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Aircraft Double-Spool Single Jet Engine with Afterburning System ALEXANDRU NICOLAE TUDOSIE Avionics Department, Faculty of Electrical Engineering University of Craiova 105-107 Decebal Blvd., Craiova, Dolj ROMANIA
[email protected],
[email protected], http://www.elth.ucv.ro/catedre/avionica/, CONSTANTIN LUCIAN SEPCU Avionics Department, Faculty of Electrical Engineering University of Craiova 105-107 Decebal Blvd., Craiova, Dolj ROMANIA
[email protected],
[email protected], http://www.elth.ucv.ro/catedre/avionica/, Abstract: -In this paper the authors have studied an aircraft engine for high thrust, the double spool single jet engine with afterburning. The authors have identified the control and the controlled parameters of the integrated propulsion system and they have also established the non- dimensional mathematical model for the integrated system (jet engine+afterburning), starting from earlier determined models; based on it, one has built the system’s block diagram with transfer functions. Some simulations were performed, considering this kind of system description, concerning the system time behavior, such as step response of its main outputs (engine speeds and temperatures, as well as the estimated augmented thrust), for different inputs (throttle’s positioning and flight regime’s variation). Key-Words: Jet-engine, Double-spool, Afterburner, Control, Fuel flow rate, Speed, Flight regime, Temperature. engines can be completed with afterburning, which is one of the most effective aircraft engine’s thrust augmentation method, being the controlled fuel injection and burning in a special kind of combustor, called “afterburner”, mounted after the engine’s last gas turbine, before the exhaust nozzle. Such a jet engine is represented in fig.1, its main parts being: a) the air inlet; b) the low pressure spool (L.P. compressor + L.P. turbine); c) the high pressure spool (H.P. compressor + H.P. turbine); d) the engine’s combustor; e) the afterburner with fuel injectors and flame stabilizers; f) the adjustable exhaust nozzle. This kind of engine assures higher thrust because of a higher π c* , as well as because of the afterburning.
1. Introduction Modern aircraft engines, especially the combat aircraft engines must assure high level of thrust, low time response and maneuverability. For a high thrust level, there are necessary high compressors’ pressure ratios π c* and/or high combustors’ temperatures T3* . For high values of π c* the compressor must be “split” in two or more parts, coupled to the same number of gas turbines (turbo-compressor groups or spools), which means that the engine become a multi-shaft or a multi-spool one; each one of these spools has its own rotation speed, all of them being gas-dynamic, but not mechanical, related (coupled). In order to obtain a supplementary thrust, these
flame stabilizers
fuel
H
1 air inlet
11
21
low pressure compressor
12
22
3
31
42
high pressure turbine
41
32
2
high pressure compressor
combustor
4
afterburner
actuator insulation
low pressure turbine
4p
5p nozzle's flaps
injectors
fuel
burned gases jet
fuel
air
n1 n1 low pressure spool
n2
n2
high pressure spool
Fig. 1. Double-spool jet engine with afterburning
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ISBN: 978-960-474-193-9
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the fuel pump plate (or the discharging valve) positioning [11,15]. Meanwhile, the exhaust nozzle’s opening is controlled by a follower system [2]. The afterburner is supplied by a fuel pump (turned round by the high-pressure shaft) and controlled by an actuator, which co-relates the fuel flow rates to the engine’s operating regime and the flight regime. The afterburning control system [12] is integrated to the engine’s control system, because it uses the same control parameters ( n 2 and A5 ), that means the unique input parameter, which is the throttle’s position, as well as the flight regime (considered as disturbance), given by the inlet air pressure p1* . Fuel’s flow rate of the afterburner determines the level of the afterburning temperature T4*p .
2. System presentation As fig. 1 and 2 shows, the system consists of two interconnected jet propulsion engines: a double-spool turbo-engine and an afterburner, each one of them having its own control and controlled parameters. As control parameters (inputs) for a single-jet double spool engine, only two inputs can be identified: the fuel flow rate Qc (which is the most important control parameter) and the exhaust nozzle opening A5 [4],[13]. The controlled parameters are, obviously, the spools’ speeds ( n1 and n 2 ); meanwhile, the combustor’s temperature T3* is a limited controlled parameter, its limitation being realized through the fuel flow-rate control [13]. The engine has, eventually, a single input parameter, which is the throttle’s position α . The throttle is the unique command that the pilot can use, but it generates, by a complex input mechanism, the two input signals (presetting the reference signals for Qc
Eventually, system’s outputs are: spools’ speeds ( n1 and n 2 ) and afterburning temperature T4*p . The afterburner’s fuel pump is permanently turned round by the engine’s high pressure shaft, but the afterburning operates only by the pilot’s command; so, when the afterburning is switched-off, the fuel pump only re-circulates the fuel, without injection and burning.
and A5 ). Engine’s fuel pump’s rotor is turned round by the high pressure shaft, but the centrifuge speed transducer is turned round by the low pressure shaft; the execution element is an actuator, which realizes H,V (p*1)
Flight regime H,V (p*1)
A5
n2 y(y-ye) Fuel pump
Qc
Qc Low pressure n1 n2 rotor (spool) n1
Qc
_ ye
Σ
T3*controller
y
Actuator (servoamplifier) Feed-back
Double-spool turbo-jet engine
T3*
n1 n2
Afterburning Sub-system
Afterburning Qcp Fuel Pump
Afterburner A5
T3*
T3*
Throttle
Complex Rotation speed n -n u (~n ) x input signal n 1i 1i 1 centrifuge Σ + forming + transducer block uv(~A5i) Fuel flow rate control sub-system n1
x-z
Σ
z
A5
Positioning mechanism
yv
yv yv
Nozzle's flaps positioning sub-system
Actuator x -v v v (servoamplifier)
+ xv
Σ vv
-
Slide valve
Position Feed-back
α
u (~A ) uv-wv + v 5i
Σ
wv
Inner Feed-back
yv
wv(~A5)
Fig. 2. Block diagram of the double-spool single-jet engine with afterburning
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* T4p
n1
A5
+ y
High pressure n2 rotor (spool)
n2
n2
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For the temperature T3* one can obtain the form:
3. System’s mathematical model The studied system’s mathematical model consists of the motion equation for each of its parts, brought to an acceptable form for studying and simulations. A simplified mathematical model, based on the joining of the engine and afterburning simplified models, can be obtained and used for further studies. The authors have used the models presented in [2], [4], [8], [11] and [13], as follows: a) for the double-spool engine
(τ r1s + 1)n1 = k c1 Qc + k1n 2 n2 + k1 A A5 − k p1e p1* , (τ r 2s + 1)n2 = k c 2 Qc + k 2 n1 n1 ,
T3* = k 3c Qc + k 3n1 n1 + k 3n 2 n 2 ,
⎛ n ⎞⎛ ∂T * ⎞ ⎛ n ⎞⎛ ∂T * ⎞ k 3n1 = ⎜⎜ 10* ⎟⎟⎜⎜ 3 ⎟⎟ , k 3n 2 = ⎜⎜ 20* ⎟⎟⎜ 3 ⎟ , (5) ⎟ ⎜ ⎝ T30 ⎠⎝ ∂n 2 ⎠ 0 ⎝ T30 ⎠⎝ ∂n1 ⎠ 0 or, the new form
(1)
T3* =
(2)
Qc 0 ni 0
⎛ ∂M Ti ⎜⎜ ⎝ ∂Qc
⎞ Q ⎟⎟ , k1 A = K R1 c 0 n10 ⎠0
n20 n10
⎡⎛ ∂M T 1 ⎞ ⎛ ∂M C1 ⎞ ⎤ ⎟⎟ ⎥, ⎟⎟ − ⎜⎜ ⎢⎜⎜ ⎣⎢⎝ ∂n2 ⎠ 0 ⎝ ∂n2 ⎠ 0 ⎦⎥
k 2 n1 = K R 2
n10 n20
⎡⎛ ∂M T 2 ⎞ ⎛ ∂M C 2 ⎞ ⎤ ⎟⎟ ⎥ , ⎟⎟ − ⎜⎜ ⎢⎜⎜ ⎢⎣⎝ ∂n1 ⎠ 0 ⎝ ∂n1 ⎠ 0 ⎥⎦
⎛ p * ⎞⎛ Q ⎞ k p1e = k c1 ⎜⎜ 10 ⎟⎟⎜⎜ c* ⎟⎟ , ⎝ Qc 0 ⎠⎝ p1 ⎠ H =0,V =0
(6)
2
m2 = k3cτ r1τ r 2 ,
m0 = k 3c + k 3n1 (k c1 + k c 2 k1n 2 ) + k 3n 2 (k c 2 + k c1k 2 n1 ) .(7) If the flight regime must be taken into account, it shall affect the LPS equation, because it is given by the terms containing the pressure in the front of the compressor p1* . It has a direct influence above the low pressure compressor’s inlet, as well as above the low pressure turbine’s exhaust, which explains why the flight regime affects only the LPS.
( )
b) for the afterburning
(
)
k n (τ n s + 1)n − k p1a p1* = a 2 s 2 + a1s + a0 Qi ,
(8)
where the involved co-efficient are:
(
)
k n = k Q k pn k pi + k iu k ui , τ n = k ypB τ y ,
(3)
(
)(
)
k HV = k Bx k x1 k CQ k py k ypB − k CB k pi + k iu k ui ,
their expressions being explained in [4]. Both speeds have similar expressions, but some important particularities are occurred. Each spool can be assimilated to an independent spool of a single-jet single-spool engine, but operating as a couple, being gas-dynamic bounded. Consequently, analyzing the (1) and (2) forms, an observation can be made, concerning the existence of a mutual influence between the spool’s speeds, accomplished by the coefficient k1n 2 and k 2n1 which appear in (1) and (2). These co-efficient are the mutual co-efficient and have a lot of influence in the engine’s stability (as controlled object); these mutual co-efficient are not constant, but they depends on the flight regime (altitude H and speed V). Another observation is that the exhaust nozzle’s opening A5 influences only the LPS speed, as (1) shows, being the consequence of the burned gas flow rate’s dependence on the above mentioned parameter ( A5 ) .
ISSN: 1790-5117
)
+ m1s + m0 Qc + (r1s + r0 )A5 , τ r1τ r 2 s + (τ r1 + τ r 2 )s + (1 − k1n 2 k 2 n1 ) 2
2
m1 = k 3c (τ r1 + τ r 2 ) + k 3n1k c1τ r 2 + k 3n 2 k c 2τ r1 ,
⎛ ∂M T 1 ⎞ ⎟⎟ , ⎜⎜ ⎝ ∂Qc ⎠ 0
k1n 2 = K R1
(m s
with
where the used annotations are: πJ 1 K Ri = ,τ ri = i K Ri , i = 1,2 , 30 ⎛ ∂M Ci ⎞ ⎛ ∂M Ti ⎞ ⎟⎟ ⎟⎟ − ⎜⎜ ⎜⎜ ⎝ ∂ni ⎠ 0 ⎝ ∂ni ⎠ 0
k ci = K Ri
(4)
⎞ ⎟ , ⎟ ⎠0
⎞⎛ ⎟⎜ ⎟⎜ ∂Q ⎠⎝ c
⎛Q k 3c = ⎜⎜ c*0 ⎝ T30
where
∂T3*
[ ( − k (k
)
a2 = k i k ui k ypB k rc k yrc − k CQ k py k ypC + 1 − ypC
(
CQ k py k ypB
)[
)]
− k CB ,
( )] )] − k k + 1) +
a1 = k rc k yrc − k CQ k py k ypC + 1 k ypB + k i k ui 1 + k ypB −
(k
CQ k py k ypC
)[
(
− k CB k ypC + k i k ui k ypC + k CB
[(
a0 = (1 + k i k ui ) k rc k yrc − k CQ k py k ypC
(
)]
ic
ypB
k CB k CQ k py k ypB − k CB − k ic ,
,
(9)
their expressions being explained in [12] and[14].
[(
)
(
)
T4*p = b2c s 2 + b1c s + b0c Qc − b2 As 2 + b1 As + b0 A A5 +
(
) ][
( )k ],
+ b2 p s 2 + b1 p s + b0 p Q p : k p1eτ r1τ r 2 s 2 + k p1eτ r1 +
τ r 2 k p1q )s + (k p1e − k1n 2 k 2 n1
p1q
(10)
where the co-efficient are the one in [14]. An observation can be made, concerning the presence of the afterburner, which affects the LP turbine’s exhaust, so the Eq. (1) shall be completed with a term containing the afterburner’s fuel flow
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p*1
Flight regime
Double Spool Turbo-jet Engine
kp1e _ kuvα
uv
+ _
Σ
α Throttle
kunα Complex Input Signal Forming Block
A5
ρsv
A5
A5
kuα kes + n1i
Σ
n1
Σ Σ
x-z
+ _
z
1 τsns
y
+
+ +
kc1
Qc Q c +
x
k1A5 Qc
Nozzle's Flaps Positioning Sub-system
un
_
1
τsvs
Σ
k1qp Qp
τr1s+1
+ k2n1
Σ
k1n2
+ kc2
+
n1
1
+
1
Σ
τr2s+1
n1
n2
kp1e
n1
_
n2
n2
kn(τns+1)
+
T3* 1 a2s2+a1s+a0
Σ
n2 Qc
Qp
b2ps2+b1ps+b0p
b2cs2+b1cs+b0c
n2 Qp
kpn2 kpy
A5
ρsn
b2As2+b1As+b0A
_+
+
Σ
c2s2+c1s+c0 Afterburning
Fuel Flow-rate Control Sub-system
Fig. 3. System’s block diagram with transfer functions
rate: − k1qp Q p . The value of the co-efficient k1qp is a
at their new values with static errors, so the system is a static-one. However, the static errors are acceptable, being fewer than 5% for each output parameter. LP shaft’s speed n1 has a static error bigger than the HP shaft’s speed n2 ; both of them are stabilizing after 5…6 s, characteristic for a slow engine. The afterburner’s fuel flow rate Q p has a
small one, being ten times smaller than the other coefficient involved in (1), so, for preliminary studies it can be neglected. Based on the above presented equations, which are the system’s simplified mathematical model, one has built the block diagram with transfer functions, as fig. 3 shows.
similar behavior, but the temperature T4*p has an initial decreasing, because of the initial exhaust nozzle’s opening and of the air flow rate initial growing (due to the speeds’ growing).
4. System’s quality As one can observe in fig. 2 and 3, the system has two effective inputs: a) engine’s throttle’s position; b) aircraft flight regime (given by the inlet inner pressure p1* ). So, the system should operate in case of any changing affecting one or both of the input
When the throttle is immobile, for a step input of p1* (high speed modifying of the flight regime) system’s behavior is similar ( asymptotic stability, see fig. 5), but the static errors are negative and their absolute values are a little lower, being around 4.5 % for n1 and 2% for n2 . Both the afterburner’s fuel flow rate
( )
parameters α , p1* . A study concerning the system quality was realized (using the co-efficient values for a jet engine, R-11F300, presented in [8] and [14]), by analyzing its step response (system’s response for step input for one or for both above-mentioned parameters). As outputs, one has considered the engine’s speeds ( n1 and n 2 ),
Q p and the temperature T4*p have similar behavior, being also asymptotic stables. When both of the input parameters have step variations, the effects are overlapping, so system’s behavior is the one in fig. 6. In this case, some parameters have a smaller or a bigger override, which gives a periodic stability. One can observe that, because of the initial n1 decreasing, both
the afterburner’s temperature T4*p , as well as the afterburning fuel flow-rate and the total thrust F p . Concerning the system’s step response for throttle’s step input (high speed pushed throttle), when the flight regime is constant (see fig.4), one can observe that all the output parameters are stabiles ( n1 , n 2 , T4*p ), so the system is a stabile-one (even an
initial decreasing; however, the temperature T4*p is
asymptotic stabile one, according to the shapes of the curves in fig. 4). All output parameters are stabilizing
more sensitive, so its initial decreasing is bigger (3% static error), but eventually it reaches the 2% level of
ISSN: 1790-5117
parameters
(the
temperature T4*p
158
fuel
flow
rate
Q p and
the
) have similar behavior, with an
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T4p*
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T4p* Qp
n2
n1
0.05
0.06
Qp
0.04
0.05
n1
0.04
0.03 0.02
n2
0.03
T4p*
0.01
t [s]
0.02 0 0.01
-0.01
t [s] 0 0
1
2
3
4
5
6
7
8
9
-0.02
10
0
1
2
3
4
5
6
7
8
9
10
Fig. 4. System step response for constant flight regime and step input of throtlle's displacement * t [s] T4p
n2
n1 0 -0.005
Qp
t [s]
0
-0.005
n2
-0.01
T4p*
-0.01
-0.015 -0.02
-0.015
-0.025
-0.02
-0.03
-0.025
n1
-0.035
Qp
-0.03
-0.04 -0.035
-0.045 -0.05
0
1
2
3
4
5
6
7
8
9
-0.04
10
0
1
2
3
4
5
6
7
8
9
10
Fig. 5. System step response for constant throtlle's position and step input of the flight regime
T4p* Qp
n2
n1
0.06
0.04
n2
0.05
Qp
0.03
0.04
0.02
0.03
0.01
n1
0.02
0
0.01
-0.01
t [s] 0
-0.02
-0.01
-0.03
t [s]
T4p*
-0.04
-0.02 0
1
2
3
4
5
6
7
8
9
0
10
1
2
3
4
5
6
7
8
9
10
Fig. 6. System step response for combined throtlle's step displacement and step input of the flight regime
the static error. Another important output parameter is the total
ISSN: 1790-5117
thrust, which means the combined thrust of both of the studied propulsion systems (double-spool engine
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Area. Proceedings of the 17th International Symposium on Naval and Marine Education, Constanta, May 24-26, 2001, Sect. III, pp. 26-35. [3] Lungu, R. Flight Apparatus Automation. Publisher Universitaria, Craiova, 2000. [4] Lungu, R., Tudosie, A., Dinca, L. Double-Spool Single Jet Engine for Aircraft as Controlled Object. NAUN International Journal of Mathematical Models and Methods in Applied Sciences, Issue 4, Vol. 2, 2008, pp.553-562. [5] Mattingly, J. D. Elements of Gas Turbine Propulsion. McGraw-Hill, New York, 1996. [6] Pimsner, V. Aero-jet Engines, Didactical and Pedagogic Publishing House, Bucharest, 1982. [7] Stevens, B.L., Lewis, E. Aircraft Control and Simulation, John Willey Inc. N. York, 1992. [8] Stoenciu, D. Aircraft Engine Automation. Aircraft Double-spool Single-jet Engines as Controlled Objects. Publisher of Military Technical Academy, Bucharest, 1981. [9] Stoenciu, D. Aircraft Engine Automation. Catalog of Automation Schemes. Publisher of Military Technical Academy, Bucharest, 1986. [10] Stoicescu, M., Rotaru, C. Turbo-jet Engines. Characteristics and Control Methods. Publisher Military Technical Academy, Bucharest, 1999. [11] Tudosie, A., Jet Engine Rotation Speed Hydromechanical Automatic Control System, Proceedings of the Scientific Session “25 Years of High Education in Arad”, Arad, 30-31 october, 1997, section 8, pp. 177-184. [12] Tudosie, A. Mathematical Model for an Afterburning Automatic Control System. Proceedings of the 17th International Symposium on Naval and Marine Education, Constanta, May 24-26, 2001, Sect. III, pp. 270-275. [13] Tudosie, A. N., Two Shaft Single Jet Engine’s Rotation Speed Automatic Control, Annals of the University of Craiova – Electrical Engineering Series, no. 30, Craiova, 2006, pp. 344-349, ISSN 1842-4805. [14] Tudosie, A. Aerospace propulsion systems automation. Inprint of Univ. of Craiova, 2005. [15] Tudosie, A. N., Jet Engine’s Speed Controller with Constant Pressure Chamber, Proceedings of the WSEAS International Conference on Automation and Information ICAI’08, Bucharest, June 26-28 2008, pp. 229-234.
Fp 0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 combined flight regime&throttle step input
0.01
constant throttle, flight regime step input constant flight regime, throttle step input
0.005
t [s]
0 -0.005 -0.010 -0.015
0
1
2
3
4
5
6
7
8
9
10
Fig. 7. Total thrust of the combined propulsion system
and afterburning). Using the formula given by [14] F p = l 2c s 2 + l1c s + l0 c Qc − ls 2 + l1 As + l0 A A5 +
(
[(
)
) ][
(
( )k ],
)
+ l 2 p s 2 + l1 p s + l0 p Q p : k p1eτ r1τ r 2 s 2 + k p1eτ r1 +
τ r 2 k p1q )s + (k p1e − k1n 2 k 2 n1
p1q
(11)
one has obtained the behavior in fig.7. As the figure shows, the total thrust tends to decrease when the flight regime becomes more intense (dashdot curve); meanwhile, it grows when the throttle is pushed up, as consequence of all involved parameters growing (speeds, fuel flow rates and temperatures), see the dash curve. For a combined input, one can observe that the flight regime’s influence is smaller then the throttle’s positioning influence, total thrust’s behavior (continuous curve) being very similar to the throttle’s positioning behavior. The above-described engine’s model can be extended to other multi-spool engines, such as turbofans, or twin-jet engines.
References: [1] Abraham, R. H. Complex Dynamical Systems. Aerial Press, Santa Cruz, California, 1986. [2] Aron, I., Tudosie, A. Automatic Control System for the Jet-engine’s A5 Exhaust Nozzle’s Section
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