INTRODUCTION
A gas turbine as part of a cogeneration plant delivers power at constant shaft
speed and hot exhaust gas at optimum heat recovery steam boiler inlet temperature
from the combustion of a gas or liquid fuel. Main components of a typical singleshaft
gas turbine are as indicated in Fig. 1. The corresponding
process diagram is given in Fig. 2. The air from the atmosphere
is first compressed (process 12) to high pressure in the compressor. The compressed
air is then mixed with fuel and burned (process 23’) in the combustion
chamber. The hot gas then expands through the turbine (process 3’4) resulting
in rotation work on the turbine shaft. In an industrial gas turbine, the shaft
is connected to a generator shaft through a speed reduction gear box.
In general, gas turbines have fast starting characteristics. The improved dynamic
load characteristic has made them the preferred choice for industrial applications.
In the state of the art designs they are equipped with variable geometry compressors
which allow wider surge and stall margins in part load applications. Topic of
concern is a gas turbine working in synchronous with another gas turbine to
provide the required load demand and at the same time delivering high temperature
gas to a heat recovery steam generator.
During starting and for speeds lower than about 65% of the design speed, the
Inlet Guide Vanes (IGVs) and Variable Stator Vanes (VSVs) are at minimum opening
position; at a speed of about 98%, they open fully with the opening starting
at around 65% of design speed. The latter region is identified as a variable
geometry region where the IGVs and VSVs are manipulated to vary the air flow
rate.

Fig. 2: 
Process diagram for a singleshaft gas turbine 
In a cogeneration design, the gas turbine is also expected to deliver high
temperature exhaust gas. For loads higher than 50% of the rated value this requirement
is achieved by a temperature controller. In this region, IGVs and VSVs are manipulated
to vary the air flow rate into the compressor as per the load demand. Fuel flow
rate
is also varied to keep temperature at the inlet of the third stage of the gas
turbine (T_{5}) to the required set point. For a load demand lower than
50%, the gas turbine is on speed and load control. During this period the variable
geometry (θ_{IGV}) is at fully open position and the state can
be considered as fixed geometry region. Figure 3 shows plots
of main parameters in their normalized formeach divided by their maximum value.
Dynamic studies of a gas turbine require thermodynamic models of the gas turbine
components as well as details of the control system. Overall performance map
instead of component performance maps are usually provided by the manufacture
that made it difficult to proceed with the dynamic studies. For a variable geometry
compressor, it is not practical to provide performance maps corresponding to
the whole working region of the IGV and VSV settings. In energy and exergy focused
parametric studies, compressor and turbine efficiencies may be assumed constant
(Abam et al., 2011; Mahmoudi
et al., 2009) which neglects the effect of IGV. In power plant stability
studies, the simplest approach yet is to use empirical models to describe the
thermodynamic part (Rowen, 1983; Zhang
and So, 2000). In the study of Kakimoto and Baba (2003),
the pressure ratio in the compressor and turbine are described as a function
of design pressure ratio and air flow rate while the efficiencies are assumed
constant. The effect of rotational speed on the pressure ratio was latter included
in the models by Suzaki et al. (2000).

Fig. 3: 
Actual Operating schedule for a Singleshaft Gas Turbine Generator
from Univeriti Teknologi PETRONAS GDC Plant 
An alternative approach is to use a scaling method for fixed geometry compressors
and turbines (Gaudet, 2008) with extrapolation techniques
applied at lower speeds. Stagestacking techniques based on generalized stage
performance curves are another widely used method (Kim
et al., 2001). Recent techniques from computational intelligenceneural
network (Zayandehroodi et al., 2010) and fuzzy
logic (Radaideh, 2003) are also getting frequent use
in the area of thermal plants. They can be applied to capture thermodynamic
characteristics of the system over the whole operating region. In dynamic studies,
a model needs to be simple to consume short calculation time and good enough
to accommodate variable operating conditions. It must also allow the study of
change in parameters like compressor inlet temperature, efficiency of the compressor,
efficiency of the turbine etc.
The objective of this study is to develop compressor and turbine maps for a
5.2 MW rated industrial gas turbine whose design specifications are partially
known. The research proposes the use of scaling method in the fixed geometry
regions and stagestacking method in the variable geometry region. Generalized
stage and overall component performance curves from literature are used in presenting
the performance characteristics. The reference map for compressors is selected
after preliminary tests on the suitability of three different generalized maps.
It is hoped that the resulting maps will be suitable enough to perform steady
state performance and dynamic studies.
GAS TURBINE THERMODYNAMIC MODEL
In the simulation of a gas turbine the compressor and turbine are represented
by performance maps. For a given inlet
conditions and inlet fuel flow
thermodynamic equations are applied to estimate properties of the working fluid
at following state points. For the compressor, the temperature at the outlet
(T_{2}) and work input requirement ,
respectively are calculated by:
where, h is enthalpy, PR_{c} and η_{c} are pressure ratio
and efficiency, respectively, of the compressor corresponding to the given mass
flow rate. The two values are read from specific compressor maps usually described
as PR_{c} = f (m_{air}, N) and η_{c} = f (PR_{c},
N). Where, N is the compressor shaft speed.
The compressed air from the compressor is mixed with fuel and burned in the
combustion chamber before it expands through the turbine. The temperature at
the outlet of the combustion chamber (T_{3'}) can be estimated using
combustion charts (Razak, 2007) or by an iterative approach.
The energy balance around the combustor is expressed as:
where, h is the enthalpy, LHV is lower heating value of the fuel and FAR is
the fuel to air ratio.
There is a pressure loss in the compressor due to the chamber resisting air
flow, high level of turbulence required for combustion and heat addition. This
pressure loss is considered in the model applying the following equation:
where, PLF is the pressure loss factor; M_{23'} is the dimensionless
flow; K_{1} and K_{2} are constants for the cold loss and hot
loss of the combustor, respectively. Then from the pressure loss and inlet pressure:
The hot gas temperature leaving the turbine (T_{4}) and the power transferred
to the turbine shaft are expressed as:
where, h is enthalpy of the hot gas; η_{t} = f_{1} (PR_{t},
N) and PR_{t} are isentropic efficiency and pressure ratio, respectively,
of the turbine. The pressure at the outlet of the turbine P_{4} is limited
to less than 105.825 kPa. Using this value, the pressure ratio is calculated
and used in (m_{air}+m_{f}) = f_{2} (PR_{t},
N) to check if the mass balance is satisfied.
The following is assumed for the electric power at the generator terminal:
where, η_{m} and η_{ele} are mechanical efficiency
and electrical efficiency, respectively.
In the first step of performance map calculation, Eq. 18
are applied iteratively to determine estimated design point parameters. At each
state point variation of specific heats with temperatures are taken into consideration.
Most of the time, Heat Rate (HR), Lower Heating Value (LHV) of the fuel and
exhaust flow rate
are provided. In that case, the following equations are applied to obtain the
fuel flow rate and air mass flow rate, respectively:
Once the component design parameters are determined, scaling and stagestacking
methods are applied to generate the performance maps. Offdesign analysis is
just solving the same set of Eq. 18 satisfying
the matching conditionsconservation of mass and power.
Scaling method: For a compressor, there are three available performance
calculation techniques: scaling method, stagestacking method and blade element
method (Johnsen and Bullock, 1965). Among the three,
scaling method is the easiest. But the inherent simplicity has the drawback
of demanding a suitable reference map. Besides, it is not suitable for studying
variable geometry compressors from the point of view of understanding the stage
interaction inside the compressor. Scaling method is good enough if the interest
of the analyst is to understand the compressor as a black box. This is true
for the case of fixed geometry compressors. As will be demonstrated in the results
and discussion section, the accuracy of the method is largely dependent on the
shapes of the reference maps. Scaling method overlooks compressibility effect.
Modified versions of the scaling method have been suggested recently (Kong
and Ki, 2007; Kong et al., 2003). The main
idea in the scaling method is that performance map of a compressor can be generated
if the design conditions
are known. The basic equations for the scaling method are:
Pressure ratio:
Mass flow rate:
Efficiency:
where, PR_{map, des} and PR_{map} are design pressure ratio
and offdesign pressure ratio for the reference map ;
and
are design mass flow rate and offdesign mass flow rate for the reference map;
η_{isen, des} and η_{isen, map} are design efficiency
and offdesign efficiency, respectively, of the reference map. For generating
the compressor performance maps using the scaling method, multistage compressor
maps from three different sources (Converse and Giffin, 1984;
Johnsen and Bullock, 1965; Saravanamuttoo
and MacIsaac, 1983) will be considered. The first set of maps were used
to study a twospool industrial gas turbine (Zhu and Saravanamuttoo,
1992).
For turbines, the easiest method is to assume that the turbine is choked and
the corresponding efficiency is constant. If that is not good enough, Stodola
ellipse equation (Dixon, 2005) or onedimensional nozzle
equation (Ordys et al., 1994) may be applied. The
third option is to use Ainley and Mathieson (1951) method.
This method, however, requires geometric data which is hardly available. In
this paper, scaling method that demands relatively little information at design
point is considered.
Stagestacking method: In the stage stacking method, generalized stage
performance curves are used in order to develop overall compressor performance.
Based on a flow coefficient φ calculated at the inlet of a stage, pressure
ratio ψ and efficiency η are read from generalized curves and used
to estimate for outlet pressure and temperature of the stage. The total pressure
and temperature at the outlet of the preceding stage are considered as an input
to the following stage. The calculations are repeated covering all the stages
in the compressor. The properties at the outlet of the last stage will be considered
as the properties of the compressor. The relevant equations are as described
below:
Flow coefficient:
Pressure coefficient:
Temperature coefficient:
Efficiency:
where, C_{a} is the stage inlet axial velocity, U is the tangential
blade speed, T_{os} is stage inlet air total temperature, ΔT_{os}
is stage total temperature rise, C_{p} is specific heat at constant
pressure and γ is ratio of specific heats.
Stagestacking procedure is suitable to develop performance maps for fixed
as well as variable geometry compressors. The first proposed stage stacking
method was based on the stage performance map of a single stage compressor or
fan. Latter it was improved by introducing generalized stage performance curves.

Fig. 4: 
Graph of stage normalized efficiency versus ratio between
temperature coefficient and flow coefficient (Howell
and Bonham, 1950) 

Fig. 5: 
Plot of normalized pressure coefficient versus normalized
flow coefficient ψ* = f_{ψ} (φ*, SF) (Spina,
2002) 
Ratios of each characteristic parameter (Eq. 1417)
and the value of the same parameter at the point of maximum efficiency are used
to define a point on the generalized curves. Figure 4 shows
the generalized efficiency curve as developed by Howell
and Bonham (1950). The other curve required for stagestacking calculation
is the ψ* verse φ* plot (Fig. 5). This was developed
by Muir (1988) assembling a number of openly published
stage data on pressure ratio. The concept of Shape Factor (SF) was then introduced
by Spina (2002) to obtain a number of generalized curves
covering all the data points calculated by Muir (1988).
The corresponding equation becomes:
where:
The normalized isentropic efficiency and total temperature equations are given
by:
In order to obtain stage characteristics of a given compressor the shape factor
is estimated by minimizing the sum of squares of errors between calculated and
given performance parameter, for instance compressor pressure ratio.
In a variable geometry compressor, IGVs and VSVs are adjusted to cope with
the variation of speed and gas turbine load. In that case, each new setting
of the blades results in a new stage characteristic. From velocity triangle
of a rotating compressor blade (Fig. 6a) and assuming constant
axial flow velocity and mean rotor speed and from equation of energy balance
for the rotor:
Assuming that the IGVs and VSVs are adjusted such that the relative flow angles
are constant and equal to the design point angles (Fig. 6b),
Eq. 20 can be written in a form suitable to find generalized
performance curves at offdesign IGVs and VSVs setting:
The same set of assumptions leads to the conclusion that efficiency is constant.
This makes sense for the pressure losses across the rotor blades are at minimum
if the angle of incidence is adjusted such that the flow angle is unaffected.
The resulting equation between φ and ψ will be:

Fig. 6: 
Compressor blades: (a) velocity triangles (b) IGV setting 
RESULTS AND DISCUSSION
The results from application of the proposed methods are presented here. The
first sets of maps are modeled by scaling method. As mentioned in the discussion
of the scaling method, data from three different sources will be used as reference
data. Fig. 7 shows the difference in the shapes of the three
selected reference maps. At the corrected design speed (N/N_{d} = 1),
the three maps seem overlapping in a certain region. For speeds other than the
design point, the maps are hardly close to each other. These set of maps are
chosen arbitrarily to see if random selection of a compressor map in a certain
application is acceptable at all the times.
The first axial compressor considered for testing the suitability of the three
reference maps is taken from (Budinger and Thomson, 1952).
The compressor is a 10 stage axial flow compressor with mass flow rate ,
pressure ratio PR_{4} = 12.4, efficiency η_{d} = 0.86 and
shaft speed N_{d} = 14000 rpm. The comparison between experimental data
of the compressor and the maps generated based on the three reference maps are
shown in Fig. 8. The scaled maps predicted different points
except for the few points closely predicted by Johnsen’s Map in the lower
speed regions.
The second axial compressor selected for examining the same set of reference
maps is obtained from (Geye et al., 1953) . The
compressor is designed for a mass flow rate of
, pressure ratio of PR_{d} = 10.26, isentropic efficiency of η_{d}
= 0.86 and rotational speed of N_{d} = 13380 rpm. As presented in Fig.
9, at the design speed of the compressor, the prediction by the three reference
maps are somehow in agreement. For the offdesign speed, the only reference
map that may be considered is the Johnsen’s map.
The third test compressor map is a 16 stage compressor taken from Medeiros
et al. (1951).

Fig. 7: 
Normalized reference compressor performance maps 
It has the design flow capacity of
and pressure ratio of PR_{d} = 8.75 for a speed of N_{d} = 6100
rpm and efficiency η_{d} = 0.83. Similar to the previous compressors
the same set of reference compressors maps were applied to predict offdesign
characteristics of the compressor. The resulting curves are depicted in Fig.
10. At 80% of the design speed, Saravanamutto’s map seems predicting
the performance accurately.

Fig. 11: 
Performance maps of a 3stage turbine: (a) Normalized efficiency
versus normalized pressure ratio (b) Normalized mass flow rate versus normalized
pressure ratio

Except that, the rest of the predictions are far from being acceptable for
further analysis.
From the presented tests, it is observed that even for pressure ratios closer
to the reference map pressure ratio the resulting prediction may not be as accurate
as expected. Therefore, having this problem from the simplest technique known
and with the scant information available from manufactures catalogue, the task
of finding a suitable map using this method is too much of an out of option
procedure than other wise.
Table 1: 
Design specification of solar gas turbine 

^{*}Calculated values 
The present work suggested scaling method for the fixed geometry region and
stage stacking method for the variable geometry region. The stage stacking method
is more involved as it follows a design procedure based on carefully decided
design parameters. However, as demonstrated latter, the accuracy is much better
than scaling method.
After studying the importance of choosing a geometrically suitable reference
map for predicting the performance of an axial compressor based on the scaling
method, the technique was applied to the components of a 5.2 MW Solar gas turbine.
Manufacturer supplied overall parameters as indicated in Table
1 are used for the calculations. To determine the missing design information
regarding the 12 stage axial flow compressor and the 3 stage turbine, Eq.
110 were solved in an iterative way with the objective
of minimizing sums of Squares of Relative Errors (SSRE) given by the following
equation:
where, P_{2a} is design point compressor discharge pressure and
is the power at the generator terminal. The error is minimized varying isentropic
efficiencies of the compressor and turbine, respectively. The design point discharge
pressure is read from overall performance map of the gas turbine provided by
the manufacturer. Results of the optimization calculation are given in Table
1.
From the optimized result, the turbine appeared to have exhaust gas flow rate
of
and a pressure ratio of Pr_{d} = 10 while the compressor is featured
by a mass flow rate of and
pressure ratio of PR_{d} = 11. The performance maps developed based
on the calculated design values are portrayed in Fig. 11
and 12.

Fig. 12: 
Performance maps of a 12stage compressor: (a) Normalized
pressure ratio versus normalized mass flow rate (b) Efficiency versus normalized
mass flow rate 
As a reference, a turbine map from Converse (1984) is
chosen for applying the scaling method attributed closer design pressure ratio
to the pressure ratio of the Solar Turbine.

Fig. 14: 
Stage inlet flow area of a 16stage compressor (Muir,
1988) 

Fig. 15: 
Plot of normalized pressure ratio versus normalized mass flow
rate for a 3stage axial compressor (Kashiwabara et
al., 1986) 
In applying the stagestacking method, Eq. 22 and 23
are first validated by experimental data taken from (Kim
et al., 2001). As indicated in Fig. 13, the results
from the two equations are in good agreement with the experimental data.
Stagestacking method relies on geometric data. However, this information is
not available for proprietary reasons. In this paper, a general design procedure
with design conditions decided based on data from well established literature
are adapted. The following parameters are assumed for design point calculation.
• 
Axial velocity 
: 
150 m sec^{1} 
• 
Inlet temperature 
: 
288.15 K 
• 
Inlet pressure 
: 
101.325 kPa 
• 
Polytropic efficiency 
: 
0.89 
• 
Inlet flow angle 
: 
15° 
For validating the geometry calculation, actual dimensions for 16 stage axial
compressor taken from Muir (1988) is compared with the
geometry calculated using to the above input parameters. As indicated in Fig.
14, the real design and the calculation result match quite closely. The
stagestacking method, before it is applied to the Solar Turbine, is also tested
by predicting offdesign performance characteristics of a 3 stage axial flow
compressor taken from Kashiwabara et al. (1986).
As shown in Fig. 15, the calculated result agrees well with
the experimental data as compared to other similar work (Song
et al., 2000).
CONCLUSION
In this study, we have developed gas turbine component maps based on partially
known design information. At the start, three reference maps from different
sources have been tested to verify if random selection of reference maps could
be used for generating characteristic curves through scaling method. After testing
the reference maps by three other actual compressor maps, the scaling method
is applied to Solar Taurus 60s gas turbine whose overall design information
is partially known. Based on the cases tested, the following conclusions have
been reached:
• 
Having partially known overall design information, the application
of scaling method requires care in the selection of the reference map. It
was observed that even for the case where the design pressure ratios of
the reference and the new compressor are closer, accurate off design point
performance prediction may not be granted 
• 
As compared to the scaling method, the stage stacking method well matches
the experimental data. The only drawback to this method is the need for
design point calculation 
The next task to the presented work is to test the generated performance maps
in actual overall performance prediction of the gas turbine generator.
ACKNOWLEDGMENT
The authors acknowledge the support by Universiti Teknologi PETRONAS for providing
all the resources sought by the research.