AiMT
Advances in Military Technology
Vol. 6, No. 1, June 2011
Gas Turbine Engine Off-Design Calculations
Using Matlab
J. Pečinka1*, M. Poledno1
Department of Airspace and Rocket Technologies, University of Defence, Brno, Czech Republic
The manuscript was was received on 6 October 2010 and was accepted after revision for
publication on 15 February 2011
Abstract:
It is a common practise not only in turbomachinery to use lookup tables in engine
models to describe some characteristics. These characteristics are often available in
graphical form of a diagram with multiple plots which were formerly used in
calculations for manual reading. Using an example of centrifugal compressor map
calculation, this paper shows how a diagram with multiple plots can be transformed into
a lookup table that is used in automatic calculation.
Keywords:
Centrifugal compressor characteristic, gas turbine engine, off-design performance,
lookup table, structured variable
1. Introduction
During the design and life cycle of a gas turbine engine, several types of calculating
models are used. At first a detailed nonlinear static model is built to investigate engine
characteristics in all possible working conditions. Later a nonlinear dynamic model for
engine transient and controller performance analysis is developed from the static
model. For controller design and testing, simplified piecewise linear dynamic models
and real time dynamic models are used [1]. The models mentioned above use, in
addition to algebraic and differential equations, also graphs or tables to some extent.
Graphs are used to describe properties of constituent components of the engine or the
gas turbine engine as a whole. They easily and intelligibly describe even complicated,
nonlinear and multivariable dependences. Formerly used for manual reading,
nowadays characteristics are implemented into the calculation in the form of lookup
*
Corresponding author: Department of Airspace and Rocket Technologies, University of
Defence, Kounicova 65, CZ-662 10 Brno, Czech Republic, phone: +420 973 445 281, E-mail:
jiri.pecinka@unob.cz
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J. Pečinka, M. Poledno
tables which enable to cover nonlinearities, multiple variable dependences and
concurrently keeping the computation time low.
This paper describes how common computation methods can be automated in
Matlab including transformation of multiple plot diagrams into lookup tables.
An example of a calculation technique of a new centrifugal compressor
characteristic is described in Chapter 2. This method includes plenty of manual
readings from diagrams and several iteration steps as well and thus it is rather
laborious. In Chapter 3 it is shown how diagrams with multiple parametric plots can be
used in such a calculation, digitalized and transformed to multidimensional structured
matrix variables that are utilized in an automatic calculation of a new compressor map.
The definition of structured variables and automatic calculation was performed in a
Matlab environment. Chapter 4 describes further possibilities of automated
computations using multidimensional structured matrix variables in gas turbine engine
(GTE) off-design considerations.
2. Centrifugal Compressor Characteristics Calculation
Characteristics of the main engine components (i.e. inlet, compressor, combustion
chamber, turbine end exhaust nozzle) are necessary for execution of the off-design
performance calculations. However, these components do not exist in the early stage
of the design process and hence their real maps are not accessible. The plain
characteristics, e.g. of the exhaust nozzle, could be calculated relatively simply, from
the main dimensions which are defined by the design cycle of the gas turbine engine;
nevertheless the operation of a compressor or a turbine is much more complicated and
thus their characteristics are also quite complex. For determination of approximate
GTE compressor or turbine map, the theory of similarity is used according to which
the flow in components with comparable design has to be similar and their
characteristics have to be alike as well.
The calculation procedure is based on diagrams compiled from the number of
real (measured) compressor characteristics. Nondimensional parameters that describe
the flow in geometrically similar compressors were derived from their measured
dependencies and presented in available references [2, 3]. With support of these
reference diagrams and with the defined design point of the arising compressor given
by mass flow rate, pressure ratio and efficiency, it is possible to get its rough
characteristics that would be similar to the measured ones.
In the computation, separate parts of the compressor (impeller, diffuser, exhaust
manifold) are not taken into account individually but they are considered as one
whole, whose work is defined by parameters at the inlet and the discharge.
At first, the mean axial velocity ratio is defined as
cˆa 
ca
 ca opt
,
(1)
 
where c a is the mean axial flow velocity in the compressor and c a
opt
is the optimal
mean axial velocity which is velocity at the point where the compressor has the
highest efficiency which is in proximity of the design point. Dimensionless velocity
ratio c a is defined by both the mass flow rate and the compressor pressure ratio (  kc ).
Gas Turbine Engine Off-Design Calculations Using Matlab
101
101
The relative rotational speed of the compressor is defined similarly by
dimensionless speed ratio
n
n
T1c
nd
,
(2)
T1c d
where T1c is the temperature at the compressor inlet and the subscript d stands for the
design point.
Further, total of five diagrams describing variation of compressor efficiency ratio
kc kc max versus mean axial velocity ratio cˆa for different compressor relative
rotational speeds n and engine pressure ratios  kc [3], specific work ratio We We opt
as function of velocity ratio cˆa for different pressure ratios  kc [3], maximum-design
compressor efficiency ratio kc max kc d versus relative compressor speed n [3],
efficiency change at the surge line kc kc n 1  f (n ) [2] and mean axial velocity
ratio at the surge margin cˆa versus the compressor speed ratio n [3] are defined. As
an example, efficiency ratio versus velocity ratio dependence for different speed ratios
is shown in Fig. 1.
The calculation combines diagrams readings with algebraic thermodynamic
equations solutions.
 In the first step of the computation, utilizing the diagrams, specific work We ,
pressure ratio  kc , efficiency kc and mass flow rate Qm at the surge margin for
the design speed n  1 are figured out.
 Subsequently, employing the design speed surge point parameters, the points at
the surge line for different compressor speeds are calculated and hereby the
surge line is defined.
 Finally the different speed lines are carried out [4].
Fig. 1 Compressor efficiency ratio versus mean axial velocity ratio [3]
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J. Pečinka, M. Poledno
The result of the calculation is the compressor map given by pressure ratio versus
mass flow rate dependence with a defined surge line and design point and adequate
efficiency mass flow rate dependence (Fig. 2)
Fig. 2 Compressor characteristics [4]
3. Calculation Technique in Matlab
The computation routine described in chapter 2 was programmed in Matlab. The
calculation was performed for small compressor with mass flow rate of 1.1 kg/s and
pressure ratio 4.1, for speed range n  0.7,1.05 . The resultant characteristic is shown
in Fig. 2.
A script that defines the computation procedure was written, its flow chart is
shown in Fig. 3. “Read” blocks in the diagram represent readings from digitalized
characteristics. Within the reading process, the interpolation between discrete points in
one line and also between neighbouring lines in the case of multiple curve diagrams is
performed.
Ahead of the script execution, the characteristics are loaded into operational
memory from data files described later in this chapter. The calculation runs in cycles.
In each cycle one point of the compressor characteristics is carried out and saved into
a structured variable.
Gas Turbine Engine Off-Design Calculations Using Matlab
103
103
n 1
Read cˆa  f  n  ,
We
We opt
 f  cˆa  ,

kc
kc max
 f  cˆa , n 
 n 1 ,  Qm n 1 , kc
 n 1
Calculate  kc
Determination of surge
point at nominal speed
n=1
Start
Read
Next n

kc
kc n 1
Determination of surge
points for different
speed lines
n 0.7 ,1.05
n  nmin
 f n 
 , Qm , kc

Calculate  kc
false
n  nmax
true
n  nmin
 f  n  , cˆa  f  n 
Next n
Read
We
We opt
cˆa  cˆa  cˆa
 f  cˆa  ,
kc
kc max
 f  cˆa , n 
Calculate  kc , Qm , kc
false
cˆa  cˆa,max
true
false
n  nmax
true
End
Fig.3 Centrifugal compressor map calculation flow chart
… and for all speed lines
of the compressor map
kc max
kc d
Calculation of discrete
points of current speed line
Read
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J. Pečinka, M. Poledno
3.1. Structured Variable Description
As it was mentioned above for the automated calculation purposes, the graphical
dependences described in Chapter 2 have to be digitalized.
Single curve diagrams can be described by n  2 matrix, where the number of
rows n is the number of points of the curve, hence each row represents coordinates of
one curve point.
If the diagram consists of more curves (e.g. diagram in Fig. 1), it can be
described by a structured variable. This variable is a matrix of k elements where each
element represents one curve in the diagram. The elements of the matrix are made of a
structure of two fields. The first field contains an identifier of the curve, the second
one contains a matrix of curve points coordinates. The structure of the variable is
shown in Fig. 4.
id1


  x11 y11 
  x12 y12 





  x1n y1n 

id 2
 x21
x
 22


 x2 n
id k
y21 
y22 



y2 n 
 xk 1
x
 k2


 xkn


yk 1  
yk 2  



ykn  
Fig.4 Schematic view of structured variable
3.2. Data File Format
Digitalized diagrams are saved in ASCII data files, one file for each diagram. Single
curve diagrams can be saved in a simple form of rows of points coordinates. Such a
file can be easily loaded into a Matlab workspace. In the case of structured variables
apart from the curves coordinates, number of curves in the diagram (k), curve
identifiers and number of points on a specific curve (n,k) have to be saved.
Fig.5 Example of a complex data-file format [4]
Gas Turbine Engine Off-Design Calculations Using Matlab
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An example of a data file of a structured variable representing the digitized
diagram from Fig. 1 is shown in Fig. 5. In this case, the curve identifier is the
rotational speed ratio. Note that each line can be described by a different number of
points. Similarly, simple or structured variables from all of the seven diagrams
described in Chapter 2 were created and then they were used for data reading in the
calculation.
The number on the first line represents the number of curves in the diagram
which is corresponding to the number of data blocks in the file. At the beginning of
each data block, there is an identifier describing a given curve. In this case, we use the
value of the dimensionless speed ratio as the identifier. This identifier is followed by a
figure representing the number of points that create a given curve. After this the
coordinates of all the points are listed.
4. Conclusion
In the GTE off-design performance calculation, or in GTE modelling if you like,
multiple plot diagrams are often used for component or engine properties description.
Because performance of these elements is changing with many parameters, like engine
speed, cruise speed, flight altitude or ambient temperature, it is not always possible or
suitable to describe it just by one curve.
Using an example of a compressor map calculation performed in Matlab, it was
shown how multiple plot 2-D graphs widely used in different references can be
digitized, interpolated and used in an automated computation. This technique can be
used for many other GTE off-design calculations, for example turbine map calculation,
thrust and specific fuel consumption versus flight Mach number or altitude
dependencies, engine operating line definition or transient performance (acceleration,
deceleration) calculations. Generally, the same principle can be used for any two
variable dependencies and can be of course expanded for more parameters by
increasing the structured variable matrix dimension.
The paper shows a possible way how known and well-established computation
procedures based on repeated diagram readings can be automated by transforming the
diagrams into lookup tables defined by structured matrix variables saved in simple
ASCII files.
References
[1] KULIKOV, GG. and THOMPSON, HA. Dynamic Modelling of Gas Turbine,
London : Springer, 2004.
[2] RŮŢEK, J. and KMOCH, P. Aircraft engine theory I (in Czech). Brno : VAAZ,
1979. 373 p. U-1275/I.
[3] KMOCH, P. Aircraft engine theory calculation exercises I (in Czech). Brno :
VAAZ, 1972. 56 p. U-2335/I.
[4] POLEDNO, M. Design of aircraft power unit (in Czech). Brno: Brno University
of Technology, 2010. 117 p.