On calculation of ideal thrust augmenter

ISSN 1068-7998, Russian Aeronautics (Iz.VUZ), 2012, Vol. 55, No.1, pp. 50–55. © Allerton Press, Inc., 2012. Original Russian Text © V.I. Panchenko, R.R. Bikbulatov, 2012, published in Izvestiya VUZ. Aviatsionnaya Tekhnika, 2012, No. 1, pp. 36 –39.

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AIRCRAFT AND ROCKET ENGINE THEORY

On Calculation of Ideal Thrust Augmenter V. I. Panchenko and R. R. Bikbulatov

Tupolev National Research Technical University, Kazan, Russia Received September 1, 2011

Abstract—In this paper, a simple technique for calculating the ideal ejector thrust augmenter of a jet engine is presented. It is based on the well-known method for averaging the parameters of a nonuniform gas flow with the entropy of the initial flow being unchanged. Making use of the technique presented, we compare the calculated data on the pressure ratio in the gas being ejected as well as the thrust boost coefficient with those obtained by the other authors. Also shown is the existence of the thrust boost maximum in the ideal and real thrust augmenters when the value of the ejection coefficient is approximately equal to 100.

DOI: 10.3103/S1068799812010096

Keywords: ejector, ejector thrust augmenter, ideal thrust augmenter, calculation, thrust boost limit.

As is seen, for instance, in [1, 2], the gas ejectors have received wide acceptance in many fields of engineering including the aeroindustry by virtue of their design simplicity, reliability, and the positive effect which can be attained by using a properly designed ejector. So, the history of aircraft equipment development is indicative of numerous examples when the efforts have been made to use the ejectors as thrust augmenters. Calculations and experiments reveal that the ejector is capable to realize such values of the ejection coefficient n that make it possible to attain, under definite conditions, the considerable gain in the engine thrust and total efficiency η.

And really, if the ejector is located downstream of the engine propulsive nozzle, when the ejecting gas (active flow) is the exhaust jet and the ejected gas is the atmospheric air, it is possible in some cases to augment the system jet thrust without any expenditure of additional energy by redistributing the initial energy of an active flow to the larger mass of gas. This reasoning is qualitative in essence and thrust augmentation at suction of outside air to the ejecting jet can be explained by the fact that additional forces appear on the ejector units; the resultant of these forces directed along the flow axis is added with the jet thrust of the nozzle. The basic of these forces (that determines a gain in thrust) is the out-of-balance force of outside pressure acting on the inlet flared end (air intake) of the ejector. Its appearance is caused by decompression on the flared end walls at the air ejected influx [2].

In this paper, we consider the design of an ideal ejector the notion of which was originally introduced in [3]. By the term “ideal ejector” we will mean the ejector that is free from shock and friction losses. We assume that there is no heat exchange through the ejector walls and the compressible media are used as active and passive flows. Based on these assumptions, we can conclude that the total gas entropy in such an ejector remains a constant value.

A large scope of work aimed at analyzing the operation of the ideal thrust augmenter (ideal ejector) was conducted by the authors [4]. In the framework of this investigation, use was made of the sophisticated mathematical techniques for calculating the ejector thrust and extended considerably the very theory of ideal thrust augmenter. Alongside the high-pressure gas work with respect to the low-pressure one, the authors were the first who assumed the presence of heat exchange in the ideal ejector. They showed also that the thrust boost coefficient was in this case even larger than when there was no heat exchange between flows. Also obtained in [4] were the conditions for the optimal active and passive jets efflux, needed to provide an ideal process of the ejector operation.

ON CALCULATION OF IDEAL THRUST AUGMENTER

RUSSIAN AERONAUTICS Vol. 55 No. 1 2012

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The main goal of this study is the development of a simple calculation technique as well as qualitative and quantitative assessment of the value of thrust boost in an aircraft jet engine equipped with the ideal thrust augmenter (ideal ejector). The quantitative assessment will make it possible to reveal the maximum possible parameters of this idealized ejector operation.

Figure 1 presents schematically the placement of this ejector thrust augmenter downstream of the ABE propulsive nozzle, where the exhaust engine jet will generate the active flow in the ejector, and the ambient atmosphere—the passive one.

Fig. 1.

Initial parameters for calculation are the following (notation is fully borrowed from [4]):

Mass flowrate of active jet 1nG G= ..................................................................1.8 kg/s

Pressure drop at the exhaust nozzle exit * *0 1 2p pΠ = .................................................. 5

Gas density ratio * *0 1 2r = ρ ρ ....................................................................................... 0.5

Active nozzle diameter 1nD d= .......................................................................... 0.098 m

Isentropic gas exponent in the active jet 1k .............................................................. 1.33

Static pressure of the ejected medium Hp ..................................................... 101 325 Pa

Static temperature of the ejected medium HT .....................................................288.2 K

Thermal capacity at constant pressure of the ejected medium 2pc ............ 1005 J/(kg⋅K)

Isentropic gas exponent in the ejected medium 2k ..................................................... 1.4

Gas constant of the ejected medium R ..................................................... 287.3 J/(kg⋅K)

Mach number (object’s flight speed) MH ............................................................ 0 – 1.5

From here on, we will denote parameters of the exhaust nozzle exit (used in the TJ100s engine) by the subscript n; these parameters are actually the parameters of the active nozzle exit denoted by subscript 1. The subscript H denotes the parameters of undisturbed air and the subscript 2—the parameters of the gas jet ejected.

In order to compare the data obtained, we additionally calculated the designs of ejectors with the ejection coefficients 2,n = 5, 7, 8, 10, 12, 14, and 20 [4].

CALCULATION OF THE IDEAL THRUST AUGMENTER

In [1], we have already presented the technique for calculating the ejectors, which is based on the application of the nonuniform flow parameter-averaging method; the essence of this technique consists in replacement of the initial flow (nonuniform at the cross-section) by the uniform flow (one-dimensional and averaged). According to the conditions of the problem we face, an additional restriction is imposed

PANCHENKO, BIKBULATOV


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on the simultaneous equations from [1], more exactly, the constancy of gas total entropy prior to and after averaging. This condition constS = leads to the fact that the pulse of the entropy-averaged flow (proportional to the value of the gasodynamic function ( ),z λ ) will be in the problem posed larger than

the total pulse of the initial nonuniform flow. In other words, the pulse is const,I ≠ which leads, in turn,

to an increase of the total gas-mixture pressure *3p downstream of the mixing chamber and a decrease of

the mixture efflux velocity 3λ (in the subsonic flow) or to its growth (in the supersonic flow) as

compared with the GEI method. The very name of this calculation technique “GEI method” suggests that when making calculations, the integral parameters const,G

Σ= const,E

Σ= and const .I

Σ= are kept

constant in the resultant flow. In this case, the values of the total flowrate and gas energy will remain unchanged, i.e., they will be the same as in the initial nonuniform flow. In connection with the foregoing, we present the procedure of calculating the ideal ejector and ideal thrust augmenter using as the base the GES averaging method and the following conditions const,G

Σ= const,E

Σ= const,S

Σ= and const .I

Σ≠ The

condition for entropy constancy in the averaged and real flows can be written in the form [2]:

( ) ( )* *1 1

* *ln ln ,

k k

k k

G

T TGR R dG

p p

− −

= ∫ (1)

or

( )**

* ** *

ln ln ln .1F

p qm k Tp p dF

G k T T

λ⎛ ⎞= −⎜ ⎟

−⎝ ⎠∫ (2)

If we integrate expression (2), we will obtain the formula for determining the averaged total pressure of the gas mixture at the mixing chamber exit:

( ) ( )* ** *

1 1 2 2* * *1 1 2 21 2* ** *

1 21 2

ln ln ln ln ln .1 1

p q p qk T k Tmp p p

G k kT TT TΣ

⎛ ⎞λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟= − + −⎜ ⎟ ⎜ ⎟⎜ ⎟− −⎝ ⎠ ⎝ ⎠⎝ ⎠

(3)

In doing it, the equations for determining the values of total flowrate and gas energy remain practically the same as those used in the GEI method:

( ) ( ) ( )* * *

1 1 1 2 2 21 2* * *

1 2

;F F

mp q m p q m p qG dG dF F F

T T TΣ

λ λ λ= = = +∫ ∫ (4)

( )

( ) ( )

( ) ( )1 2

*

1 2

* * * *1 1 1 2 2 2

* 1 2

1 1 2 21 2

* *1 2

.F F

p p

F F

p q T dF p q T dFE E E

Tp q p qc G c G G

dF dFT T

Σ

Σ

λ + λ

+= = =

λ λ++

∫ ∫

∫ ∫ (5)

Similarly to the GEI method, we can find all the other parameters of the flow:

*2

;1cr

k RTa

k=

+

(6)

( )*

*.

G Tq

mp FΣ

λ = (7)

Here, the averaged values of ,m ,R and k can be determined in a way similar to that in [2]. The value of

( )2q λ is the gasodynamic function, cra is the averaged critical velocity of sound at the outlet section of the ejector mixing chamber.



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When solving this system of equations, it is possible to calculate the value of total pressure in the gas mixture downstream of the ejector mixing chamber, its velocity, total flowrate and temperature, and, hence, all other gas mixture parameters in the ejector. The values obtained can be treated as the initial data needed for calculations of a diffuser or a shaped nozzle the ejector is equipped with [2], as well as for determination of the thrust values for the “jet engine–ejector” system.

So, we have

* *4 3 ;dp p= σ (8)

( ) *4 4 4 ;p pπ λ = (9)

* *4 3 ;T T= (10)

*

34 3

2;

1cr cr

kRTa a

k= =

+

(11)

4 4 4;crw a= λ (12)

( )

( )34

3 4

,d

qFf

F q

λ= =

λ σ (13)

where dσ is the coefficient of total pressure recovery in the diffuser; ( )π λ is the gasodynamic function;

cra is the critical sound velocity at the corresponding section; f is the ratio of output device area

expansion, and 4F is the area of ejector outlet section.

The value of jet thrust for the “engine–ejector” system can be determined in this case from the following equation [2]:

( ) ( )*4 4 4 4 1 2 ,H HP p F f p F G G w= λ − − + (14)

where ( )f λ is the gasodynamic function.

In order to compare the data obtained, it is necessary to calculate the value of thrust in the jet engine (without ejector) [2]:

( )*0 1 1 1 1 1 .H HP p F f p F G w= λ − − (15)

The thrust boost value can be determined by the ratio of these values: 0 .P Pδ = (16)

RESULTS OF CALCULATING THE IDEAL THRUST AUGMENTER

Figure 2* presents some graphs of variations in the gas mixture total pressure ratio * *3 2p p at section 3

(downstream of the ejector mixing chamber), while Figure 3 those of the thrust boost coefficient variation depending on the ejection coefficient n; one of these graphs was obtained with by the GEI method and the other one—by the GES method for the starting regime ( )0Hw = provided that the static pressures in

the ejector are equal ( 1 2p p= ). Proceeding from the well-known considerations, i.e., that the positive

thrust boost for the “engine–ejector” system at the starting regime is maximal, the graphs (Figs. 2 and 3) describe the maximal number of all possible parameters for the values of the ejection coefficient n from 0

up to 20. As is seen from these graphs, the maximum of pressure ratio * *3 2p p in the ejector gas mixture

falls on 1n = (by the GES method) and is equal to * *3 2 13.p p = This fact can be explained by the heat

exchange effect between the gas flows, while the drop of this value at further growth of n—by the prevailing influence of the secondary flow (by mass).

* In Figs. 2---6 the scale n is nonuniform.

PANCHENKO, BIKBULATOV


54

Fig. 2. Fig. 3.

As is seen from our calculations, the determining parameter of the ejector, which influences significantly upon the thrust boost in the “engine–ejector” system, is nonetheless the coefficient of ejection n: the value of thrust boost increases with the increase of the ejection coefficient. This brings up the question: what is the limiting value of the thrust boost coefficient at the definite coefficient of ejection both with account of losses (GEI method) and in their absence (GES method). Figure 4 presents the dependences of the thrust boost coefficients obtained by both GEI and GES methods on the coefficient of ejection n. As we see from this graph, for the preset operating parameters, there exist the limiting values of the thrust boost coefficient for the ejectors in the “engine–ejector” system. In cases when the real ejector is calculated with the use of the GEI method, the limiting value will be 2;δ = as to the use of the GES method that presents the maximally possible trust boost in the ideal ejector, it showed that the trust cannot exceed, in the ideal case in this system with such preset parameters, the tenfold value of the initial trust in the engine having the same working gas parameters. These statements on the thrust boost limit in the “engine–ejector” system are valid for any ejectors equipped with the cylindrical mixing chamber.

Fig. 4. Fig. 5.

As is known, the positive effect of the ejector use (thrust boost) in the “engine–ejector” system decreases as the flight speed increases. Figure 5 presents some comparative graphs of the thrust boost coefficient variation depending on the coefficient of ejection n for different flight Mach numbers MH ;



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some of them are constructed with the use of the GEI method, while the others—by the GES method. These characteristics are well agreed with data presented in literature [2, 4]. In order to verify such correspondence, we conducted calculations of some ejectors with parameters similar to those examined in [4]. Figure 6 also presents a number of comparative graphs of variations in the thrust boost coefficient δ depending on the coefficient of ejection n for different flight Mach numbers MH ; some of them are

constructed by the GES method (shown by solid lines), while the other ones were obtained by the technique borrowed from [4] (shown by graphic characters). The graphs show that the calculated data obtained are well agreed with those presented in literature; the exception is the variant M 1.5,H = which

can be explained by a possibility of shock wave appearance (which the GES method fails to correct for) at the “ideal” ejector inlet.

Fig. 6.

In such a way, the results of the studies presented showed that calculations of the ideal thrust augmenter can be carried out by using the technique for averaging the nonuniform flow parameters provided that const .S = As applied to the ideal ejector with parameters described, the pressure ratio of

the resultant flow * *3 2p p exceeds the ratio of the active flow decompression * *

1 2p p (at 7.n < ) Also

shown is the existence of the thrust boost maximum in the real and ideal thrust augmenters (at )100 .n ≈

REFERENCES 1. Panchenko, V.I. and Bikbulatov, R.R., Application of the Parameter-Averaging Method to Calculations of Gas

Ejectors with Cylindrical Mixing Chambers, Izv.Vuz. Av. Tekhnika, 2011, vol. 54, no. 4, pp. 29–32 [Russian Aeronautics (Engl.Transl.). vol. 54, no.4, pp. 367–371].

2. Abramovich, G.N., Prikladnaya gazovaya dinamika (Applied Gas Dynamics), Moscow: Nauka, 1991. 3. Heiser, W.H., Thrust Augmentation, Trans. ASME. Ser. A, Journal of Engineering for Gas Turbines and

Power, 1967, vol. 89, no. 1, pp. 75–81. 4. Efremov, N.L. and Kraiko, A.N., Theory of an Ideal Thrust Augmentor, Izv. RAN. Mekhanika Zhidkosti i Gaza,

2004, vol. 39, no. 4, pp. 130–142 [Russian Aeronautics (Engl.Transl.). vol. 54, no. 4, pp. 367–371].

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On calculation of ideal thrust augmenter