Aeronautics and Astronautics Part 9 pdf

Aircraft Gas-Turbine Engine’s Control Based on the Fuel Injection Control 309 The system operates by keeping a constant pressure in chamber 10, equal to the preset value (proportional to the spring 16 pre-compression, set by the adjuster bolt 15). The engine’s necessary fuel flow rate i Q and, consequently, the engine’s speed n, are controlled by the co-relation between the c p pressure’s value and the dosage valve’s variable slot (proportional to the lever’s angular displacement  ). An operational block diagram of the control system is presented in figure 4. TURBO-JET ENGINE FUEL INJECTION DOSAGE VALVE FUEL INJECTION PUM P PRESSURE CONTROL LER ( p c =const.)   n n y Q p Q i p c p c i (z) x  * 1 * 1 ,, TpVH Flight regime THROTTLE p c  Fig. 4. Constant pressure chamber controller’s operational block diagram 3.2 System mathematical model The mathematical model consists of the motion equations for each sub-system, as follows: a. fuel pump flow rate equation (,) pp QQn y  , (5) b. constant pressure chamber equation ipA QQQ   , (6) c. fuel pump actuator equations 2 2 4 A AdA cA d Qpp    , (7)  0 d d dd A As A A s p QQ V S yy tt    , (8)  2 2 ddy d d f sBcAA y mkyySpSp t t    , (9) d. pressure sensor equations 0 2 () snn A Qdzxpp      , (10) 2 12 2 () 4 n mc A e d lS p l p lk z x   , (11) e. dosing valve equation Aeronautics and Astronautics 310 1 2 s ii cCA Qb pp      , (12) f. jet engine’s equation (considering its speed n as controlled parameter)   ** 11 ,, i nnQ p T , (13) where , , , p iAs QQQQ are fuel flow rates, c p -pump’s chamber’s pressure, A p - actuator’s A chamber’s pressure, CA p -combustor’s internal pressure, 0 p -low pressure’s circuit’s pressure, dA  , n  , i  -flow rate co-efficient, , An dd -drossels’ diameters, , AB SS -piston’s surfaces, AB SS , m S -sensor’s elastic membrane’s surface, , fe kk-spring elastic constants, 0A V -actuator’s A chamber’s volume,  -fuel’s compressibility co-efficient,  -viscous friction co-efficient, m-actuator’s mobile ensemble’s mass,  -dosing valve’s lever’s angular displacement (which is proportional to the throttle’s displacement), x-sensor’s lever’s displacement, z-sensor’s spring preset, y-actuator’s rod’s displacement, ** 11 , p T -engine’s inlet’s parameters (total pressure and total temperature). It’s obviously, the above-presented equations are non-linear and, in order to use them for system’s studying, one has to transform them into linear equations. Assuming the small-disturbances hypothesis, one can obtain a linear form of the model; so, assuming that each X parameter can be expressed as   2 0 1! 2! ! n XX X XX n      , (14) (where 0 X is the steady state regime’s X-value and X  -deviation or static error) and neglecting the terms which contains  ,2 r Xr   , applying the finite differences method, one obtains a new form of the equation system, particularly in the neighborhood of a steady state operating regime (method described in Lungu, 2000, Stoenciu, 1986), as follows:   AAc A Qkp p   , (15) ii icc Qk k p     , (16) sSAAs s Qk p kxkz     , (17) ipA QQQ    , (18) 0 dd dd AsA AA QQV p S y tt        , (19)  1 2 emc l kxz S p l     , (20) 2 2 dd 1 d d f cA Ae e k m y ypp Sk kt t        , (21) Aircraft Gas-Turbine Engine’s Control Based on the Fuel Injection Control 311 where the above used annotations are  2 10 00 2 2 , 8 ic A AdA i cA b p d kk pp         ,  00 0 0 2 2 nn A SA A dx z p k p      ,  10 0 0 0 22 , 2 is c nn A ic s c b p d p kk p         . (22) Using, also, the generic annotation 0 X X X   , the above-determined mathematical model can be transformed in a non-dimensional one. After applying the Laplace transformer, one obtains the non-dimensional linearised mathematical model, as follows   s1 s PA A A y cx cz c k py kx kz p   , (23) cx cz zxc c kx kz k p  , (24)   22 0 s2 s1 A yy y AC c A kT T ykpp  , (25) p cc py A kk p k yp   , (26) cQ c Q i kp k Q   . (27) For the complete control system determination, the fuel pump equation (for p Q ) and the jet engine equation for n (Stoicescu & Rotaru, 1999) must be added. One has considered that the engine is a single-jet single-spool one and its fuel pump is spinned by its shaft; therefore, the linearised non-dimensional mathematical model (equations 23÷27) should be completed by pp n py Qknk y , (28)  * 1 s1 MciHV nkQ k p    . (29) For the (23)÷(29) equation system the used co-efficient expressions are   0 00 00 0 00000 ,,,,,,, ASA c Ae ss A AC PA cx cz A y Ay A AAC ccASAAcAc kk pSyky kx kz V kk kk k pkkppkkkpSp         0 0 1 0 20 ,, , , 2 smc i yzxc eye e AA kS p k ml Tkk kTk klkp           0 0 0 00 0 ,,, p Aic ic c i p cQ p QcQ c AAC AA i i Q kk kp k kkkkk kk kp Q Q      . (30) Aeronautics and Astronautics 312 Based on some practical observation, a few supplementary hypotheses could be involved (Abraham, 1986). Thus, the fuel is a non-compressible fluid, so 0   ; the inertial effects are very small, as well as the viscous friction, so the terms containing m and  are becoming null. The fuel flow rate through the actuator A Q is very small, comparative to the combustor’s fuel flow rate i Q , so p i QQ  . Consequently, the new, simplified, mathematical model equations are: - for the pressure sensor: lc z xkp kz, (31) where 00 0 1 0020 0 0 , cc m lz ce pp Sz xl x kk x p xlk x z           , (32) or, considering that the imposed, preset value of c p is 0 0 0 c z ci lc p kz pz k p z       , one obtains   lci c xk pp   ; (33) - for the actuator:   s1 y x y kx  , (34) where   00 0 00 0 2 0 00 4 , 2 Af c f A yAcf s A fnn dAA Skyp ky S pp k y Q Q kdxd yy                  , 0 00 0 0 0 2 0 0 00 4 2 s nnA f x s A nn dAA y Q dp ky xx y k x Q Q dx d yy                    . (35) Simplified mathematical model’s new form becomes  * 1 s1 MciHV nkQ k p    , (36) i pp n py QQ knk y   , (37)   lci c xk pp   ; (38) Aircraft Gas-Turbine Engine’s Control Based on the Fuel Injection Control 313   s1 y x y kx   , (39) 1 ci pp k pQ kk    . (40) One can observe that the system operates by assuring the constant value of c p , the injection fuel flow rate being controlled through the dosage valve positioning, which means directly by the throttle. So, the system’s relevant output is the c p -pressure in chambers 10. For a constant flight regime, altitude and airspeed ( const., const.HV   ), which mean that the air pressure and temperature before the engine’s compressor are constant   ** 11 const., const.pT , the term in equation (36) containing * 1 p becomes null. 3.3 System transfer function Based on the above-presented mathematical model, one has built the block diagram with transfer functions (see figure 5) and one also has obtained a simplified expression:   2 s1 1 s1 rpy rpy y Mc p n y Mc p nc pp kk kk kk kk p kk                          s1 s1 s1 py r y Mcpn Mci pp kk k kk p kk        , (41) where rxl kkk . 1s M  k c  + n py k pn k i Q i Q 1 p k  + y + x x k 1s y  l k  l k z k z ci p _ c p c n p c p + p k  k  _ Fig. 5. System’s block diagram with transfer functions So, one can define two transfer functions: a. with respect to the dosage valve’s lever angular displacement   sH  ; b. with respect to the preset reference pressure ci p , or to the sensor’s spring’s pre- compression z,   s z H . While  angle is permanently variable during the engine’s operation, the reference pressure’s value is established during the engine’s tests, when its setup is made and Aeronautics and Astronautics 314 remains the same until its next repair or overhaul operation, so 0 ci zp   and the transfer function   s z H definition has no sense. Consequently, the only system’s transfer function remains     2 s1 s1 s s1 1 s1 yMcpn p r py r py yM cpn y M cpn pp k kk k H kk kk kk kk kk                      , (42) which characteristic polynomial’s degree is 2. 3.4 System stability One can perform a stability study, using the Routh-Hurwitz criteria, which are easier to apply because of the characteristic polynomial’s form. So, the stability conditions are 0 yM    , (43)  11 0 rpy cpn y M p kk kk k         , (44) 10 rpy cpn p kk kk k  . (45) The first condition (43) is obviously, always realized, because both y  and M  are strictly positive quantities, being time constant of the actuator, respectively of the engine. The (40) and (41) conditions must be discussed. The factor 1 cpn kk is very important, because its value is the one who gives information about the stability of the connection between the fuel pump and the engine’s shaft (Stoicescu&Rotaru, 1999). There are two situation involving it: a. 1 cpn kk  , when the connection between the fuel pump and the engine shaft is a stable controlled object; b. 1 cpn kk  , when the connection fuel pump - engine shaft is an unstable object and it is compulsory to be assisted by a controller. If 1 cpn kk  , the factor 1 cpn kk  is strictly positive, so   10 cpn y kk    . According to their definition formulas (see annotations (35) and (30)), ,, rppy kkk are positive, so 0 rpy p kk k  and 10 rpy M p kk k       , which means that both other stability requests, (44) and (45), are accomplished, that means that the system is a stable one for any situation. Aircraft Gas-Turbine Engine’s Control Based on the Fuel Injection Control 315 If 1 cpn kk  , the factor 1 cpn kk  becomes a negative one. The inequality (44) leads to   1 1 cpn My rpy p kk kk k          , or  1 1 rpy p y M cpn kk k kk          , (46) which offers a criterion for the time constant choice and establishes the boundaries of the stability area (see figure 6.a). Meanwhile, from the inequality (45) one can obtain a condition for the sensor’s elastic membrane surface area’s choice, with respect to the drossels’ geometry ( , An dd) and quality   , nA   , springs’ elastic constants   , ef kk , sensor’s lever arms   12 ,ll and other stability co-efficient  ,, cpnpy kk k   2 00 2 10 22 1 4 nn dAA f cpn e m fpy nA dx d ky kk k l S kl k dp       . (47) Another observation can be made, concerning the character of the stability, periodic or non- periodic. If the characteristic equation’s discriminant is positive (real roots), than the system’s stability is non-periodic type, otherwise (complex roots) the system’s stability is periodic type. Consequently, the non-periodic stability condition is  2 11 4 0 rpy rpy cpn y M yM pp kk kk kk kk                   , (48) which leads to the inequalities     222 2 12 2 prpy p pcpn rpy p y M rpy p rpy kkk kk k k kk kkk kkk            , (49)     222 2 12 2 prpy p pcpn rpy p y M rpy p rpy kkk kk k k kk kkk kkk         , (50) representing two semi-planes, which boundaries are two lines, as figure 6.b shows; the area between the lines is the periodic stability domain, respectively the areas outside are the non- periodic stability domains. Obviously, both time constants must be positive, so the domains are relevant only for the positives sides of y  and M  axis. Both figures (6.a and 6.b) are showing the domains for the pump actuator time constant choice or design, with respect to the jet engine’s time constant. Aeronautics and Astronautics 316 The studied system can be characterized as a 2 nd order controlled object. For its stability, the most important parameters are engine’s and actuator’s time constants; a combination of a small y  and a big M  , as well as vice-versa (until the stability conditions are accomplished), assures the non periodic stability, but comparable values can move the stability into the periodic domain; a very small y  and a very big M  are leading, for sure, to instability. UNSTABLE SY STEM STA B L E SY STEM NON-PERIODIC STA BI L I TY        M pyrppyr ppyrpncp y kkkkkk kkkkkkkkk p pyrp     2 21 2 222       M pyrppyr ppyrpncp y kkkkkk kkkkkkkkk p pyrp     2 21 2 222 NON-PERIODIC ST A B I L I TY PERIODI C STABILITY   M pnc p pyr y kk k kk  1 1             M  M  y  y a) b) Fig. 6. System’s stability domains 3.4 System quality As the transfer function form shows, the system is static one, being affected by static error. One has studied/simulated a controller serving on an engine RD-9 type, from the point of view of the step response, which means the system’s behavior for step input of the dosage valve’s lever’s angle  . System’s time responses, for the fuel injection pressure c p and for the engine’s speed n are   1 1 1 1 c rpy p cpn k p tt kk k kk              , (51)    1 cr pcpnrpy kkk nt t kkkkk     , (52) as shown in figure 7.a). One can observe that the pressure c p has an initial step decreasing, (0) c p k p k   , then an asymptotic increasing; meanwhile, the engine’s speed is continuous asymptotic increasing. One has also performed a simulation for a hypothetic engine, which has such a co-efficient combination that 1 cpn kk  ; even in this case the system is a stable one, but its stability happens to be periodic, as figure 7.b) shows. One can observe that both the pressure and the speed have small overrides (around 2.5% for n and 1.2% for c p ) during their stabilization. Aircraft Gas-Turbine Engine’s Control Based on the Fuel Injection Control 317 The chosen RD-9 controller assures both stability and asymptotic non-periodic behavior for the engine’s speed, but its using for another engine can produce some unexpected effects. 0 0.04 0.08 -0.04 -0.08 -0.12 -0.16 012 3 4 5 p c n n p c t [s] 0 0.05 0.10 -0.05 -0.10 -0.15 -0.20 012 3 4 5 p c n n p c t [s] 6 a) b) Fig. 7. System’s quality (system time response for  -step input) 4. Fuel injection controller with constant differential pressure Another fuel injection control system is the one in figure 8, which assures a constant value of the dosage valve’s differential pressure ci p p  , the fuel flow rate amount i Q being determined by the dosage valve’s opening. As figure 8 shows, a rotation speed control system consists of four main parts: I-fuel pump with plungers (4) and mobile plate (5); II-pump’s actuator with spring (22), piston (23) and rod (6); III-differential pressure sensor with slide valve (17), preset bolt (20) and spring (18); IV-dosage valve, with its slide valve (11), connected to the engine’s throttle through the rocking lever (13). III IV n I 3 4 5 6 7 8 9 10 11 12 0 0  y FUEL TANK i p 0>z>0 0<x<0 c p II 2 1 13 16 15 14 17 19 20 21 23 2425 A B A p B p r Q sB Q sB Q sA Q 22  ea k A Q B Q A Q S T O P i Q i p fuel (to the combustor i njectors) i Q i p p Q B Q c p 0 >  > 0 18  es k Fig. 8. Fuel injection controller with constant differential pressure rci p pp   Aeronautics and Astronautics 318 The system operates by keeping a constant difference of pressure, between the pump’s pressure chamber (9) and the injectors’ pipe (10), equal to the preset value (proportional to the spring (18) pre-compression, set by the adjuster bolt (20)). The engine’s necessary fuel flow rate i Q and, consequently, the engine’s speed n, are controlled by the co-relation between the rci p pp differential pressure’s amount and the dosage valve’s variable slot opening (proportional to the (13) rocking lever’s angular displacement  ). 4.1 Mathematical model and transfer function The non-linear mathematical model consists of the motion equations for each above described sub-system In order to bring it to an operable form, assuming the small perturbations hypothesis, one has to apply the finite difference method, then to bring it to a non-dimensional form and, finally, to apply the Laplace transformer (as described in 3.2). Assuming, also, that the fuel is a non-compressible fluid, the inertial effects are very small, as well as the viscous friction, the terms containing m,  and  are becoming null. Consequently, the simplified mathematical model form shows as follows     s1 p BA px pp kx  , (53) BAAB pp k y  , (54)  1 iz ci p ic p ic k xpp z kk  , (55)  1 ci p Qx Qp p pQkkx k      , (56) ip QQ k    , (57) pp n py Qknk y . (58) The model should be completed by the jet engine as controlled object equation  * 1 s1 MciHV nkQ k p    , (59) where, for a constant flight regime, the term * 1HV k p becomes null. The equations (53) to (59), after eliminating the intermediate arguments ,,, ABc ppp ,, ,, iip p QQ y x , are leading to a unique equation:     s1 s1 s1 s1 Mcpn MAB AB pic Qx p iz pic Qx p px c py c px py kk kk k kk nkzkk kkkk kk               .(60) [...]... C ( 199 9) Turbo-Jet Engines Characteristics and Control Methods, Military Technical Academy Printing House, ISBN 97 3 -98 940-5-4, Bucuresti, Romania Tudosie, A N (20 09) Fuel Injection Controller with Barometric and Air Flow Rate Correctors, Proceedings of the WSEAS International Conference on System Science and Simulation in Engineering (ICOSSSE' 09) , pp 113-118, ISBN 97 8 -96 0-474-131-1, ISSN 1 790 -27 69, ... ISBN-13: 97 8-1-60086-705-7, USA Lungu, R.; Tudosie, A ( 199 7) Single Jet Engine Speed Control System Based on Fuel Flow Rate Control, Proceedings of the XXVIIth International Conference of Technical Military Academy in Bucuresti, pp 74-80, section 4, Bucuresti, Romania, Nov 13-14, 199 7 Lungu, R (2000) Flying Vehicles Automation, Universitaria, ISBN 97 3-8043-11-5, Craiova, Romania 330 Aeronautics and Astronautics. .. short flight time, high speed and altitude would be promising to military aircraft and missiles An air-breathing hypersonic vehicle operates in multiple engine cycles 332 Aeronautics and Astronautics Altitude, m 45,750 90 0 0 4000 5250 6500 2750 127 50 61,000 Ttotal(K) 1500 76,250 0.05 0.10 0.25 0.50 0.70 0 .95 Dynamic Pressure, atm and modes to reach scramjet operating speeds and could be used for the development... 1 790 -27 69, Genova, Italy, October 17- 19, 20 09 12 Plasma-Assisted Ignition and Combustion Andrey Starikovskiy1 and Nickolay Aleksandrov2 2Moscow 1Princeton University Institute of Physics and Technology 1USA 2Russia 1 Introduction The history of application of thermally-equilibrium plasma for combustion control started more than hundred years ago with IC engines and spark ignition systems The same principles... Craiova, Romania 330 Aeronautics and Astronautics Mattingly, J D ( 199 6) Elements of Gas Turbine Propulsion, McGraw Hill, ISBN 1-56347-7 79- 3, New York, USA Moir, I.; Seabridge, A (2008) Aircraft Systems Mechanical, Electrical and Avionics Subsystems Integration, Professional Engineering Publication, ISBN-13: 97 8-1-56347 -95 2-6, USA Stoenciu, D ( 198 6) Aircraft Engine Automation Catalog of Automation Schemes,... the drossels diameter’s choice, correlated to its membrane and its spring elastic properties However, it has a simple shape, consisting of simple and reliable parts and its operating is safe, as long as the drossels and the mobile parts are not damaged Air flow-rate corrector’s using is more spectacular, especially for the unstable engines and/ or for the periodic-stable controller assisted engines;... E/n ~ 0.1 Td for atomic gases and to E/n ~ 1 Td for molecular gases (1 Td = 10-17 V  cm2) Fig 15 Characteristic electron energy He and Ar – [Dutton, 197 5]; H2, N2 and CO2 – [Huxley&Crompton, 197 4] Nonequilibrium electron energy distribution function (EEDF) can be found from a solution of the Boltzmann equation In the simplest case, the EEDF is time- and space-independent and depends only on the local... both of the correctors, the (13)-lever equation results overlapping (73) and ( 79) -equations, which leads to a new form     * * 2 kyR pR  kyp pp  kyH p1  ktf p *  k1t p1  Ty s 2  20Ty s  1 y  y , f (81) 328 Aeronautics and Astronautics which should replace the (65)-equation in the mathematical model (equations (63) to ( 69) ) The new block diagram with transfer functions is depicted in figure... the Mach 3 to 6 regime along the isolator capability limit to avoid inlet unstart and to remain within the structural limits (Figure 2) [Andreadis, 2005] As the vehicle continues to accelerate beyond Mach 7, the combustion process is unable to separate the flow and the engine operates in scramjet mode 334 Aeronautics and Astronautics with a pre-combustion shock-free isolator The inlet shocks propagate... hypersonic microwave ignition (top) and the timeline of operation of experimental setup (bottom) Pressures and temperatures as well as times of discharge and fuel injection are marked in the scheme Waveforms of stagnation temperature and emission from the discharge during combustion of a propane–air mixture for various air–fuel ratios, r (a) r = 0, (b) r = 5 .9, (c) r = 14.4, and (d) air flow without propane . equation (for p Q ) and the jet engine equation for n (Stoicescu & Rotaru, 199 9) must be added. One has considered that the engine is a single-jet single-spool one and its fuel pump is. engine’s tests, when its setup is made and Aeronautics and Astronautics 314 remains the same until its next repair or overhaul operation, so 0 ci zp   and the transfer function   s z H. between the fuel pump and the engine’s shaft (Stoicescu&Rotaru, 199 9). There are two situation involving it: a. 1 cpn kk  , when the connection between the fuel pump and the engine shaft