The modelling of thrust force for generalised nozzles using supply coefficients.

Cuciumita, Cleopatra ; Silivestru, Valentin ; Stanciu, Virgil 等

1. INTRODUCTION

The studies conducted in the field of propulsion systems revealed that all the components participate, more or less, in the development of the thrust force. It was also established that thrust force is obtained through the modification of the parameters of the flow, the most important being mass, thermal, mechanic (pressure) and geometric ones. Therefore, each component of a propulsion system can be considered a generalized nozzle, capable of generating a thrust force or even an active force. The designated literature offers some models for the assessment of the thrust force these current evaluating models are time consuming and not able to actually reveal how the thrust force is produced. The goal of elaborating this model was to determine a simplified expression of the thrust force developed by a gasodynamic and geometrical configuration in which the fluid is substantially modified. This research was conducted as part of the University "Politehnica" of Bucharest PhD scholarships Project ID 5159.

2. CONTENT

The model is based on fundamental thrust force expression, which derives from the conservation of impulse equation applied to a volume which includes the propulsion system (Carafoli & Constantinescu, 1984).

Independently of the flow regime, the acting force of a fluid is given by:

F = [F.sub.fc1] - [F.sub.fc2] (1)

where 1 is the index for inlet, 2 for outlet and [F.sub.fc] represents the flow force function (Pimsner, 1984)

[F.sub.fc] = [??]V + S(p - [p.sub.H]) (2)

The variables involved represent:

* [??], mass flow rate;

* V, velocity;

* [p.sub.H], exterior pressure;

* p, static pressure;

* S, area.

In equation 2, the thrust function can be pointed out:

[F.sub.t] = [??]V + Sp (3)

If the variables are expressed as a function of the speed coefficient, [lambda], after some calculus, the thrust function becomes:

[F.sub.t] = k + 1/k [[??]a.sub.cr] Z{[lambda]) (4)

In this equation, k represents the adiabatic factor, [a.sub.cr] the critical speed of sound and Z ([lambda]) is the gasodynamic function of thrust, given by the following expression (Rogers & Mayhew, 1992):

Z([lambda]) = 1/k ([lambda] + 1/[lambda]) (5)

Following similar steps, the mass flow rate can be obtained,

[??] = a [p.sup.*]/[square root of [T.sup.*]] Sq([lambda]) (6)

where mass flow rate gasodynamic function is:

q([lambda]) = [lambda] [(k + 1/2).sup.1/k - 1] [(1 - k - 1/k + 1 [[lambda].sup.2]).sup.1/k - 1] (7)

The fundamental idea of the supply coefficients model is to express the performances of a propulsion system dependant on the basic variables which can generate thrust force (mass, geometrical, thermal, and mechanical). In order to obtain such an expression, a connection has been established between the two gasodynamic functions, Z ([lambda]) and q ([lambda]), separately for the subsonic and supersonic regimes due to different boundary conditions (Carafoli & Oroveanu, 1955). The rate of these two curves can be seen in figures 1 and 2.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

2.1 Subsonic regime

[FIGURE 3 OMITTED]

The scope is to find a function [F.sub.sS] to satisfy the equation 8 and taking into account the boundary conditions:

* [lambda] = 1, q([lambda]) = 1, Z([lambda]) = 1, [F.sub.sS][q([lambda])] = 1;

* [[lambda] = 1, dZ([lambda])/d[lambda] = d{[F.sub.sS][q([lambda])]/dq([lambda]) = 0;

* [lambda] [right arrow] 0, [F.sub.sS][q([lambda])] [right arrow] [infinity].

Z([lambda]) [approximately equal to] [F.sub.sS][q([lambda])] (8)

Considering the rate of the two gasodynamic functions, the following approximation function is suggested:

[F.sub.sS][q([lambda])] = [C.sub.1]q([lambda]) + [C.sub.2]/q([lambda]) + [C.sub.3] (9)

The rate of the newly obtained curve is represented in figure 3.

The constants involved are fluid dependant. For example, in case of air, when k=1.4,

* [C.sub.1] = 0.215, [C.sub.2] = 0.79, [C.sub.3] = - 0.005.

This approximation is valid for 0.05 [less than or equal to] [lambda] [less than or equal to] 1, which includes the speeds usually encountered in jet propulsion engines.

2.2 Supersonic regime

Just like in the previous case, a function [F.sub.SS] to correlate the two gasodynamic functions is required. In this case,

* [lambda] = 1, q([lambda]) = 1, [F.sub.SS][q([lambda])] = 1;

* [lambda] = [[lambda].sub.max], q ([[lambda].sub.max] = 0, [F.sub.SS](0) = z ([[lambda].sub.max]).

[F.sub.SS][q([lambda])] = Z([[lambda.sub.max]) + [1 - Z([[lambda].sub.max])][[q([lambda])].sup.[alpha]] (10)

For air, k=1.4, [alpha] = 0.59 and Z([[lambda].sub.max]) = [k/[square root of [k.sup.2] - 1].

2.3 The expressions of the flow force function

Taking into account all the above, the expressions of the two flow force functions become:

[F.sub.fcsS] = [[alpha].sub.s] + [[??].sup.2][T.sup.*]/[p.sup.*]s + [[beta].sub.s][p.sup.*]S + [[gamma].sub.s][??] [square root of [T.sup.*]] - [[delta].sub.S]S (11)

[F.sub.fcSS] = [[alpha].sub.s] [[??].sup.[alpha]+1][([T.sup.*]).sup.[alpha]+1/2] + [[gamma].sub.S][??] [square root of [T.sup.*]] - [[delta].sub.S] S (12)

The constants involved in these two functions represent:

* [[alpha].sub.s] = [square root of 2 k+1/k] R [square root of [T.sup.*]] [C.sub.1]q([[lambda].sub.1]);

* [[beta].sub.s] = [square root of 2 k+1/k] R [square root of [T.sup.*] [C.sub.2] 1/q([[lambda].sub.1]);

* [[gamma].sub.s] = [square root of 2 k+1/k R [square root of [T.sup.*]] [C.sub.3];

* [[alpha].sub.s] = [square root of 2 k+1/k] R [square root of [T.sup.*]] [1 - Z([[lambda].sub.max])] 1/[a.sup.[alpha]];

* a = [square root of k/R[(2/k+1).sup.k+1/k-1];

* [[gamma].sub.s] = [square root of 2 k+1/k] R [square root of [T.sup.*]] Z ([[lambda].sub.max]);

* [[delta].sub.s] = [[delta].sub.s] = [p.sub.H].

2.4 The expressions of the specific thrust force

Applying the relations 11 and 12 for the main sections of the flow, inlet and outlet, based on equation 1, results the expressions of the specific thrust force (Pimsner, 1983) for the subsonic and supersonic regime:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

These expressions are valid in the hypothesis that the fluid doesn't suffer changes in composition, which means that all the parameters depending on its nature are constant.

It can be noticed that, regardless of the regime, the specific thrust force can be expressed as a function of the supply coefficients involved:

* [??], mass supply coefficient;

* [bar.[T.sup.*]], thermal supply coefficient;

* [bar.[p.sup.*]], mechanical supply coefficient;

* [bar.S], geometrical supply coefficient.

The expressions 13 and 14 can be singularized for simple nozzles (mass, geometrical or thermal) according to the particular values of these coefficients and the nature of the fluid involved.

3. CONCLUSION

The main advantages of the supply coefficients model are the fact that is easy to be applied and has a practical character. The whole process of modelling the trhust force of a system was simplified to one single expression. The established formulas, for each regime, are simple enough to allow discussions with respect to the influence of each modification made on the system. The values of mass, thermal, mechanical or geometrical contributions of the fluid to the obtaining of thrust force can be calculated.

This paper can be extended to determine the influence of each type of contribution for both regimes studies but also to apply this model for least common "nozzles", as turbines and compressors can be considered.

However, this model is limitated by the correlation function between the gasodynamic functions of thrust and mass flow rate, as its values is an approximate one. Therefore, the results must be validated in practice.

4. REFERENCES

Carafoli, E. & Oroveanu, T. (1955). Mecanica fluidelor (Fluid Mechanics), Editura Academiei, B000WSDV5Q, Bucharest

Carafoli, E. & Constantinescu, V. N. (1984). Dynamics of compressible fluids, Editura Academiei R.S.R., B00144KASI, Bucharest

Pimsner, V. (1983). Motoare aeroreactoare (Aero-jet engines), Editura Didactica si Pedagogica, 3521983IPCLUMO, Bucharest

Pimsner, V. (1984). Teoria si constructia sistemelor de propulsie (The theory and construction of propulsion systems), Institutul Politehnic Bucuresti, Bucharest

Rogers, G.F. & Mayhew V.R. (1992). Engineering thermodynamics: Work and Heat transfer, Longman Group UK Limited, ISBN-978-0-582-04566-8, London