A mathematical model of a small class turboprop engine Текст научной статьи по специальности «Математика»

УДК

JOZEF POLACEK, LUBOS VARGOVCIK

Unis a.s, Czech Republic

A MATHEMATICAL MODEL OF A SMALL CLASS TURBOPROP

ENGINE

Results and process of a development of a mathematical model of a small turboprop engine are presented in this article. The presented model is based on engine characteristics and other necessary input data which came from the EUproject ESPOSA of an engine. This engine is currently under the development and testing by present European manufacturer. The model is derived as generic, dynamic and nonlinear\ There is chosen a conception, where final set of governing equations is based on a connection of submodels governing equations of elementary engine parts, i.e. compressor-, turbine, nozzle. The lumped parameters method is applied to these submodels where their properties are described with characteristics, which are standard in the aircraft industry. 1st and 2nd derived mathematical models are comprised of a dynamic part which respects the rotating mass dynamics only. Result of this assumption is a system of equations, both differential and nondifferential. Dynamic part of 3th derived model respect also gas temperature and pressure dynamics. Its system of equations is fully differential.

Key words: mathematical model, small turboprop engine, ESPOSA.

1. Fundamentals of turboprop engine modeling philosophy

Problems in the development of mathematical models for turboprop engines are of high interest in today's control industry. There are several methods how to build a mathematical model of the arbitrary gas turbine engine in order to its control system synthesis, see [1], [2].

There were some initially failed trials of this activity at our company in history, aiming how to reach a usable result in relatively short time. Finally, an industrial standard of mathematical model design which is based on connection of submodels governing equations of basic engine parts, i.e. compressor, turbine, nozzle was accepted. The lumped parameters method is applied on these submodels where their properties are described by well-known kind of characteristics.

1.1 Model requirements

The main purpose of engine model creation is its application in the process of engine control software design and testing. Therefore, the basic set of requirements which must be met by the mathematical model of an engine is given:

— determinism — the results are dependent only on the engine inputs and have to be-same in each case where the inputs are same;

— continuity in time — its desirable discretization is one of next steps;

— two-parts architecture: static model (so called engine deck) and dynamic model;

© Jozef Polacek, Lubos Vargovcik, 2015

ISSN 1727-0219 Вестник двигателестроения №2/2015 - 73 -

— interactivity — responds to an impulse from an engine environment;

— must run in real time;

1.2 Model assumptions

Physical nature of any industrial power plant should be correctly described by its distributed parameters. This can be realized via system of partial differential equations. This task is solved by various software based on finite volume methods (Ansys—Fluent, Comsol—Multyphysics or others) in the common industrial level. This approach is excessive and impractical in order to control algorithm design. Anyway, in order to finding of acceptable equilibrium between an accuracy and a complexity, there must be accepted following assumptions and restrictions:

— The engine = distributed physical system. Its parameters are continuously distributed in a space and time ( Y=f(x,y,z,t) ). Accepted assumption is a discretization of chosen control volume for particular components described by lumped parameters, see fig. 1.

— Simplification of physical properties. Thermal properties of air (dry air) and products of combustion of air with hydrocarbon fuels are described only as a function of temperature for the sake of simplicity.

— 1st and 2nd generation of his mathematical model considers only the dynamics of rotating masses. There are neglected temperature and pressure dynamics. These dynamics are considered in a 3th model generation.

— Effect of a heat accumulation inside the metal components of the engine is not considered.

— A very simple model of combustion chamber is applied. It is set as a constant of a pressure drop and combustor effectivity

— Characteristics of the engine components are described in a form of a pressure drop and adiabatic effectivity as a functions of reduced variables. The well-known theory of similarity is applied.

— Only the adiabatic processes are considered.

1.3 Model composition

Process of the model component architecture is composed of the following steps. The first step is a decomposition of the whole power plant into the main groups and components in next step, see fig. 1. Model of the engine is assembled from components which are ordered along the engine gas path, see fig. 2

Fig. 1. Scheme of typical engine components modeled as discrete control volumes

(3) (4) (5)

Inlet system Rotor of the Dtfusser of the compresser compressor

(12) Transient etiaonel

(13)

Exhaust nozzle

Fig. 2. Scheme of the engine gas path components

CAY è flèarHOCTMKa

These components are joined together in the sense of serial/parallel block connections which are subsequently merged into a static and dynamic model, see fig. 3.

2. Derivation of the engine model

Borders between components in the gas path are numbered according to the aerospace standard SAE ARP755C, see fig. 4.

Fig.3. Scheme of the model with static and dynamic parts

Fig. 4. Station numbering of the engine

2.1 Static model

Governing equations for temperatures, pressures and mass flows were assembled and adjusted in these numbered stations. The common laws of energy and mass conservation and various semi-empirical estimations of dimensionless values of efficiencies, pressures and speed ratios of mass flow parameters were applied.

2.2 Dynamic model

A dynamic model were assembled with the help of the static model. The governing equations were rewritten from equations of the static model, see two examples below.

1. Dynamic equilibrium of gas generator rotating mass, [rpm/s]:

dnm AP„„

dx

n J

f V n

2. Dynamic equilibrium of free turbine rotating mass, [rpm/s]:

dn

prop

dx "

AP

ipt

nprop (Jlptksiprop + J prop )

fn}1 v30,

2.3 Component characteristics

Chosen block values are incorporated in the engine model. This characteristics deals with maps of pressure ratios and efficiency which are dependent on a speed ratio and mass flows parameters. These characteristics are used in calibrated forms, in the 2nd and 3rd model, see fig. 5.

3. Results of calculation of the engine model

Calculation of important values (pressures, temperatures, ...) are realized only in the numbered stations, see fig. 4.

Fig. 5. Example of a calibrated compressor characteristic

3.1 Calculation of the steady state series

Calculations of above-mentioned values in the steady state are processed via the static model, so called engine deck. Examples of pressure and temperature gas path profiles are depicted in fig. 6. These values were calculated by the 1st engine

model. The 2nd engine model is more precise. It enables to calculate similar set of steady state values - examples of static characteristics as a function of steady state fuel flow and Mach number, see fig. 7. Tab.1 shows calculated steady-state series of chosen values in the points of equilibrium.

0.5 1 1.5 2 25 3 3.5 4 4.5 5 5.5 6 65 7 75 9

Fig. 6. Engine pressure and temperature profiles

Tab. 1

Example of a rig test at referential altitude 4000 m (ISA). The steady-state series of chosen values are

calculated as a function of steady-state fuel flow

CASE

<D

• •• H.e 4000 (m) *** delt.Te 0 <K> *** MO.e O(l) *** • •• ***

mf.F mf.2 beta.C n.GG n.LPT n.PP T.3 T.4 T.41 T.42 T.44 T.5 P.3 p.42 p.5 fi.PP

(kg/s) (kg/s) (1) (rpm) (rpm) (rpm) (K) <K> (K) (K) <K) (K) (Pa) (Pa) (Pa) (rad)

1 50 1,070989 0,8747896 35064,64 30829,04 2019,5899 441,45404 972,68538 947,98481 782,42996 775,28734 706,93319 278033,32 104435,89 69549,805 0.1745329

2 50,4375 1,0776499 0,8748213 35167,479 31029,826 2032,7432 442,44971 974,81891 950,06746 783,66451 776,51291 707,35987 280087,38 104880,07 69562,623 0,1745329

3 50,875 1,0843967 0,8748293 35270,925 31240,338 2046,5338 443,44976 976,89891 952,09909 784,84248 777,68294 707,66911 282158,49 105319,8 69574,907 0,1745329

4 51,3125 1,0910916 0,8748595 35373,323 31450,274 2060,2865 444,43802 978,97646 954,12787 786,02927 778,86147 707,97991 284218,95 105758,84 69586,584 0,1745329

5 51,75 1,0974817 0,8749942 35471,576 31639,868 2072,7067 445,3855 981,14403 956,24095 787,34308 780,1643 708,51758 286217,73 106204,05 69597,546 0,1745329

6 52,1875 1.103552S 0,875235 35565,616 31807,077 2083,6605 446,29123 983,40605 958,44254 788,78917 781,59658 709,30176 288151,82 106656.17 69607,826 0,1745329

7 52,625 1,1093987 0,8755487 35656,732 31958,369 2093,5715 447,16668 985,72751 960,69965 790,32233 783,11403 710,2548 290040,17 107113,04 69617,496 0,1745329

8 53,0625 1,1151253 0,8759006 35746,375 32101,46 2102,9453 448,02437 988,07021 962,97631 791,89321 784,6683 711,28668 291903,93 107572,11 69626,617 0,1745329

9 53,5 1,1207841 0,8762746 35835,366 32240,241 2112,0368 448,87066 990,41574 965,25517 793,47768 786,23576 712,35253 293753,56 108032,17 69635,219 0,1745329

10 53,9375 1,1264043 0,8766628 35924,244 32376,891 2120,9886 449,70929 992,75402 967,52677 795,0623 787,80324 713,42723 295594,96 108492,57 69643,318 0,1745329

11 54,375 1,1320069 0,8770597 36013,454 32512,917 2129,8996 450,54299 995,07799 969,78447 796,63752 789,36139 714,49309 297432,42 108952,92 69650,924 0.1745329

12 54,8125 1,1376062 0,8774626 36103,375 32649,295 2138,8336 451,37372 997,3B307 972,02399 798,19695 790,90397 715,53846 299268,87 109413,01 69658,043 0,1745329

13 55,25 1,1432044 0,8778725 36194,252 32785,989 2147,7884 452,20212 999,6691 974,24519 799,73983 792,4302 716,56291 301104,8 109872,96 69664,68 0,1745329

14 55,6875 1,1495797 0,8780559 36296,412 32819,75 2150 453,11407 1001,659 976,18678 800,91243 793,59357 716,852 303098,1 110314,5 69671,019 0,1767501

15 56,125 1,1561732 0,8781816 36401,782 32819,75 2150 454,0461 1003,5535 978,03807 801,97227 794,64636 716,92616 305135,51 110751,39 69676,892 0,1796699

IS 56,5625 1,1627308 0,8783181 36505,755 32819,75 2150 454,9684 1005,4423 979,88345 803,03574 795,7026 717,02028 307165,23 111189,42 69682,237 0.1825615

17 57 1,1692559 0,8784564 36607,223 32819,75 2150 455,8798 1007,3228 981,72044 804,10159 796,76107 717,13007 309187,73 111627,98 69687,055 0,185424

18 57,4375 1,175746 0,8785909 36705,247 32819,75 2150 456,77878 1009,1949 983,54881 805,17114 797,82307 717,25446 311202,3 112066,4 69691,347 0,1882547

19 57,875 1,1821895 0,8787211 36799,072 32819,75 2150 457,66326 1011,062 985,37177 806,24982 798,89391 717,39876 313206,4 112504,29 69695,111 0,1910458

20 58,3125 1,1885778 0,8788524 36888,634 32819,75 2150 458,53229 1012,927 987,19206 807,34133 799,97724 717,56701 315198,22 112941,59 69698,35 0,1937924

21 58,75 1,1949113 0,8789997 36974,799 32819,75 2150 459,38676 1014,7904 989,01023 808,44522 801,0726 717,75903 317177,91 113378,62 69701,069 0.1964966

22 59,1875 1,2011901 0,8791759 37058,318 32819,75 2150 460,2275 1016,653 990,82693 809,56123 802,17971 717,97478 319145,57 113815,74 69703,274 0,1991609

23 59,625 1,2074129 0,8793917 37139,753 32819,75 2150 461,05511 1018,5157 992,64315 810,68971 803,29892 718,21492 321100,98 114253,26 69704,97 0,2017875

24 60,0625 1,2135773 0,8796563 37219,545 32819,75 2150 461,86999 1020,3797 994,46018 811,83151 804,43106 718,48062 323043,72 114691,47 69706,162 0,204378

25 60,5 1,2196802 0,8799782 37298,048 32819,75 2150 462,67245 1022,2468 996,27957 812,98789 805,57738 718,77352 324973,17 115130,66 69706,855 0,2069333

26 60,9375 1,2257175 0,880365 37375,558 32819,75 2150 463,46269 1024,1188 998,10304 814,16039 806,73936 719,09561 326888,58 115571,12 69707,055 0,2094539

27 61,375 1,2316879 0,8808143 37452,289 32819,75 2150 464,24142 1025,9967 999,93173 815,34948 807,91749 719,44737 328789,71 116012,79 69706,769 0,2119406

28 61,8125 1,2375863 0,8812973 37528,182 32819,75 2150 465,0098 1027,8836 1001,7686 816,55719 809,11377 719,8304 330675,71 116454,96 69706,004 0,2143947

29 62,25 1,2434093 O,881783 37603,175 32819,75 2150 465,76902 1029,782 1003,616 817,78488 810,32955 720,24547 332545,99 116896,86 69704,767 0,2168177

30 62,6875 1,2491567 0,882243 37677,271 32819,75 2150 466,52033 1031,6928 1005,475 819,03249 811,56481 720,69184 334400,63 117337,88 69703,067 0,2192116

31 63,125 1,2548306 0,8826517 37750,514 32819,75 2150 467,26501 1033,6162 1007,3457 820,29896 812,8185 721,16766 336240,18 117777,45 69700,91 0,221579

32 63,5625 1,2604352 0,8829854 37822,985 32819,75 2150 468,00435 1035,5515 1009,2275 821,58244 814,08885 721,67022 338065,51 118215,06 69698,304 0,2239229

33 64 1,2659756 0,8832219 37894,783 32819,75 2150 468,73964 1037,4975 1011,1194 822,88063 815,3736 722,1963 339877,77 118650,26 69695,254 0,2262466

34 64,4375 1,2714578 0,8833397 37966,021 32819,75 2150 469,47213 1039,4527 1013,02 824,19094 816,67022 722,74234 341678,24 119082,61 69691,766 0,2285535

35 64,875 1,2768886 0,8833296 38036,813 32819,75 2150 470,20261 1041,415 1014,9273 825,51048 817,97588 723,3047 343468,27 119511,96 69687,842 0,2308469

36 65,3125 1,2822732 0,883213 38107,234 32819,75 2150 470,93048 1043,3817 1016,8386 826,83714 819,2885 723,88121 345248,79 119938,92 69683,485 0,2331283

37 65,75 1,2876146 0,8830153 38177,322 32819,75 2150 471,65485 1045,3508 1018,7522 828,16965 820,60682 724,47078 347020,27 120364,18 69678,697 0,2353983

38 66,1875 1,2929148 0,8827613 38247,095 32819,75 2150 472,37483 1047,3209 1020,6664 829,5073 821,93012 725,0729 348782,96 120788,41 69673,482 0,2376576

39 66,625 1,2981744 0,8824753 38316,548 32819,75 2150 473,0895 1049,2908 1022,5802 830,84987 823,25818 725,68764 350536,88 121212,23 69667,841 0,2399059

AO 67,0625 1,3033928 0,8821818 38385,657 32819,75 2150 473,79792 1051,2601 1024,4932 832,19768 824,5913 726,31565 352281,78 121636,23 69661,779 0,2421412

41 67,5 1,3085682 0,881905 38454,379 32819,75 2150 474,49909 1053,2287 1026,4052 833,5516 825,9303 726,95821 354017,17 122061 69655,298 0,2443589

42 67,9375 1,3136973 0,8816694 38522,65 32819,75 2150 475,19197 1055,1971 1028,3167 834,91301 B27,27655 727,61726 355742,28 122487,12 69648,404 0,2465544

43 68,375 1,3187749 0,8814965 38590,381 32819,75 2150 475,87554 1057,1668 1030,2289 836,2842 828,63227 728,29568 357455,95 122915,07 69641,105 0,2487227

Fig. 7. Values of engine gas generator speed (nGG), compressor air mass flow (mf2), -propeller speed (nPP) and propeller blade angle (fiPP) as the function of steady-state fuel flow and Mach number

3.2 Calculation of the transient state series

Transient states of important values were calculated for many cases of input values combinations. These calculations were processed using MathCAD software as well as MATLAB/Simulink, see fig. 8.

The MathCAD calculations are more convenient for the algorithm debugging. The MATLAB/Simu-link model is advantageous for large and real-time data processing.

Fig. 8. Example of time series of compressor air mass flow (mf2) and gas generator speed (nGG) as a reaction to fuel mass

There was made an attempt to model development in a GSP software (Gas turbine Simulation Program) in order to get some kind of spectacular verification. An approximate example of this cal-

culation is placed in fig. 9. This attempt for verification failed due to unclarity of input and other parameters setting.

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I Project ? » □ fl x IN №. X Hi a id * * 1 7T: H * ► ' pnletKi 3 6 w

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Q| Save Copy ££ Printer Tools C Refresh ? Help Z None - 22 None

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10.9 10 B 10 7 Re(Wodcl : Caws 1 G SPT«sl1 jet603FF .inxl GSP 11 12:09 Ulibepi It, 2014

105 -

10 3 102 10 1 -

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1 — 1620 —

t -

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=

M =

g 31 =

00 0.5 1.0 IS 2.0 2.6 3.0 3.5 4.0 4.5 5.0 Time [4]

Steady-state table | Sieoc^etale graph Transient table Transient grap h Report Edita r

Changed Input thdnged

Fig. 9. Jet engine example of a GSP results sheet. There is calculated time series of compressor pressure ratio (PRC), temperature after combustor (T4) and compressor mass flow (W2)

4. Possibilities for the next development

Development of the 3rd revision of the engine model successfully started in the first quarter of 2015. This approach was chosen in response to the problems with an implementation of previous two models into the dSPACE hardware-in-the-loop simulation platform. The 3rd generation of model is based on fully differential set of governing equations which is the main difference in comparison with the previous model generations. Previous models were constituted of mixed set of both differential and nondifferential governing equations. This mixed system contained algebraic loops as a source of problems. The major expectations for the 3rd model generation are connected with a significant shortening of calculation time in order to fulfilling of real-time requirements. Unfortunately, the lower accuracy in calculated results is expected as a

penalty of additional simplifications. This drawback has been accepted in order to fast processing of the 3rd engine model algorithm.

Development of the 3rd engine model algorithm is being finalized nowadays in MathCAD software. In order to a future progress, there is a need to start the transition and testing of the model in MATLAB/Simulink.

Literature

1. G. G. Kulikov — H. A. Thompson : Dynamic modelling of gas turbines - Identification, simulation, condition monitoring and optimal control. Springer Verlag London Limited, 2005.

2. L. C. Jaw - J. D. Mattingly : Aircraft engine controls, Design, system analysis, and Health Monitoring. American institute of Aeronautics and Astronautics, Inc. 2009.

Поступила в редакцию 16.06.2015

Jozef Polacek, Lubos Vargovcik. Математическая модель турбовинтового малоразмерного двигателя

В данной статье представлены результаты и описывается процесс разработки математической модели турбовинтового малоразмерного двигателя. Представленная модель основана на характеристиках двигателя и других необходимых входных данных, полученных по проекту Европейского Союза ESPOSA. Данный двигатель в настоящее время находится на стадии разработки и испытаний европейским изготовителем. Полученная модель является типовой, динамической и нелинейной. Используется концепция, согласно которой окончательный набор определяющих уравнений основан на связи определяющих уравнений субмоделей элементарных узлов двигателя, например, компрессора, турбины, соплового аппарата. Для этих субмоделей используется метод сосредоточенных параметров, где их свойства описываются характеристиками, которые являются стандартными в авиации. Первая и вторая производные математические модели состоят из динамической части, которая распространяется только на динамику вращающихся масс. Результатом этого ограничения является система как дифференциальных, так и недифференциальных уравнений. Динамическая часть третьей производной модели также распространяется на динамику температуры и давления. Ее система уравнений является полностью дифференциальной.

Ключевые слова: математическая модель, малые турбовинтовые двигатели, ESPOSA

Jozef Polacek, Lubos Vargovcik. Математична модель турбогвинтового малорозм!рного двигуна

Уданш статт1 представлено результати i описуеться процесрозробки математично1 модели турбогвинтового малорозмiрного двигуна. Представлена модель заснована на характеристиках двигуна i тших необхiдних вхiдних даних, отриманих за проектом €вропейського Союзу ESPOSA. Даний двигун у цей час перебувае в стади розробки i випробувань европейським виготовлювачем. Отримана модель е типовою, динамiчною i нелшйною. Використовуеться концепщя, вiдповiдно до яко1 остаточний набiр виз-начальних рiвнянь засновано на зв'язку визначальних рiвнянь субмоделей елементарних вузлiв двигуна, наприклад, компресора, турбти, соплового апарата. Для цих субмоделей використовуеться метод зосереджених параметрiв, де ¿х властивостi описуються характеристиками, ят е стандартними в авiацii. Перша i друга похiдноi математично1 модели складаються з динамiчноi частини, що поширюеться тыьки на динамшу обертових мас. Результатом цього обмеження е система як диференщальних, так i недиференщальних рiвнянь. Динамiчна частина третьоi похiдноi моделi також поширюеться на динамку температури i тиску. Ii система рiвнянь е повшстю диференщальною.

Ключов1 слова: математична модель, малорозмiрний турбогвинтовий двигун, ESPOSA.

ISSN 1727-0219 Вестник двигателестроения №2/2015

- 79 -