$

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:

$$\mathcal{C}\_p \, dT + V \, dV = 0 \tag{6}$$

The integration between the stagnation state (*V0 ≈ 0, T0*) and supersonic state (*V, T*) gives:

$$V^2 = \mathcal{Z}\,H(T)\tag{7}$$

Where

422 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

are responsible for this deviation. Our flow is regarded as perfect, permanent and nonrotational. The gas is regarded as calorically imperfect and thermally perfect*.* The theory of

To determine the application limits of the perfect gas model, the error given by this model is

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 &

Using relationship between *CP* and *γ* [*CP=γR/(γ-1)]*, the equation (1) can be written at the

The integration of the relation (2) gives the adiabatic equation of a perfect gas at high

*dρ dT ρ T T* 

*dP*

 *dP dT <sup>ρ</sup> R RT <sup>d</sup><sup>ρ</sup> <sup>d</sup><sup>ρ</sup>*

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

The equation of the energy conservation in differential form (Anderson, 1988 & Moran,

The integration between the stagnation state (*V0 ≈ 0, T0*) and supersonic state (*V, T*) gives:

 

*d*

2

*a*

The differentiation of the state equation of a perfect gas gives:

[ ( ) 1]

*entropy cons t* tan

<sup>0</sup> *C RT <sup>P</sup> dT d<sup>ρ</sup> γ ρ* (1)

(2)

(3)

(4)

<sup>2</sup> *a T*( ) ( ) *γ T RT* (5)

*C dT V dV <sup>P</sup>* 0 (6)

perfect gas does not take account of this temperature.

compared with our results.

following form:

temperature.

ratio *γ(T).*

2007) is written as:

**2. Mathematical formulation**

Bilbarz, 2002, Zebbiche, 2010a, 2010b).

The sound velocity is (Ryhming, 1984),

$$H(T) = \int\_{T}^{T\_0} \mathbf{C}\_p(T) \, dT \tag{8}$$

Dividing the equation (6) by *V2* and substituting the relation (7) in the obtained result, we obtain:

$$\frac{dV}{V} = -\frac{C\_P(T)}{2}\,dT\tag{9}$$

Dividing the relation (7) by the sound velocity, we obtain an expression connecting the Mach number with the enthalpy and the temperature:

$$M(T) = \frac{\sqrt{2 \ H(T)}}{a(T)} \tag{10}$$

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):

$$dV\,dV + \frac{dP}{\rho} = 0\tag{11}$$

Using the expression (3), the relationship (10), can be written as:

$$\frac{d\rho}{d\rho} = F\_{\rho}(T) \cdot dT \tag{12}$$

Where

$$F\_{\rho}(T) = \frac{C\_P(T)}{a^2(T)}\tag{13}$$

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*):

$$\frac{\rho}{\rho\_0} = \exp\left(-\int\_{T\_{-}}^{T\_0} F\_\rho(T) \, dT\right) \tag{14}$$

The pressure ratio is obtained by using the relation of the perfect gas state:

$$\frac{P}{P\_0} = \left(\frac{\rho}{\rho\_0}\right) \left(\frac{T}{T\_0}\right) \tag{15}$$

Effect of Stagnation Temperature on Supersonic

(J/(KgK) <sup>γ</sup>(T) T (K) CP

722.205 1080.005 1.362 1888.872 1243.883 1.300 777.761 1093.370 1.356 1999.983 1250.305 1.298

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

Table 2. Coefficients of the polynomial *CP(T)*.

Table 1. Variation of *CP(T)* and *γ(T)* versus the temperature for air.

**3. Calculation procedure**

T (K) CP

Flow Parameters with Application for Air in Nozzles 425

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.

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

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):

*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

1 2 3 4 5 6 7 8 9 10 ( ) ( ( ( ( ( ( ( ( ))))))))) ( *C T a Ta Ta Ta Ta Ta Ta Ta Ta Ta <sup>P</sup>* (23)

(J/(Kg K) γ(T) T (K) CP

J/(Kg K) γ(T)

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

$$
\rho \,\, V \,\, A \,\, = \,\, const \,\, \tan t \,\, \tag{16}
$$

The taking logarithm and then differentiating of relation (16), and also using of the relations (9) and (12), one can receive the following equation:

$$\frac{dA}{A} = F\_A(T) \,\,\, dT \tag{17}$$

Where

$$F\_A(T) = C\_P(T) \left[ \frac{1}{a^2(T)} - \frac{1}{2H(T)} \right] \tag{18}$$

The integration of equation (17) between the critical state (*A\*, T\**) and the supersonic state (*A, T*) gives the cross-section areas ratio:

$$\frac{A}{A\_{\bullet}} = \operatorname{Exp}\left(\int\_{T}^{T\_{\bullet}} F\_{A}(T) \, \, dT\right) \tag{19}$$

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.* 

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

$$\frac{m}{A\_\ast \, \rho\_0 \, a\_0} = \int\_{\cdot A} \left( \frac{\rho}{\rho\_0} \right) \left( \frac{a}{a\_0} \right) M \cos(\theta) \frac{dA}{A\_\ast} \tag{20}$$

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:

$$\frac{\dot{m}}{A.\ \rho\_0\ a\_0} = \left(\frac{\rho\_\*}{\rho\_0}\right)\left(\frac{a\_\*}{a\_0}\right) \tag{21}$$

The determination of the velocity sound ratio is done by the relation (5). Thus,

$$\frac{a}{a\_0} = \left[\frac{\chi(T)}{\chi(T\_0)}\right]^{1/2} \left[\frac{T}{T\_0}\right]^{1/2} \tag{22}$$

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.
