AIAA 2011-6623
AIAA Modeling and Simulation Technologies Conference
08 - 11 August 2011, Portland, Oregon
A Non-Iterative Aeroengine Model Based on
Volume Effect
Xiangxing Kong 1 , Xi Wang 2 , Daoliang Tan3, Ai He4
Laboratory of Jet Informatics
Department of Aviation Propulsion, Beihang University
Beijing, China, 100191
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In traditional aeroengine modeling, the nonlinear equations of engine model
are generally solved through iterative algorithm. However, due to the strong
nonlinear characteristics of the equations, the iterative model often fails to
converge at some points of the full envelope and has a poor real time
performance. In order to solve the problems, this paper proposes a non-iterative
modeling method based on volume effect. In this method, several variables and
differential equations of volume dynamics in aeroengine are introduced to the
nonlinear equations, as a result, the whole set of equations becomes closed-form
and can be solved without iteration. The non-iterative model of a gas turbine
engine is written in matlab code. Furthermore, an open-loop simulation is
carried out in matlab/simulink, both under groud and altitude condition.
Meanwhile, a comparison is made between the non-iterative model and a
previous iterative model. The results illustrate that the non-iterative model
provides better performance than iterative model both in the stability of
solutions and real time performance.
Nomenclature
N
= rotate speed (r/min)
J
= rotor inertia (kg•m2)
R
= air/gas constant (kj/kg/K)
Cp
= specific heat at constant pressure (kj/kg/K)
Cv
= specific heat at constant volume (kj/kg/K)
K
= adiabatic exponent
π
= pressure ratio
v
= velocity of airflow
h
= air/gas enthalpy (kj/kg)
u
= air/gas internal energy (kj/kg)
V
= volume cubage (m3)
Wf
= main fuel (kg/s)
W
= air/gas flow (kg/s)
Wcol
= cooling air (kg/s)
1
Doctor Graduate Student, Department of Aviation Propulsion, E-mail:kongxx@sjp.buaa.edu.cn.
2
Professor, Department of Aviation Propulsion, E-mail:xwang@buaa.edu.cn.
1
American Institute of Aeronautics and Astronautics
Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
T
= total temperature (K)
P
= total pressure (Pa) or power (j/s)
Ps
= static pressure (Pa)
BPR
= Bypass Ratio
FP
= propulsive force (N)
SFC
= Specific Fuel Consumption (kg/N/h)
PLA
= Power Location Angle (degree)
I. Introduction
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T
o establish an exact numerical aeroengine model is the most essential work for the control system
design and research. Aeroengine model has been extensively applied in engine dynamic
performance research, control plan constituting and control system designing. In traditional aeroengine
modeling, the nonlinear equations of engine model are generally solved through iterative algorithm.
But it requires much computer’s memory to achieve the solutions. Generally, the iterative steps are
restricted less than 5, so that the model can execute in real time. However, the restriction of the
iterative steps results in the loss of calculation accuracy. Moreover, the iterative model often fails to
converge at some points of the full envelope. As a result, we have the necessity for attempting an
approach to establish a non-iterative aeroengine model.
In recent ten years, the non-iterative method plays an important role in aeroengine modeling
domain, since many problems brought by iteration can be eliminated. During this period, an amount of
research has been devoted to achieving a fast non-iterative real time model. For example, Khary I.
Parker and Ten-Heui Guo 5 developed a turbofan engine in a graphical simulation environment in 2003.
Dean K. Frederick 2 described a new method for constructing fast models of jet engines in Simulink.
George Mink and Al Behbahani 8 accomplished a real time engine model for Air Force Research
Laboratory in 2009. As reported, they all obtained good simulation results to validate their models.
However, in their papers there is a lack of detailed engine equations and simulation results obtained in
the full flight envelope.
This paper makes an exhaustive research on non-iterative aeroengine modeling. The purpose of this
paper is to establish a non-iterative model of aeroengine so as to overcome the drawbacks of the
iterative model. Furthermore, different from the simulations of the non-iterative model in most papers,
the engine equations are listed in detail and the simulation is operated not only under the sea-level,
standard-day condition, but also over the full flight envelope. Last but not least, this paper proposes a
unique and original modeling measure to realize the gas flow mix of the bypass and the core, contrast
to the traditional modeling, it makes several improvements in the bypass and the mixer components.
This paper is organized as follows. First, we introduce the modeling method in Section Ⅱ,
including non-iterative principle, volume dynamics and component equations. Then, an open-loop
simulation involving two experiments is operated in Section Ⅲ. The simulation results are analyzed in
this section. Finally, according to the simulation results, three conclusions are concluded in Section Ⅳ.
II. Modeling Method
In this section, we begin with a general description of the non-iterative principle and follow up with
a detailed description of the volume dynamics. In the end, the equations of an aeroengine are listed in
detail.
A. Non-Iterative Principle
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In the traditional modeling method, aerothermodynamic equations are utilized to represent the
components of the engine. Then, the equilibrium relationships of aerothermodynamic and shaft
dynamics unite the equations of components as a whole. The progress of solving the aeroengine model
is to solve the equilibrium equations.
However, during the calculation, several interim variables, such as N H , N L , π F , π C etc. , are
unknown quantities. Consequently, a set of guess values must be given at first and then iteration is
needed to solve the equilibrium equations.
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As a simplification, this paper takes the following nonlinear equations as an example,
⎧ x1 = f1 ( x 2 , a )......(1)
⎪
⎨ x 2 = f 2 ( x1 ).........(2)
⎪ x = f ( x ).........(3)
3
2
⎩ 3
where a is a known quantity; x1、x 2、x3 are unknown quantities. It is not difficult to find that the
equations above need to be solved by iteration. This paper introduces the volume dynamic differential
equations to avoid iteration.
For the equations listed above, assume a differential equation, x& 4 = f 4 ( x 4 ) , can be introduced in,
and x1 in the former equation (2) can be substituted by x 4 . Ultimately, the nonlinear equations
changes to the following form,
⎧ x& 4 = f 4 ( x 4 ).........(4)
⎪ x = f ( x ).........(5)
⎪ 2
2
4
⎨
⎪ x1 = f 1 ( x 2 , a )......(6)
⎪⎩ x3 = f 3 ( x 2 ).........(7)
In substance, the principle of avoiding iteration is to change the equations (1)(2)(3) to equations
(4)(5)(6)(7). To analyze the equations in the viewpoint of engine modeling, functions of f 1 and f 2
represent the aerothermodynamic functions of two adjacent components. While the volume effect3,7 is
not taken into account, the inlet and outlet parameters of the volume between components are
equivalent. The output of the front component acts as input of the next component, which is as equation
(2) expressed; while the volume effect is considered, the inlet and outlet parameters of the volume are
no longer equivalent. The volume dynamic differential equation (4) is introduced to calculate the
volume outlet parameters, which act as the input of the next component, as equation (5) shown.
The differential equation (4) is calculated through Euler scheme3. Then the non-linear equations
changes to be closed-form. The equations can be worked out explicitly in turn so the iteration is not
needed.
B. Volume Dynamics
As part A discussed, the volume dynamics is introduced to avoid iteration. Hence, the volume
effect is necessarily described before deducing the component equations.
An aeroengine consists of numerous small chambers called volumes. A volume is shown in Fig.1.
The volume effect denotes the effect of the volume dynamics. The volume effect indicates that in
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dynamic progress each volume is capable of storing thermal energy and air/gas masses. As a result, the
inlet and outlet parameters of volume are no longer equivalent. Generally, the volume dynamics
involves mass equation, energy equation and momentum equation.
For this case, mass equation and energy equation are considered so as to avoid iteration.
⎛ RT ⎞
⎟ m& (Ref. 7)
⎝ V ⎠0
mass equation: P& = ( ρR ) 0 T& + ⎜
⎛ RT ⎞
⎛ RT2 ⎞
⎟⎟ (minhin − mouthout) − ⎜⎜
& (Ref. 7)
⎟⎟ m
⎝ PV ⎠0
⎝ cv PV ⎠0
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energy equation: T& = ⎜⎜
where P& =
dP
dT
dm
= Pin − Pout ； T& =
= Tin − Tout ； m& =
= min − mout 。
dt
dt
dt
Define H& = min hin − mout hout . By equation of ideal gas state, mass equation and energy equation
can be simplified as follows,
dT
T
= (k − 1) ⋅
⋅ ( H& − um& ) (Ref. 4)
dt
PV
dP P dT RT
= ⋅
+
⋅ m& (Ref. 4)
dt T dt
V
where u represents the internal energy of air/gas. T and P denote the total temperature and pressure
of the volume respectively.
C. Component Equations
The schematic of a turbofan aeroengine is shown as Fig.2. Note that the volume element(marked by
①②③④) is inserted in the engine to allow for the mass and energy storages. Then, the equations of
the components are deduced in turn (The footnotes of the parameters which denote engine sections are
all defined in Fig.2.).
1)Fan
W2 T2
P2
= f FAN (
η F = f FAN (
N1
T2
⎛ N1 ⎞
⎜
⎟ ,π F )
⎜ T ⎟
⎝ 2 ⎠ des
N1
T2
⎛ N1 ⎞
⎜
⎟ ,π F )
⎜ T ⎟
⎝ 2 ⎠ des
f FAN denotes the 2 × 2 map function of the component characteristics. The fan inlet flow rate W2
and efficiency η F are calculated by characteristic interpolation.
Fan pressure ratio,
πF =
P13
P2
Fan exit temperature,
T13 = T2 × (1 + (π F a
(k -1)/k a
- 1)/η F )
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2)Compressor
The compressor inlet flow rate W25 and efficiency η C are calculated by characteristic
interpolation.
W25 T25
P25
= f COMPRESSOR(
T25
N2
η C = f COMPRESSOR (
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N2
T25
⎛ N2 ⎞
⎜
⎟ ,π C )
⎜ T ⎟
⎝ 25 ⎠des
⎛ N2 ⎞
⎜
⎟ ,π C )
⎜ T ⎟
⎝ 25 ⎠ des
T25 = T13
P25 = σ 25 P13
πC =
P3
P25
( k −1) / k a
T3 = T25 × (1 + (π C a
− 1) / η C )
The exit flow rate,
W3 = W25 − W27 col
3)Combustor Chamber
Calculate the inlet and exit flow rate,
W31 = W3 − W3col
W4 = W41 − W41col
Combustor calculation demands high precision, so volume dynamic equations adopted in this
component are not simplified.
Rg ⋅T4 ⎡
hg
hg
dT4
1 ⎤
=
⎢Wf ⋅(ηb ⋅ Hu +hf − ) +W31 ⋅(ha − ) −W4 ⋅(1− )⋅ hg⎥(Ref. 7)
dt P4 ⋅V⋅ DHg ⎣⎢
kg
kg
kg ⎦⎥
T4 = T4 +
dT4
⋅ Δt
dt
dP4 R g ⋅ T4
P dT
=
⋅ (W f + W31 − W4 ) + 4 ⋅ 4 (Ref. 7)
dt
V
T4 dt
P4 = P4 +
dP4
⋅ Δt
dt
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⎡ h (T + 1) hg (T4 − 1) ⎤
where DH = 0.5 × ⎢ g 4
−
⎥；
g
k
k
g
g
⎣⎢
⎦⎥
P31 = P4 / σ CC
σ CC denotes the combustor total pressure coefficient.
P41 = P4
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T31 = T3
P3 = P31
T41 =
W4 C pg T4 + W41col C pa T3
W41C pg
4)High Pressure Turbine
The inlet flow rate W 41 and efficiency η HT are calculated by characteristic interpolation.
W41 T41
P41
η HT = f HT (
π HT =
N2
= f HT (
T41
N2
T41
, π HT )
, π HT )
P41
P43
T43 = T41 × (1 - (1 - π HT
(1- kg)/kg)
× η HT )
W43 = W41
5)Section 43-45 Volume
The volume effect between the two turbines is taken into account. The volume dynamics is
introduced to calculate the pressure and temperature of section 45.
W44 = W43 + W44col
T44 =
W43C pg T43 + W44col C pa T44col
W44 C pg
H& = W44 C pg T44 − W45 C pg T45
m& = W44 − W45
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u 45 = Cvg T45
dT45
T
= ( k g − 1) ⋅ 45 ⋅ ( H& − u 45 m& )
dt
P45V45
T45 = T45 +
dT45
⋅ Δt
dt
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dP45 P45 dT45 R g T45
=
⋅
+
⋅ m&
dt
T45 dt
V45
P45 = P45 +
P45
P44 =
P43 =
dP45
⋅ Δt
dt
σ 45
P44
σ 44
6)Low Pressure Turbine
The inlet flow rate W45 and efficiency η LT are calculated by characteristic interpolation.
W45 T45
P45
= f LT (
η LT = f LT (
π LT =
N1
T45
⎛ N1 ⎞
⎜
⎟ , π LT )
⎜ T ⎟
⎝ 45 ⎠ des
N1
T45
⎛ N1 ⎞
⎜
⎟ , π LT )
⎜ T ⎟
⎝ 45 ⎠ des
P45
P5
T5 = T45 × (1 - (1 - π LT
(1- kg)/kg
) × η LT )
W5 = W45
W51 = W5 + W5col
T51 =
W5 C pg T5 + W5col C pa T27 col
W51C pg
7)Section 6-63 Volume
Section 6-63 is a volume between the low pressure turbine and core nozzle.
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H& = W6 C pg T6 − W63C pg T63
m& = W6 − W63
u63 = Cvg T63
dT63
T
= (k g − 1) ⋅ 63 ⋅ ( H& − u63 m& )
dt
P63V63
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T63 = T63 +
dT63
⋅ Δt
dt
dP63 P63 dT63 Rg T63
=
⋅
+
⋅ m&
dt
T63 dt
V63
P63 = P63 +
P6 =
P5 =
dP63
⋅ Δt
dt
P63
σ 63
P6
σ 51
W6 = W5 + W51col + W6 col
8)Mixer
The exit total pressure,
P65 = σ 65 P63
The exit total temperature,
T65 =
W63C pg T63 + W18C paT18
W65C pg
9)Core Nozzle
The exit total pressure,
P8 = σ e P65
The critical pressure ratio,
kg
⎛ k + 1 ⎞ kg −1
P
⎟
π e ,cr = 8 = ⎜⎜ g
Pcr ⎝ 2 ⎟⎠
Nozzle may work under three working states,
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if
P8
P
P
< π e ,cr , P8 s = PH , π (λ8 ) = 8 s = H ,through π (λ8 ) work out λ8 ,then calculate
PH
P8
P8
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q ( λ8 ) ;
If
P8
= π e ,cr , P8 s = PH , q(λ8 ) = 1 ;
PH
If
P8
> π e ,cr , q(λ8 ) = 1 , P8 s > PH .
PH
W8 = K q'
P8 A8 q (λ8 )
T65
v8 = σ e λ8 a cr = σ e λ8
2k g
kg +1
R g T65
10)Bypass Volume
The volume effect is taken into account in bypass volume.
W13 = W2 − W25 + W13col
W16 = W18
H& = W13 C pa T13 − W16 C pa T16
m& = W13 − W16
u16 = CvaT16
dT16
T
= ( k a − 1) ⋅ 16 ⋅ ( H& − u16 m& )
dt
P16V16
T16 = T16 +
dT16
⋅ Δt
dt
dP16 P16 dT16 Ra T16
=
⋅
+
⋅ m&
dt
T65 dt
V16
P16 = P16 +
P13 =
dP16
⋅ Δt
dt
P16
σ 16
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P18 = P16
11)Bypass Nozzle
Traditionally, in iterative modeling, the static pressures for bypass exit Ps ,16 and low pressure
turbine exit Ps ,5 are assumed equivalence. But, in terms of non-iterative modeling, this can result in
the bypass, low pressure turbine and mixer do not co-operate with each other. As the solution, Ps ,18 is
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assumed the same as the atmosphere pressure PH . Then, the section 16-18 is considered as a bypass
nozzle. The airflow of the bypass duct is ejected through the bypass nozzle and then is mixed together
with the gas flow of the core duct.
P18 = σ e P16
kg
⎛ k + 1 ⎞ kg −1
P
⎟
π e ,cr = 18 = ⎜⎜ g
Pcr ⎝ 2 ⎟⎠
If
P18
P
P
< π e ,cr , P18 s = PH , π (λ18 ) = 18 s = H ,through π (λ18 ) work out λ18 ,then calculate
PH
P18
P18
q(λ18 ) ;
If
P18
= π e ,cr , P18 s = PH , q(λ18 ) = 1 ;
PH
If
P18
> π e ,cr , q(λ18 ) = 1 , P18 s > PH .
PH
W18 = K q'
P18 A18 q(λ18 )
T16
v18 = σ e λ18 a cr = σ e λ18
2k g
kg +1
R g T16
W63 = W8 − W18
12)Power Equations
PFan = W2 C pa (T13 − T2 ) × 10 3
PC = (W3 C paT3 + W27 col C p T27 − W25 C pa T25 ) × 10 3
PHT = (W41C pg T41 − W43 C pg T43 ) × 10 3
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PLT = (W45 C pg T45 − W5 C pg T5 ) × 10 3
PFan , PC , PHT and PLT separately denote the power of fan, compressor, high and low pressure
turbines.
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13)Rotor Dynamic Equations
dN H
π
= (η NH PHT − PC ) /[( ) 2 J H N L ]
dt
30
dN H
NH = NH +
⋅ Δt
dt
dN L
π
= (η NL PLT − PFan ) /[( ) 2 J L N L ]
dt
30
dN L
NL = NL +
⋅ Δt
dt
All these equations constitute the aeroengine model. The differential equations in this part are
calculated through Euler scheme of single step, and other equations can be calculated explicitly. Finally,
the non-iterative model is obtained. In addition, it should be pointed out that the algebraic loops in the
model are eliminated by unit delay block in matlab/simulink.
III. Simulation Result
The aeroengine model is established under matlab/simulink software environment. In this paper, we
are only concerned about the open-loop performance of the model. The open-loop simulation is carried
on both under the sea-level, standard-day condition and over the full flight envelope. The purpose of
the simulation is to validate whether the non-iterative model is of notable advantages in stability of
solution and real time performance. In order to achieve the purpose, two experiments are operated,
including ground experiment and altitude experiment.
A. Ground Experiment
The first open-loop experiment is operated under the sea-level, standard-day condition. The PLA
command is given as Fig.3. Over the entire range of operation, totally eighteen static operating states of
aeroengine are presented in the dynamic progress. And the simulation time in matlab/simulink is 1100
seconds. The simulation is capable of reflecting both the transient and the steady-state performance of
the model. Furthermore, the data from ground test is used to evaluate the accuracy of the non–iterative
model, as well as a previous iterative model is included in this simulation. Finally, a comparison
between the two models is made.
Fig.4 shows the performance curves of N H , P3 , P 6 and T 6 ( N H , P3 , P 6 and T 6 respectively
denote the rotate speed of the high pressure rotor, total pressure of the compressor exit, total pressure
and total temperature of the low pressure turbine exit). The outputs are divided by the parameters of
design point so as to eliminate the dimensions. The curves show that the solutions of iterative model
bear great instability at some points in the low power rating state. Meanwhile, the solutions of noniterative model are of high stability. The errors of the solutions are plot in Fig.5. As Table 1 described,
on the whole, the average errors of non-iterative model are a bit higher, except N H . And none of them
exceeds 5%. However, the maximum and minimum errors of the iterative model are evidently much
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higher(at some points, it exceeds 100%), which illuminates that the solutions of the non-iterative model
are of better stability. What’s more, the calculating time of the model is sharply shortened due to noniteration, from 2473.1 seconds to 175.6 seconds. To sum up, in ground experiment, the non-iterative
model has good calculating accuracy; And the non-iterative model is of better performance in terms of
computational stability and efficiency.
B. Altitude Experiment
In this part, the second open-loop experiment is operated over the full flight envelope so as to
evaluate the non-iterative model’s performance at altitude. For achieving our aim, the PLA command
remains the same as the ground experiment, and the flight trajectory is shown in Fig.6. (The iterative
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model fails to converge at some points of the full envelope. So, it is not included in this experiment.)
Then, the non-iterative model is validated by the following approach. First, convert the test data
from ground test to altitude using similarity theory3, then make a comparison between the test data and
the calculation results.
Fig.7 shows the simulation results of N H 、 P3 、 P 6 、 T 6 . The outputs are divided by the
parameters of design point so as to eliminate the dimensions. The curves generated by non-iterative
model appear good consistent with the test data. Fig.8 implies the errors between the calculation results
and the test data. The simulation results are summarized in Table 2.
As table 2 shown, the average errors of the non-iterative model calculation results are no more than
5%. And, the calculating time of the model is 182.4 seconds. It indicates that the non-iterative model is
also of favorable accuracy and real time performance in the full envelope.
IV. Conclusion
According to the open-loop simulation results, the following conclusions can be concluded,
(1) The component equations of aeroengine deduced in this paper are of high validity and
applicability;
(2) The non-iterative model provides better performance than iterative model both in the stability
of solutions and real time performance;
(3) The non-iterative model has a high accuracy in the full flight envelope.
References
1
Hua Qing. A Realtime Simulation Model of a Typical Turbofan Engine in Full Flight Envelope. Aeroengine,
2002,01（12）:47~52.
2
Dean K. Frederick. A New Method for Constructing Fast Models of Jet Engines in Simulink. 45th
AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit August 2009, AIAA 2009-5419.
3
Luo Guangqi,Sang Zengchan. Numerical Methods for Aviation Gas Turbine Engine Simulation. National
Defence Industry Press. 2007.4.
4
Tang Shijian,Tong Wanjun. Gear Driving Turbofan Engine Dynamic Modeling Study based on Volume Method.
Jet Propulsion Control conference. 2010.11.
5
Khary I. Parker and Ten-Heui Guo. Development of a Turbofan Engine Simulation in a Graphical Simulation
Environment. NASA/TM—2003-212543.
6
Sonny Martin, Iain Wallace and Declan G. Bates. Development and Validation of an Aero-engine Simulation
Model for advanced controller design. 2008 American Control Conference . June 11-13, 2008.
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American Institute of Aeronautics and Astronautics
7
Gurevich, Gore Bigger（Russia）. Turbofan Aero-Engine control Method and Transient characteristicSimulation.
China Aviation Motor Control System Institute. 1998.
8
George Mink. The AFRL ICF Generic Gas Turbine Engine Model. 41st AIAA/ASME/SAE/ASEE Joint
Propulsion Conference & Exhibit . 10-13 July 2005.
9
Jeffrey S. Dalton, Al Behbahani. An Integrated Approach to Conversion, Verification,Validation and Integrity of
AFRL Generic Engine Model and Simulation. 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference &
Downloaded by BEIJING INSTITUTE OF TECHNOLOGY on October 8, 2022 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6623
Exhibit . 9-12 July 2006.
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APPENDIX: Figures and Tables
Figure 1. An air/gas volume
Figure 2. Schematic of a Turbofan Aeroengine
Figure 3. PLA Command
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Figure 4. Aeroenginge Simulation Results of Ground Experiment
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Downloaded by BEIJING INSTITUTE OF TECHNOLOGY on October 8, 2022 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6623
Figure 5. Aeroenginge Simulation Errors of Ground Experiment
Figure 6. Flight Trajectory of Altitude Experiment
Figure 7. Aeroenginge Simulation Results of Altitude Experiment
Figure 8. Aeroenginge Simulation Errors of Altitude Experiment
16
American Institute of Aeronautics and Astronautics
Figure 9. Turbofan Engine model in Matlab/Simulink
17
American Institute of Aeronautics and Astronautics
Downloaded by BEIJING INSTITUTE OF TECHNOLOGY on October 8, 2022 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6623
Model
Type
Non-Iterative
Model
Downloaded by BEIJING INSTITUTE OF TECHNOLOGY on October 8, 2022 | http://arc.aiaa.org | DOI: 10.2514/6.2011-6623
Iterative
Model
Model
Type
Non-Iterative
Model
Table 1. Summary of the Ground Experiment Simulation Results
Calculating
NH Error
P3 Error
P6 Error
T6 Error
Time
（%）
（%）
（%）
（%）
(second)
Min
-1.91
Min -13.61 Min -37.47 Min -22.07
175.6
Max
8.47
Max 14.11 Max
17.5
Max
4.42
Ave
0.80
Ave
-3.68
Ave
-1.60
Ave
-4.91
Min
-7.31
Min -127.1 Min -146.9 Min -40.05
2473.1
Max
7.48
Max 54.78 Max 76.89 Max
7.93
Ave
2.45
Ave
1.23
Ave
1.44
Ave
-2.73
Table 2. Summary of the Altitude Experiment Simulation Results
Calculating
NH Error
P3 Error
P6 Error
T6 Error
Time
（%）
（%）
（%）
（%）
(second)
Min
-1.74
Min -21.43 Min -33.63 Min -19.21
182.4
Max
7.48
Max 11.58 Max 16.54 Max
4.41
Ave
0.51
Ave
-4.67
Ave
-1.24
Ave
-4.54
18
American Institute of Aeronautics and Astronautics