Turbine Engine Starting Simulation

*Sergiy Yepifanov and Feliks Sirenko*

## **Abstract**

The process of engine control development requires the models that describe engine operation and its response on a control action. The development flow required numerous models to be engaged, like component-level non-linear model, engine-level non-linear model, linear dynamic model, etc. Models made a great progress during the recent years and became reliable tools for control engineers. However, most models are derivatives from the component-level non-linear model, which in its turn consumes the component performances. Things turn different when one addresses the starting range of engine operation. The problem here is all about the missing performances of the engine components, as it is quite hard to harvest these performances in this region as the processes that happen in the engine are transient by nature. Different scientists offered different approaches to the problem of building the component level non-linear model of the sub-idle region, but the general idea is to somehow extrapolate the known performances to the subidle region. However, there are no known reports about a model that considers all aspects of this approach and simulates the engine starting. In this chapter, you can find an alternative view on a problem of simulation of a sub-idle operation. The proposed model belongs to a group of linear dynamic models including the static model as well as simplified static model to support the dynamic model. Instead of trying to extrapolate component performances and get the full-scale componentlevel model, you will see that the canonical component performances are replaced by the direct relations between parameters that are used in the control algorithms, like gas-path parameters against the RPM. As well in this chapter, you will find the exact instructions on how to create the model and an example of the one with the real test data.

**Keywords:** turbine engine, starting, mathematical model, experimental data, identification, approximation

#### **1. Introduction**

Models are known for their benefit to reduce the price for engine development replacing the numerous experiments with the simulations. This has the biggest value for the control engineers as they need the greatest amount of experimental data. Industry came to a gold standard on how to simulate the models of engine operation within the range limited with idle on one side and the maximum on the other side. These models are well structured and validated, having the well-known thermodynamic relations under the hood. But their usage is impossible or

considerably limited in the sub-idle range, because of no experimentally registered performances of engine components. The other reason is that the required performances are challenging to be determined for the sub-idle region, especially the performance of a combustion chamber.

representation to be changed to the new one. The efficiency performance of com-

<sup>C</sup> *<sup>=</sup>*N2 � � <sup>¼</sup> f Wa � <sup>T</sup><sup>∗</sup>

i�1

i

Another method, proposed by Agrawal and Yunis [8], belongs to methods of the third group. This method requires a big number of experimental data for identifying true performances of engine components from universal ones. Relations to carry

*in=*<sup>N</sup> � <sup>P</sup><sup>∗</sup>

<sup>¼</sup> <sup>K</sup><sup>ϕ</sup> � <sup>N</sup>*<sup>=</sup>* ffiffiffiffiffiffiffiffi Tcor � � <sup>p</sup>

Tcor � � <sup>p</sup>

Tcor � � <sup>p</sup>

N*=* ffiffiffiffiffiffiffiffi Tcor � � <sup>p</sup>

" #

N*=* ffiffiffiffiffiffiffiffi Tcor � � <sup>p</sup>

" #<sup>2</sup>

<sup>¼</sup> <sup>K</sup><sup>ψ</sup> � <sup>N</sup>*<sup>=</sup>* ffiffiffiffiffiffiffiffi

<sup>¼</sup> <sup>K</sup> � <sup>η</sup>C i � <sup>N</sup>*<sup>=</sup>* ffiffiffiffiffiffiffiffi

where Kϕ, Kψ, and K are coefficients depending on a corrected rotational speed.

The last group of methods can be represented by Kong's method [9]. The point

To cope with the problem of poor convergence, Jones et al. had improved Kong's method, suggesting application of non-metering components and new criterion for

Final improvement of Kong's method was carried out by the group of scientists from Sharif University. They have suggested a new approach to scaling factor calculation. This approach provides good identification quality at all modes. All parameters necessary for compressor map identification can be calculated as follows [13]:

> Wa cor ¼ ð Þ Wa cor *univ* � Wa cor calc*=*ð Þ Wa cor calc *univ*, (7) Ncor ¼ ð Þ Ncor *univ* � Ncor calc*=*ð Þ Ncor calc *univ*, (8) π ¼ π*univ* � ð Þþ πcalc � 1 1*=*ð Þþ πcalc univ � 1 1, (9)

In addition to the above parameters, this method also allows simulation of

of the method is getting universal component performances from well-known thermodynamic relations for gas turbine engines and prior information about the engine with its further identification at modes from idle to maximum power (thrust). Here, identification must be understood as scaling factors calculating to match theoretical and true performances. This method has validated its effectiveness for the set of turboshaft engines. But the shortage of this method was poor convergence at modes with low rotational speeds, including sub-idle modes.

identification quality estimation (surge parameter). Applied measures have

N*=* ffiffiffiffiffiffiffiffi Tcor � � <sup>p</sup>

i�1

i

i�1

i

*in* � �*:* (3)

i�1

; (4)

; (5)

, (6)

i

pressor spool is expressed as follows:

*DOI: http://dx.doi.org/10.5772/intechopen.90803*

*Turbine Engine Starting Simulation*

out identification are the following:

• Air flow

• Specific work

• Efficiency

**15**

η ∗ <sup>C</sup> � <sup>L</sup><sup>∗</sup>

Wa � ffiffiffiffiffiffiffiffi Tcor <sup>p</sup> *<sup>=</sup>*Pcor � �

Wa � ffiffiffiffiffiffiffiffi Tcor <sup>p</sup> *<sup>=</sup>*Pcor � �

Lsp*=*Tcor � �

η<sup>C</sup> ð Þ<sup>i</sup>�<sup>1</sup> η<sup>C</sup> ð Þ<sup>i</sup>

torque, pressure, and temperature at compressor discharge.

improved identification, especially at sub-idle modes [12].

Lsp*=*Tcor � �

i�1

i

This relation is formed from analysis of a big number of similar engines.

Here we have reached the point that suites the most to introduce the existing models of sub-idle operation. They can go into two baskets. In the first we find the methods that deal with the extrapolation of the component performances into the sub-idle range, while in the second, methods that approximate the experimental data with various polynomials.

We can dive deeper to the first basket and discriminate four groups of models


The brightest method of the first group is based on the simplest similarity laws. The method was developed by Gaudet et al. [1]. The method is only applicable for incompressible fluid. Another method by Sexton [2] suggests obtaining new constant corrected speed lines of compressor spool using experimentally measured ones:

$$\frac{(\mathbf{W\_{a}})\_{i-1}}{(\mathbf{W\_{a}})\_{i}} = \left[\frac{\mathbf{N\_{i-1}}}{\mathbf{N\_{i}}}\right], \frac{\left(\mathbf{L\_{sp}}\right)\_{i-1}}{\left(\mathbf{L\_{sp}}\right)\_{i}} = \left[\frac{\mathbf{N\_{i-1}}}{\mathbf{N\_{i}}}\right]^2, \ (\mathbf{Power})\_{i-1}/(\mathbf{Power})\_{i} = \left[\mathbf{N\_{i-1}}/\mathbf{N\_{i}}\right]^3. \tag{1}$$

where subscript i refers to experimentally measured parameter and i � 1 refers to extrapolated parameter.

Gaudet and Gauthier improved Sexton's method by adopting it for compressible fluid. Powers in Eq. (2) are obtained by the analysis of two experimentally measured corrected constant speed lines with the lowest values of corrected rotational speed. Next, a new corrected constant speed line is calculated referring existing by the equations

$$\frac{\mathbf{(W\_{a})\_{i-1}}}{\mathbf{(W\_{a})\_{i}}} = \left[\frac{\mathbf{N\_{i-1}}}{\mathbf{N\_{i}}}\right]^{\mathbf{P}}, \frac{\left(\mathbf{L\_{sp}}\right)\_{i-1}}{\left(\mathbf{L\_{sp}}\right)\_{i}} = \left[\frac{\mathbf{N\_{i-1}}}{\mathbf{N\_{i}}}\right]^{\mathbf{q}}; (\text{Power})\_{i-1}/(\text{Power})\_{i} = \left[\frac{\mathbf{N\_{i-1}}}{\mathbf{N\_{i}}}\right]^{\mathbf{r}}.\tag{2}$$

Powers p, q, and r stay constant for all sub-idle operating range.

The second group is very similar to the first one and differs in the fact that extrapolation takes into account prior information about processes taking place during starting. This allows improving the tolerance and preventing errors.

All methods of the second group consider only separate facts from prior information (e.g. negative efficiency of compressor spool till the moment of combustion chamber lighting [5]), but there are still no known methods that are able to take into account all prior information. For example, there is a method [6, 7] to extrapolate performances of compressor or turbine taking into account the continuity equation violation at negative efficiency. Authors suggest routine performance

representation to be changed to the new one. The efficiency performance of compressor spool is expressed as follows:

$$\mathfrak{n}\_{\mathbf{C}}^{\*} \cdot \left[ \mathbf{L}\_{\mathbf{C}}^{\*} / \mathbf{N}^{2} \right] = \mathbf{f} \left( \mathbf{W}\_{\mathbf{a}} \cdot \mathbf{T}\_{in}^{\*} / \mathbf{N} \cdot \mathbf{P}\_{in}^{\*} \right). \tag{3}$$

Another method, proposed by Agrawal and Yunis [8], belongs to methods of the third group. This method requires a big number of experimental data for identifying true performances of engine components from universal ones. Relations to carry out identification are the following:

#### • Air flow

considerably limited in the sub-idle range, because of no experimentally registered performances of engine components. The other reason is that the required performances are challenging to be determined for the sub-idle region, especially the

Here we have reached the point that suites the most to introduce the existing models of sub-idle operation. They can go into two baskets. In the first we find the methods that deal with the extrapolation of the component performances into the sub-idle range, while in the second, methods that approximate the experimental

We can dive deeper to the first basket and discriminate four groups of models

• based on straight extrapolation of known performances (no physics

• "smart" extrapolation that consumes the theoretical knowledge about

• based on fitting the generic performances to the experimental data [9–11].

The brightest method of the first group is based on the simplest similarity laws. The method was developed by Gaudet et al. [1]. The method is only applicable for incompressible fluid. Another method by Sexton [2] suggests obtaining new constant corrected speed lines of compressor spool using experimentally

where subscript i refers to experimentally measured parameter and i � 1 refers

Gaudet and Gauthier improved Sexton's method by adopting it for compressible fluid. Powers in Eq. (2) are obtained by the analysis of two experimentally measured corrected constant speed lines with the lowest values of corrected rotational speed. Next, a new corrected constant speed line is calculated referring existing by

, Pow ð Þ<sup>i</sup>�<sup>1</sup>*=*ð Þ Pow <sup>i</sup> <sup>¼</sup> Ni�<sup>1</sup>*=*Ni ½ �<sup>3</sup>

; Pow ð Þ<sup>i</sup>�<sup>1</sup>*=*ð Þ Pow <sup>i</sup> <sup>¼</sup> Ni�<sup>1</sup>

Ni <sup>r</sup> *:* (1)

*:* (2)

component performances in the sub-idle region [4–7];

*Modeling of Turbomachines for Control and Diagnostic Applications*

<sup>¼</sup> Ni�<sup>1</sup> Ni <sup>2</sup>

> <sup>¼</sup> Ni�<sup>1</sup> Ni <sup>q</sup>

The second group is very similar to the first one and differs in the fact that extrapolation takes into account prior information about processes taking place during starting. This allows improving the tolerance and preventing errors.

All methods of the second group consider only separate facts from prior information (e.g. negative efficiency of compressor spool till the moment of combustion chamber lighting [5]), but there are still no known methods that are able to take into account all prior information. For example, there is a method [6, 7] to extrapolate performances of compressor or turbine taking into account the continuity equation violation at negative efficiency. Authors suggest routine performance

Powers p, q, and r stay constant for all sub-idle operating range.

performance of a combustion chamber.

data with various polynomials.

considered) [1–3];

measured ones:

ð Þ Wa <sup>i</sup>�<sup>1</sup> ð Þ Wa <sup>i</sup>

the equations

**14**

ð Þ Wa <sup>i</sup>�<sup>1</sup> ð Þ Wa <sup>i</sup>

• statistical extrapolation [8];

<sup>¼</sup> Ni�<sup>1</sup> Ni 

> <sup>¼</sup> Ni�<sup>1</sup> Ni <sup>p</sup>

to extrapolated parameter.

, Lsp i�1 Lsp i

> , Lsp i�1 Lsp i

$$\frac{\left(\mathbf{W\_{a}} \cdot \sqrt{\mathbf{T\_{cor}}} / \mathbf{P\_{cor}}\right)\_{\mathbf{i}-1}}{\left(\mathbf{W\_{a}} \cdot \sqrt{\mathbf{T\_{cor}}} / \mathbf{P\_{cor}}\right)\_{\mathbf{i}}} = \mathbf{K\_{\phi}} \cdot \frac{\left(\mathbf{N} / \sqrt{\mathbf{T\_{cor}}}\right)\_{\mathbf{i}-1}}{\left(\mathbf{N} / \sqrt{\mathbf{T\_{cor}}}\right)\_{\mathbf{i}}};\tag{4}$$

• Specific work

$$\frac{\left(\mathbf{L\_{sp}}/\mathbf{T\_{cor}}\right)\_{\mathbf{i}-\mathbf{1}}}{\left(\mathbf{L\_{sp}}/\mathbf{T\_{cor}}\right)\_{\mathbf{i}}} = \mathbf{K\_{\Psi}} \cdot \left[\frac{\left(\mathbf{N}/\sqrt{\mathbf{T\_{cor}}}\right)\_{\mathbf{i}-\mathbf{1}}}{\left(\mathbf{N}/\sqrt{\mathbf{T\_{cor}}}\right)\_{\mathbf{i}}}\right]^2;\tag{5}$$

• Efficiency

$$\frac{(\eta\_{\rm C})\_{\rm i-1}}{(\eta\_{\rm C})\_{\rm i}} = \mathbf{K} \cdot \eta\_{\rm C} \cdot \left[ \frac{\left(\mathbf{N} / \sqrt{\mathbf{T}\_{\rm cor}}\right)\_{\rm i-1}}{\left(\mathbf{N} / \sqrt{\mathbf{T}\_{\rm cor}}\right)\_{\rm i}} \right],\tag{6}$$

where Kϕ, Kψ, and K are coefficients depending on a corrected rotational speed. This relation is formed from analysis of a big number of similar engines.

In addition to the above parameters, this method also allows simulation of torque, pressure, and temperature at compressor discharge.

The last group of methods can be represented by Kong's method [9]. The point of the method is getting universal component performances from well-known thermodynamic relations for gas turbine engines and prior information about the engine with its further identification at modes from idle to maximum power (thrust). Here, identification must be understood as scaling factors calculating to match theoretical and true performances. This method has validated its effectiveness for the set of turboshaft engines. But the shortage of this method was poor convergence at modes with low rotational speeds, including sub-idle modes.

To cope with the problem of poor convergence, Jones et al. had improved Kong's method, suggesting application of non-metering components and new criterion for identification quality estimation (surge parameter). Applied measures have improved identification, especially at sub-idle modes [12].

Final improvement of Kong's method was carried out by the group of scientists from Sharif University. They have suggested a new approach to scaling factor calculation. This approach provides good identification quality at all modes. All parameters necessary for compressor map identification can be calculated as follows [13]:

$$\mathbf{W\_{a\\_cor}} = (\mathbf{W\_{a\\_cor}})\_{\textit{univ}} \cdot \mathbf{W\_{a\\_cor\\_calc}} / (\mathbf{W\_{a\\_cor\\_calc}})\_{\textit{univ}},\tag{7}$$

$$\mathbf{N}\_{\rm cor} = (\mathbf{N}\_{\rm cor})\_{\rm univ} \cdot \mathbf{N}\_{\rm cor\ calc} / (\mathbf{N}\_{\rm cor\ calc})\_{\rm univ},\tag{8}$$

$$
\boldsymbol{\pi} = \boldsymbol{\pi}\_{\text{univ}} \cdot (\boldsymbol{\pi}\_{\text{calc}} - \mathbf{1}) + \mathbf{1}/(\boldsymbol{\pi}\_{\text{calc unity}} - \mathbf{1}) + \mathbf{1},\tag{9}
$$

$$
\mathfrak{η} = \mathfrak{η}\_{\text{univ}} \cdot \mathfrak{η}\_{\text{calc}} / \mathfrak{η}\_{\text{calc\\_univ}}.\tag{10}
$$

alone. The second phase lasts till the turbine reaches the power where it can drive the compressor and accelerate the rotor on its own. The torque equation for the

The starting ends up with the phase where the starter is moved from the table, making the turbine to be the only one source of power that drives the compressor and makes all rotors accelerate to the idle or the flight idle (see **Figure 1**). The

The first and foremost problem that every simulation engineer faces when designing the model is the problem of the choice among the available model structures, as the selected structure must make a perfect fit to the problem to be solved by the model. The major goal of the proposed model is to perform the preliminary

• provide correct relations among parameters that describe the operation of the engine (as the starting is a non-steady-state process, the model must properly describe the dynamic properties of GTE including all processes that determine

• be able to be integrated with the vast majority of the models that are normally used to support the ACS design, e.g. model that describes the engine operation

• properly describe the features typical for the starting, for example, parameters

• require minimum amount of experimental data to be generated and adjusted;

• give a quantitative assessment to the required parameters that cannot be

in the range above the idle, models of governors, etc.;

M*starter* þ MT ¼ Mres þ M*inert:* (12)

MT ¼ Mres þ M*inert:* (13)

second phase is

torque balance equation is

*Turbine Engine Starting Simulation*

*DOI: http://dx.doi.org/10.5772/intechopen.90803*

engine dynamics);

measured directly.

**Figure 1.**

**17**

overlap during the starting;

*Engine starting stages and moment performances.*

tuning of the ACS. For this, the model must:

Methods of the second category accept the hypothesis about similarity of starting processes in all engines. Here appears the conclusion that any starting process can be circumscribed by universal relations between control factors and measured parameters. These methods require storing rig and in-flight experimental information to form appropriate data for identification. The polynomial relations are conserved to have the same structure from engine to engine. Only coefficients of polynomial change [14].

Wrapping up the above overview of the methods to simulate sub-idle operation of the gas turbine, one can conclude the following


## **2. Requirements to the starting simulation model**

During the starting engine speeds up from either turned off state to an idle running or windmilling to the flight idle. Based on the source of power that drives the rotor during the acceleration, the whole starting may be decomposed into three phases.

The rotor of the engine is dynamically balanced. During the first phase, the rotor is driven by the starter only, as the combustion chamber is not lighted yet. During this phase, the starter drives the rotor from turned off state to the RPM where the combustion chamber is started. The torque balance of this phase is

$$\mathbf{M}\_{\text{star}ter} = \mathbf{M}\_{\text{res}} + \mathbf{M}\_{\text{iner}},\tag{11}$$

where Mres is the total torque of resistance that is generated by compressor and turbine cascades of the rotor, by bearings and driven auxiliary units (fuel and oil pumps, electric generator, deaerator, etc.).

As soon as combustion chamber is brought into a game, the turbine starts giving the positive input to the torque balance equation. Both turbine and starter drive the rotor because the turbine does not have enough power yet to accelerate the rotor

η ¼ η*univ* � ηcalc*=*ηcalc univ*:* (10)

Methods of the second category accept the hypothesis about similarity of starting processes in all engines. Here appears the conclusion that any starting process can be circumscribed by universal relations between control factors and measured parameters. These methods require storing rig and in-flight experimental information to form appropriate data for identification. The polynomial relations are conserved to have the same structure from engine to engine. Only coefficients of

Wrapping up the above overview of the methods to simulate sub-idle operation

• most methods address the compressor map extrapolation ignoring the problem of other compressor components, especially the combustion chamber

• to identify the performances from the experimental data, the latest must be of high quality, however, as the starting is a dynamic process, the collected data may be corrupted with time lag as well as poor sensor performance at low

• getting the accurate experimental data to build the high-quality performance map is not always a good value for money as it usually needs mounting extra sensors and carrying out additional experiments (for example, low-inertia

• the methods from the second basket have extremely limited application range as they require the same starter to be used as well as the minor changes to the

During the starting engine speeds up from either turned off state to an idle running or windmilling to the flight idle. Based on the source of power that drives the rotor during the acceleration, the whole starting may be decomposed into three

The rotor of the engine is dynamically balanced. During the first phase, the rotor is driven by the starter only, as the combustion chamber is not lighted yet. During this phase, the starter drives the rotor from turned off state to the RPM where the

where Mres is the total torque of resistance that is generated by compressor and turbine cascades of the rotor, by bearings and driven auxiliary units (fuel and oil

As soon as combustion chamber is brought into a game, the turbine starts giving the positive input to the torque balance equation. Both turbine and starter drive the rotor because the turbine does not have enough power yet to accelerate the rotor

M*starter* ¼ Mres þ M*inert*, (11)

• most described methods can hardly find the usage in an every day's

polynomial change [14].

operation;

phases.

**16**

engineering practice;

operational modes;

thermocouple, strain gage, etc.);

starting control loop are allowed.

pumps, electric generator, deaerator, etc.).

**2. Requirements to the starting simulation model**

combustion chamber is started. The torque balance of this phase is

of the gas turbine, one can conclude the following

*Modeling of Turbomachines for Control and Diagnostic Applications*

alone. The second phase lasts till the turbine reaches the power where it can drive the compressor and accelerate the rotor on its own. The torque equation for the second phase is

$$\mathbf{M}\_{\text{star}ter} + \mathbf{M}\_{\text{T}} = \mathbf{M}\_{\text{res}} + \mathbf{M}\_{\text{inert}}.\tag{12}$$

The starting ends up with the phase where the starter is moved from the table, making the turbine to be the only one source of power that drives the compressor and makes all rotors accelerate to the idle or the flight idle (see **Figure 1**). The torque balance equation is

$$\mathbf{M}\_{\rm T} = \mathbf{M}\_{\rm res} + \mathbf{M}\_{\rm inert}.\tag{13}$$

The first and foremost problem that every simulation engineer faces when designing the model is the problem of the choice among the available model structures, as the selected structure must make a perfect fit to the problem to be solved by the model. The major goal of the proposed model is to perform the preliminary tuning of the ACS. For this, the model must:


**Figure 1.** *Engine starting stages and moment performances.*

### **3. Structure**

Form the vast majority of model structures, the good choice to fulfill the requirements described above is the dynamic model consisting of linear dynamic model,

$$\begin{cases} \dot{\overrightarrow{\mathbf{X}}} = \mathbf{A} \cdot \Delta \overrightarrow{\mathbf{X}} + \mathbf{B} \cdot \Delta \overrightarrow{\mathbf{U}};\\ \Delta \overrightarrow{\mathbf{Y}} = \mathbf{C} \cdot \Delta \overrightarrow{\mathbf{Y}} + \mathbf{D} \cdot \Delta \overrightarrow{\mathbf{U}}. \end{cases} \tag{14}$$

The static performance is extrapolated to the sub-idle region with a linear region and parabolic region (linking region). The performance is described with the func-

• values of the functions must be equal on the left and on the right;

Initial values of slope angle are evaluated from the next formula

• first derivative of the functions must be equal on the left and on the right.

Such a structure allows alternating the slope of the linear region within some range (see line 1 in **Figure 3**). This is a good opportunity to adjust the SSM after the

<sup>k</sup> <sup>¼</sup> ð Þ <sup>0</sup>*:*<sup>6</sup> … <sup>1</sup>*:*<sup>2</sup> Z2 � Z0

The 0.6–1.2 range determines the adjustment range of the linear region. If k = 1, then linear region matches with the line joining two points (0, Z0), (N2, Z2). Y0 is equal to a default value of a parameter to be simulated (temperature, pressure, RPM, etc.).

<sup>Z</sup>ð Þ<sup>2</sup> <sup>¼</sup> k0 <sup>þ</sup> k1N <sup>þ</sup> k2N2

Unknown coefficients k0, k1, and k2 are evaluated from the set of linear equations that is compiled from the limitations for the regions, described in this chapter:

Z0 <sup>þ</sup> kN1 <sup>¼</sup> k0 <sup>þ</sup> k1N1 <sup>þ</sup> k2N<sup>2</sup>

k0 <sup>þ</sup> k1N2 <sup>þ</sup> k2N<sup>2</sup>

k1 <sup>þ</sup> 2k2N2 <sup>¼</sup> dZ

dN 

2

N2

<sup>Z</sup>ð Þ<sup>1</sup> <sup>¼</sup> Z0 <sup>þ</sup> kN*:* (15)

k ¼ k1 þ 2k2N1; (19)

*:* (16)

*:* (17)

1; (18)

<sup>2</sup> ¼ Z2; (20)

*:* (21)

tions. They are limited on both sides of the region as

*Static model generation: 1—linear segment; 2—parabola; 3—working line.*

experiments have been carried out.

*Turbine Engine Starting Simulation*

*DOI: http://dx.doi.org/10.5772/intechopen.90803*

**Figure 3.**

**19**

Equation of linear region is described as

The linking function is described as

supported with steady-state model in simplified representation (next simplified static model, SSM).

The structure of the model is presented in a **Figure 2**.

The model of the three-spool turbofan must consider the mutual interaction of engine spools.

**Figure 2.** *Structure of the proposed starting model of a turbofan.*

## **4. Compiling of the SSM**

Static performance of GTE in a sub-idle region is nothing more than an abstraction, as in a sub-idle region, the engine does not have any steady modes. The SMM improves the precision of the starting model as well as provides the unique architecture of both sub-idle and above idle models. Let the SSM be formalized as Z! ¼ f N � �, where Z! ¼ X !<sup>T</sup> Y !<sup>T</sup> � �<sup>T</sup> ; <sup>N</sup> <sup>¼</sup> NHP cor NHP 0 is the base value that corresponds to an idle

mode.

The proposed scheme of SMM for one of parameters Y from vector in the subidle region is presented in **Figure 3**.

*Turbine Engine Starting Simulation DOI: http://dx.doi.org/10.5772/intechopen.90803*

**3. Structure**

engine spools.

simplified static model, SSM).

**4. Compiling of the SSM**

*Structure of the proposed starting model of a turbofan.*

¼ X !<sup>T</sup> Y !<sup>T</sup> � �<sup>T</sup>

idle region is presented in **Figure 3**.

f N

**18**

mode.

**Figure 2.**

� �, where Z!

model,

Form the vast majority of model structures, the good choice to fulfill the requirements described above is the dynamic model consisting of linear dynamic

> ¼ A � ΔX !

> > ¼ C � ΔY !

supported with steady-state model in simplified representation (next—

The model of the three-spool turbofan must consider the mutual interaction of

Static performance of GTE in a sub-idle region is nothing more than an abstraction, as in a sub-idle region, the engine does not have any steady modes. The SMM improves the precision of the starting model as well as provides the unique architecture of both sub-idle and above idle models. Let the SSM be formalized as Z!

The proposed scheme of SMM for one of parameters Y from vector in the sub-

NHP 0 is the base value that corresponds to an idle

; <sup>N</sup> <sup>¼</sup> NHP cor

þ B � ΔU ! ;

> þ D � ΔU ! *:*

(14)

¼

\_ X !

8 < :

*Modeling of Turbomachines for Control and Diagnostic Applications*

The structure of the model is presented in a **Figure 2**.

ΔY !

**Figure 3.** *Static model generation: 1—linear segment; 2—parabola; 3—working line.*

The static performance is extrapolated to the sub-idle region with a linear region and parabolic region (linking region). The performance is described with the functions. They are limited on both sides of the region as


Such a structure allows alternating the slope of the linear region within some range (see line 1 in **Figure 3**). This is a good opportunity to adjust the SSM after the experiments have been carried out.

Equation of linear region is described as

$$\mathbf{Z}^{(1)} = \mathbf{Z}\_0 + \mathbf{k}\overline{\mathbf{N}}.\tag{15}$$

Initial values of slope angle are evaluated from the next formula

$$\mathbf{k} = (\mathbf{0}.6\dots\mathbf{1}.2)\frac{\mathbf{Z}\_2 - \mathbf{Z}\_0}{\overline{\mathbf{N}}\_2}.\tag{16}$$

The 0.6–1.2 range determines the adjustment range of the linear region. If k = 1, then linear region matches with the line joining two points (0, Z0), (N2, Z2). Y0 is equal to a default value of a parameter to be simulated (temperature, pressure, RPM, etc.).

The linking function is described as

$$\mathbf{Z}^{(2)} = \mathbf{k}\_0 + \mathbf{k}\_1 \overline{\mathbf{N}} + \mathbf{k}\_2 \overline{\mathbf{N}}^2. \tag{17}$$

Unknown coefficients k0, k1, and k2 are evaluated from the set of linear equations that is compiled from the limitations for the regions, described in this chapter:

$$\mathbf{k}\mathbf{Z}\_0 + \mathbf{k}\overline{\mathbf{N}}\_1 = \mathbf{k}\_0 + \mathbf{k}\_1\overline{\mathbf{N}}\_1 + \mathbf{k}\_2\overline{\mathbf{N}}\_1^2;\tag{18}$$

$$\mathbf{k} = \mathbf{k}\_1 + 2\mathbf{k}\_2\mathbf{N}\_1;\tag{19}$$

$$\mathbf{k}\_0 + \mathbf{k}\_1 \overline{\mathbf{N}}\_2 + \mathbf{k}\_2 \overline{\mathbf{N}}\_2^2 = \mathbf{Z}\_2;\tag{20}$$

$$\mathbf{k}\_1 + 2\mathbf{k}\_2 \overline{\mathbf{N}}\_2 = \left(\frac{\mathbf{d}Z}{\mathbf{d}\overline{\mathbf{N}}}\right)\_2. \tag{21}$$

The last unknown to be determined from the equation set (18)–(21) is an abscissa N1 of point joining linear function with the linking function.

Set of Eqs. (18)–(21) is non-linear. Let us use Eqs. (20, 21) to solve it and express the coefficients k0 and k1 to be dependent on k2:

$$\mathbf{k}\_1 = \left(\frac{\mathrm{dZ}}{\mathrm{d\overline{N}}}\right)\_2 - 2\mathbf{k}\_2\overline{\mathbf{N}}\_2;\ \mathbf{k}\_0 = \mathbf{Z}\_2 - \left(\frac{\mathrm{dZ}}{\mathrm{d\overline{N}}}\right)\_2\overline{\mathbf{N}}\_2 + \mathbf{k}\_2\overline{\mathbf{N}}\_2^2. \tag{22}$$

From Eq. (19), we get the relation between N1 and k2:

$$
\overline{\mathbf{N}}\_1 = \frac{\mathbf{k} - \mathbf{D} + 2\mathbf{k}\_2\mathbf{N}\_2}{2\mathbf{k}\_2}.\tag{23}
$$

where Т2 cor is the temperature of airflow at compressor discharge; Wf cor�Hu

combustion chamber.

*Static model of the gas temperature.*

*Turbine Engine Starting Simulation*

*DOI: http://dx.doi.org/10.5772/intechopen.90803*

**Figure 5.**

switched in.

differential equations.

8 >>>>>><

>>>>>>:

assumptions:

**21**

above-idle operation model, respectively.

of the rotational speed: Wair cor <sup>¼</sup> <sup>C</sup> � N2

compile the SSMs of Т2 cor and airflow Wair cor .

**5. Identification of a linear dynamic model**

temperature rise caused by the amount of heat added by the fuel combustion in the

A coefficient KT considers the temperature drop that is caused by heat retraction in a high-pressure turbine and a free turbine. The coefficient is calculated to provide the equality of the exhaust gas temperatures. The first and the second values of the exhaust-gas temperature are calculated using the sub-idle operation model and the

The airflow through the engine is a priori known to be proportional to the square

HP. The dependence Т4 cor ¼ f Nð Þ HP cor is shown in **Figure 5**. The left branch (1) corresponds to the Stage 1 of starting (see **Figure 1**) when only starter rotates the rotor; the right branch presents Stages 2 and 3 when the combustion chamber is

It follows from Eq. (10) that for proper modeling of gas temperature on must

The above dynamic model (1) in the state space for the three-spool turbofan

<sup>N</sup>\_ HP <sup>¼</sup> a21 � <sup>Δ</sup>NIP <sup>þ</sup> a22 � <sup>Δ</sup>NHP <sup>þ</sup> a23 � <sup>Δ</sup>NLP <sup>þ</sup> b21 � <sup>Δ</sup>Wf <sup>þ</sup> b24 � ð Þ <sup>М</sup>*start* � <sup>М</sup>res ;

where ΔNIP ¼ NIP � NIP st, ΔNHP ¼ NHP � NHP st, ΔNLP ¼ NLP � NLP st,

The set of Eqs. (27) can be simplified by the introduction of the listed next

engine under consideration has the form of a system of linear algebraic and

<sup>N</sup>\_ IP <sup>¼</sup> a11 � <sup>Δ</sup>NIP <sup>þ</sup> a12 � <sup>Δ</sup>NHP <sup>þ</sup> a32 � <sup>Δ</sup>NLP <sup>þ</sup> b11 � <sup>Δ</sup>Wf;

<sup>N</sup>\_ LP <sup>¼</sup> a31 � <sup>Δ</sup>NIP <sup>þ</sup> a32 � <sup>Δ</sup>NHP <sup>þ</sup> a33 � <sup>Δ</sup>NLP <sup>þ</sup> b31 � <sup>Δ</sup>Wf; Δp2 ¼ c11 � ΔNIP þ c12 � ΔNHP þ с<sup>13</sup> � ΔNLP þ d11 � ΔWf; ΔТ<sup>4</sup> ¼ c21 � ΔNIP þ c22 � ΔNHP þ с<sup>23</sup> � ΔNLP þ d21 � ΔWf,

ΔWf ¼ Wf � Wf st, Δp2 ¼ p2 � p2 st and ΔТ<sup>4</sup> ¼ Т<sup>4</sup> � Т4 st. This system can be resolved by the Euler method.

Wair cor�с<sup>p</sup> is the

(27)

Putting the obtained equations into (18), we get the solution:

$$\mathbf{k}\_{2} = \frac{\left(\mathbf{k} - \left(\frac{d\mathbf{Z}}{d\overline{\mathbf{N}}}\right)\_{2}\right)^{2}}{4\left(\mathbf{Z}\_{2} - \mathbf{Z}\_{0} - \mathbf{k}\_{2}\overline{\mathbf{N}}\_{2}\right)};\\\mathbf{k}\_{1} = \left(\frac{d\mathbf{Z}}{d\overline{\mathbf{N}}}\right)\_{2} - \frac{\left(\mathbf{k} - \left(\frac{d\mathbf{Z}}{d\overline{\mathbf{N}}}\right)\_{2}\right)^{2}\overline{\mathbf{N}}\_{2}}{2\left(\mathbf{Z}\_{2} - \mathbf{Z}\_{0} - \mathbf{k}\_{2}\overline{\mathbf{N}}\_{2}\right)};\tag{24}$$

$$\mathbf{k}\_{0} = \mathbf{Z}\_{2} - \left(\frac{\mathbf{d}\mathbf{Z}}{\mathbf{d}\overline{\mathbf{N}}}\right)\_{2}\overline{\mathbf{N}}\_{2} + \frac{\left(\mathbf{k} - \left(\frac{\mathbf{d}\mathbf{Z}}{\mathbf{d}\overline{\mathbf{N}}}\right)\_{2}\right)^{2}\overline{\mathbf{N}}\_{2}^{2}}{4\left(\mathbf{Z}\_{2} - \mathbf{Z}\_{0} - \mathbf{k}\_{2}\overline{\mathbf{N}}\_{2}\right)}; \overline{\mathbf{N}}\_{1} = \frac{2\left(\mathbf{Z}\_{2} - \mathbf{Z}\_{0} - \mathbf{k}\_{2}\overline{\mathbf{N}}\_{2}\right)}{\left(\mathbf{k} - \left(\frac{\mathbf{d}\mathbf{Z}}{\mathbf{d}\overline{\mathbf{N}}}\right)\_{2}\right)} + \overline{\mathbf{N}}\_{2}.\tag{25}$$

Building up any performance, we will require turned-off parameters, idle parameters and derivative of the performance parameter against the mode parameter, e.g. RPM of HPR. An example of this map with experimental points is shown in **Figure 4**.

Because the corrected exhaust gas temperature TT cor has another physical nature, its relation is formed using another approach method.

The turbine temperature decreases to some value when the mode decreases. Further mode decrease results in a drastic temperature increase mainly because of the inefficient turbine operation and low airflow through the engine.

To consider this prior information, the turbine discharge temperature is modeled as follows:

$$\mathbf{T\_{4\text{ cor}} = T\_2 \cdot \text{cor} + K\_\Gamma \frac{\mathbf{W\_{f\text{ cor}} \cdot H\_\mathbf{u}}}{\mathbf{W\_{air\text{ cor}} \cdot \text{c}\_\mathbf{p}}},\tag{26}$$

**Figure 4.** *Relation Wf st cor* ¼ *f N*ð Þ *HP cor of the SSM.*

*Turbine Engine Starting Simulation DOI: http://dx.doi.org/10.5772/intechopen.90803*

**Figure 5.**

The last unknown to be determined from the equation set (18)–(21) is an

Set of Eqs. (18)–(21) is non-linear. Let us use Eqs. (20, 21) to solve it and express

� 2k2N2; k0 <sup>¼</sup> Z2 � dZ

N1 <sup>¼</sup> <sup>k</sup> � <sup>D</sup> <sup>þ</sup> 2k2N2 2k2

> dN

> > N2 2

Building up any performance, we will require turned-off parameters, idle parameters and derivative of the performance parameter against the mode parameter, e.g. RPM of HPR. An example of this map with experimental points is shown in **Figure 4**. Because the corrected exhaust gas temperature TT cor has another physical

The turbine temperature decreases to some value when the mode decreases. Further mode decrease results in a drastic temperature increase mainly because of

To consider this prior information, the turbine discharge temperature is

2 �

; N1 <sup>¼</sup> 2 Z2 � Z0 � k2N2

Wf cor � Hu Wair cor � с<sup>p</sup>

dN 

2

<sup>k</sup> � dZ dN 2

<sup>2</sup>

2 Z2 � Z0 � k2N2

<sup>k</sup> � dZ dN 2 <sup>þ</sup> N2*:* (25)

N2 <sup>þ</sup> k2N<sup>2</sup>

*:* (23)

N2

; (24)

, (26)

<sup>2</sup>*:* (22)

abscissa N1 of point joining linear function with the linking function.

the coefficients k0 and k1 to be dependent on k2:

2

*Modeling of Turbomachines for Control and Diagnostic Applications*

From Eq. (19), we get the relation between N1 and k2:

; k1 <sup>¼</sup> dZ

<sup>k</sup> � dZ dN 2

nature, its relation is formed using another approach method.

the inefficient turbine operation and low airflow through the engine.

Т4 cor ¼ Т2 cor þ KT

<sup>2</sup>

4 Z2 � Z0 � k2N2

Putting the obtained equations into (18), we get the solution:

k1 <sup>¼</sup> dZ dN 

> <sup>k</sup> � dZ dN 2

<sup>2</sup>

4 Z2 � Z0 � k2N2

k2 ¼

k0 <sup>¼</sup> Z2 � dZ

modeled as follows:

**Figure 4.**

**20**

*Relation Wf st cor* ¼ *f N*ð Þ *HP cor of the SSM.*

dN 

2 N2 þ *Static model of the gas temperature.*

where Т2 cor is the temperature of airflow at compressor discharge; Wf cor�Hu Wair cor�с<sup>p</sup> is the temperature rise caused by the amount of heat added by the fuel combustion in the combustion chamber.

A coefficient KT considers the temperature drop that is caused by heat retraction in a high-pressure turbine and a free turbine. The coefficient is calculated to provide the equality of the exhaust gas temperatures. The first and the second values of the exhaust-gas temperature are calculated using the sub-idle operation model and the above-idle operation model, respectively.

The airflow through the engine is a priori known to be proportional to the square of the rotational speed: Wair cor <sup>¼</sup> <sup>C</sup> � N2 HP.

The dependence Т4 cor ¼ f Nð Þ HP cor is shown in **Figure 5**. The left branch (1) corresponds to the Stage 1 of starting (see **Figure 1**) when only starter rotates the rotor; the right branch presents Stages 2 and 3 when the combustion chamber is switched in.

It follows from Eq. (10) that for proper modeling of gas temperature on must compile the SSMs of Т2 cor and airflow Wair cor .

#### **5. Identification of a linear dynamic model**

The above dynamic model (1) in the state space for the three-spool turbofan engine under consideration has the form of a system of linear algebraic and differential equations.

<sup>N</sup>\_ IP <sup>¼</sup> a11 � <sup>Δ</sup>NIP <sup>þ</sup> a12 � <sup>Δ</sup>NHP <sup>þ</sup> a32 � <sup>Δ</sup>NLP <sup>þ</sup> b11 � <sup>Δ</sup>Wf; <sup>N</sup>\_ HP <sup>¼</sup> a21 � <sup>Δ</sup>NIP <sup>þ</sup> a22 � <sup>Δ</sup>NHP <sup>þ</sup> a23 � <sup>Δ</sup>NLP <sup>þ</sup> b21 � <sup>Δ</sup>Wf <sup>þ</sup> b24 � ð Þ <sup>М</sup>*start* � <sup>М</sup>res ; <sup>N</sup>\_ LP <sup>¼</sup> a31 � <sup>Δ</sup>NIP <sup>þ</sup> a32 � <sup>Δ</sup>NHP <sup>þ</sup> a33 � <sup>Δ</sup>NLP <sup>þ</sup> b31 � <sup>Δ</sup>Wf; Δp2 ¼ c11 � ΔNIP þ c12 � ΔNHP þ с<sup>13</sup> � ΔNLP þ d11 � ΔWf; ΔТ<sup>4</sup> ¼ c21 � ΔNIP þ c22 � ΔNHP þ с<sup>23</sup> � ΔNLP þ d21 � ΔWf, 8 >>>>>>< >>>>>>: (27)

where ΔNIP ¼ NIP � NIP st, ΔNHP ¼ NHP � NHP st, ΔNLP ¼ NLP � NLP st, ΔWf ¼ Wf � Wf st, Δp2 ¼ p2 � p2 st and ΔТ<sup>4</sup> ¼ Т<sup>4</sup> � Т4 st.

This system can be resolved by the Euler method.

The set of Eqs. (27) can be simplified by the introduction of the listed next assumptions:


Concluding the just listed assumptions, the set of Eqs. (27) transforms to

$$\begin{cases} \dot{\mathbf{N}\_{\rm IP}} = \mathbf{a}\_{11} \cdot \Delta \mathbf{N}\_{\rm IP} + \mathbf{b}\_{11} \cdot \Delta \mathbf{W}\_{\rm f}; \\ \dot{\mathbf{N}\_{\rm HP}} = \mathbf{b}\_{21} \cdot \Delta \mathbf{W}\_{\rm f} + \mathbf{b}\_{24} \cdot (\mathbf{M}\_{start} - \mathbf{M}\_{res}); \\ \dot{\mathbf{N}\_{\rm LP}} = \mathbf{a}\_{31} \cdot \Delta \mathbf{N}\_{\rm IP} + \mathbf{a}\_{33} \cdot \Delta \mathbf{N}\_{\rm HP} + \mathbf{b}\_{31} \cdot \Delta \mathbf{W}\_{\rm f}; \\ \Delta \mathbf{p}\_{2} = \mathbf{d}\_{11} \cdot \Delta \mathbf{W}\_{\rm f}; \\ \Delta \mathbf{T}\_{4} = \mathbf{d}\_{21} \cdot \Delta \mathbf{W}\_{\rm f}. \end{cases} \tag{28}$$

The task of synthesizing a linear dynamic model is reduced to the problem of determining the coefficients of a linear dynamic model.

The coefficients b24, b21, and d11 are determined analytically using a priori and experimental information, followed by approximation by polynomial dependencies, the arguments of which are the rotational speed of the high-pressure rotor. The coefficient b24 was determined analytically from the following equation:

$$\mathbf{J}\_{\rm HP} \frac{\mathbf{da\_{HP}}}{\mathbf{dt}} = \mathbf{M}\_{\rm start} - \mathbf{M}\_{\rm HPC} \left(\mathbf{N}\_{\rm HP}^2\right), \tag{29}$$

where JHP is an inertia of high pressure rotor. Having transformed <sup>d</sup>ωHP dt to <sup>π</sup> <sup>30</sup> � dNHP dt , we get:

$$\mathbf{b}\_{24} = \frac{\mathbf{30}}{\pi \cdot \mathbf{J}\_{\rm HP}} \tag{30}$$

<sup>N</sup>\_ HP ¼ � <sup>1</sup>

*Coefficients of the high-pressure rotor dynamics.*

*Turbine Engine Starting Simulation*

*DOI: http://dx.doi.org/10.5772/intechopen.90803*

**Figure 6.**

experiences a gap (τHP <sup>¼</sup> <sup>π</sup>

engine operating mode.

<sup>¼</sup> a11

HP cor, ^ θ !

where θ !

<sup>i</sup>¼<sup>0</sup> pi � Ni

P<sup>2</sup>

**23**

infinity to the value at the idle mode.

τHP

ΔNHP þ

<sup>30</sup> �JHP <sup>∂</sup>МHPC <sup>∂</sup>NHP �∂МHP<sup>Т</sup> <sup>∂</sup>NHP

So, for example, the evaluation of coefficients a11 and b11 is.

^ θ !

of which is approximated with polynomials b11 cor <sup>¼</sup> <sup>P</sup><sup>2</sup>

—vector estimate θ

KW τHP

It is known that the time constant at the beginning of the rotor spinning (NHP = 0) is equal to zero. At low rotational speeds, the rotor is unstable; therefore, the value of the time constant is negative. Physically, this can be explained by the fact that the turbine in this mode of operation is inefficient, and the slope of the torque characteristic of the turbine (the dependence of the moment on rotation speed at constant fuel consumption) is greater than the slope of the torque characteristic of the compressor. As rotor speed increases, the time constant decreases until the moment when the slope of the torque characteristic of the compressor becomes equal to the slope of the torque characteristic of the turbine (<sup>∂</sup>МHPC

At this point, the function of the dependence of the time constant on the rotor speed

cannot be stable without an additional energy source or closed-loop control system. With a further increase in the rotational speed, the time constant decreases from

The coefficient KW has physical sense, therefore, the nature of its change is known. The ratio of this coefficient to the time constant shows the effect of changes in fuel consumption on the rate of NHP change. The dependences of the LDM coefficients of the high-pressure rotor on its rotational speed are shown in **Figure 6**. The analytical determination of the coefficients a11, b11, a31, a33, and b31 is impossible; therefore, they are determined by the least squares method engaging the theoretical and experimental information and with the subsequent approximation of the dependence of the obtained values on a parameter that determines the

¼ arg min J θ

! . � �!

b11 � � is a vector of coefficients to be identified (a11, b11), the change

� ΔWf ¼ a22ΔNHP þ b21ΔWf*:* (33)

¼ ∞). Thus, at low rotational speeds, the rotor

; (34)

HP cor and a11 cor ¼

<sup>i</sup>¼<sup>0</sup> qi � Ni

<sup>∂</sup>nHP <sup>¼</sup> <sup>∂</sup>МHPT

<sup>∂</sup>nHP ).

The coefficient b24 does not depend on the type or characteristics of the used starter and is constant for the entire starting.

The determination of the coefficient b21 пр showing the effect of fuel consumption on the rate of the rotational speed change for the high-pressure rotor was carried out analytically according to the following relationship:

$$\mathbf{b}\_{21\text{ corr}} = \frac{\dot{\mathbf{N}}\_{\text{HP}} - \mathbf{b}\_{24} \cdot \mathbf{M}\_{\text{start}}}{\Delta \mathbf{W}\_{\text{f}}}.\tag{31}$$

Further, the obtained values were approximated, considering the following apriori information about this coefficient at the starting region. Let us consider the following equation:

$$
\tau\_{\rm HP} \frac{d\mathbf{N\_{HP}}}{dt} + \Delta \mathbf{N\_{HP}} = \mathbf{K\_W} \cdot \Delta \mathbf{W\_{f}},\tag{32}
$$

where τHP is the time constant of the high-pressure rotor.

Let us divide both parts of this equation by time constant and move ( <sup>1</sup> <sup>τ</sup>HP ΔNHP) to the right part of the equation, and we get

*Turbine Engine Starting Simulation DOI: http://dx.doi.org/10.5772/intechopen.90803*

• as the argument of the static model and functions for determining the dependence of the coefficients of the linear dynamic model on the engine operational mode is N2, then NHP ¼ NHP st which is why ΔNHP ¼ 0.

pressure rotor affects the low-pressure rotor).

*Modeling of Turbomachines for Control and Diagnostic Applications*

8 >>>>>><

>>>>>>:

Δp2 ¼ d11 � ΔWf; ΔТ<sup>4</sup> ¼ d21 � ΔWf*:*

determining the coefficients of a linear dynamic model.

JHP

where JHP is an inertia of high pressure rotor.

starter and is constant for the entire starting.

dt to <sup>π</sup>

Having transformed <sup>d</sup>ωHP

following equation:

**22**

dωHP

carried out analytically according to the following relationship:

τHP

to the right part of the equation, and we get

dNHP

where τHP is the time constant of the high-pressure rotor.

Let us divide both parts of this equation by time constant and move ( <sup>1</sup>

• the mutual effect of the rotors is determined mainly by the interaction of the spools of the turbine; therefore, it spreads along the flow (the high-pressure rotor affects the intermediate and low-pressure rotor, and the intermediate-

Concluding the just listed assumptions, the set of Eqs. (27) transforms to

<sup>N</sup>\_ HP <sup>¼</sup> b21 � <sup>Δ</sup>Wf <sup>þ</sup> b24 � ð Þ <sup>М</sup>*start* � Mres ; <sup>N</sup>\_ LP <sup>¼</sup> a31 � <sup>Δ</sup>NIP <sup>þ</sup> a33 � <sup>Δ</sup>NHP <sup>þ</sup> b31 � <sup>Δ</sup>Wf;

The task of synthesizing a linear dynamic model is reduced to the problem of

The coefficients b24, b21, and d11 are determined analytically using a priori and experimental information, followed by approximation by polynomial dependencies, the arguments of which are the rotational speed of the high-pressure rotor. The

dt <sup>¼</sup> <sup>M</sup>*start* � <sup>М</sup>HPC N2

dt , we get:

b24 <sup>¼</sup> <sup>30</sup> π � JHP

The coefficient b24 does not depend on the type or characteristics of the used

The determination of the coefficient b21 пр showing the effect of fuel consump-

b21 cor <sup>¼</sup> <sup>N</sup>\_ HP � b24 � <sup>M</sup>*start*

Further, the obtained values were approximated, considering the following apriori information about this coefficient at the starting region. Let us consider the

ΔWf

tion on the rate of the rotational speed change for the high-pressure rotor was

HP

dt <sup>þ</sup> <sup>Δ</sup>NHP <sup>¼</sup> KW � <sup>Δ</sup>Wf, (32)

� �, (29)

*:* (31)

(28)

(30)

<sup>τ</sup>HP ΔNHP)

<sup>N</sup>\_ IP <sup>¼</sup> a11 � <sup>Δ</sup>NIP <sup>þ</sup> b11 � <sup>Δ</sup>Wf;

coefficient b24 was determined analytically from the following equation:

<sup>30</sup> � dNHP

**Figure 6.** *Coefficients of the high-pressure rotor dynamics.*

$$\dot{\mathbf{N}}\_{\rm HP} = -\frac{1}{\tau\_{\rm HP}} \Delta \mathbf{N}\_{\rm HP} + \frac{\mathbf{K}\_{\rm W}}{\tau\_{\rm HP}} \cdot \Delta \mathbf{W}\_{\rm f} = \mathbf{a}\_{22} \Delta \mathbf{N}\_{\rm HP} + \mathbf{b}\_{21} \Delta \mathbf{W}\_{\rm f}.\tag{33}$$

It is known that the time constant at the beginning of the rotor spinning (NHP = 0) is equal to zero. At low rotational speeds, the rotor is unstable; therefore, the value of the time constant is negative. Physically, this can be explained by the fact that the turbine in this mode of operation is inefficient, and the slope of the torque characteristic of the turbine (the dependence of the moment on rotation speed at constant fuel consumption) is greater than the slope of the torque characteristic of the compressor. As rotor speed increases, the time constant decreases until the moment when the slope of the torque characteristic of the compressor becomes equal to the slope of the torque characteristic of the turbine (<sup>∂</sup>МHPC <sup>∂</sup>nHP <sup>¼</sup> <sup>∂</sup>МHPT <sup>∂</sup>nHP ). At this point, the function of the dependence of the time constant on the rotor speed experiences a gap (τHP <sup>¼</sup> <sup>π</sup> <sup>30</sup> �JHP <sup>∂</sup>МHPC <sup>∂</sup>NHP �∂МHP<sup>Т</sup> <sup>∂</sup>NHP ¼ ∞). Thus, at low rotational speeds, the rotor cannot be stable without an additional energy source or closed-loop control system. With a further increase in the rotational speed, the time constant decreases from infinity to the value at the idle mode.

The coefficient KW has physical sense, therefore, the nature of its change is known. The ratio of this coefficient to the time constant shows the effect of changes in fuel consumption on the rate of NHP change. The dependences of the LDM coefficients of the high-pressure rotor on its rotational speed are shown in **Figure 6**.

The analytical determination of the coefficients a11, b11, a31, a33, and b31 is impossible; therefore, they are determined by the least squares method engaging the theoretical and experimental information and with the subsequent approximation of the dependence of the obtained values on a parameter that determines the engine operating mode.

So, for example, the evaluation of coefficients a11 and b11 is.

$$
\stackrel{\frown}{\theta}^{\hat{}} = \text{arg min } \mathsf{J}\left(\stackrel{\frown}{\theta}\right);\tag{34}
$$

where θ ! <sup>¼</sup> a11 b11 � � is a vector of coefficients to be identified (a11, b11), the change of which is approximated with polynomials b11 cor <sup>¼</sup> <sup>P</sup><sup>2</sup> <sup>i</sup>¼<sup>0</sup> qi � Ni HP cor and a11 cor ¼ P<sup>2</sup> <sup>i</sup>¼<sup>0</sup> pi � Ni HP cor, ^ θ ! —vector estimate θ ! .

**Figure 7.** *Relation of coefficient a11 on operation mode.*

Thus, the problem of the evaluation of coefficients a11, b11 reduces to the problem of determining the coefficients of polynomials approximating the change of these coefficients in the sub-idle region:

$$\begin{split} \mathbf{J}\left(\overset{\rightarrow}{\theta}\right) &= \mathbf{J}\left(\mathbf{q}\_{2}, \mathbf{q}\_{1}, \mathbf{q}\_{0}, \mathbf{p}\_{2}, \mathbf{p}\_{1}, \mathbf{p}\_{0}\right) \\ &= \sum\_{j=1}^{N} \left[\dot{\mathbf{N}}\_{\text{IP}\text{ cor }j} - \left\{ \left(\mathbf{q}\_{2}\mathbf{N}^{2}\mathbf{\_{IP}\text{ cor }j} + \mathbf{q}\_{1}\mathbf{N}\_{\text{HP}\text{ cor }j} + \mathbf{q}\_{0}\right) \Delta \mathbf{W}\_{\text{fc or }j} \right.\right. \\ &\left. + \left(\mathbf{p}\_{2}\mathbf{N}^{2}\mathbf{\_{HP}\text{ cor }j} + \mathbf{p}\_{1}\mathbf{N}\_{\text{HP}\text{ cor }j} + \mathbf{p}\_{0}\right) \Delta \mathbf{N}\_{\text{IP}\text{ cor }j} \right]^{2}. \tag{35} \end{split}$$

Unfortunately, the performance of this experiment is notably rare. Hence, the

, (39)

Fortunately, this constant can be estimated using the one-time special analysis of the experimental data. The time constant can be identified for the sub-idle modes by substituting TT in Eq. (38) using the last equation of the

> <sup>ε</sup> <sup>¼</sup> c21 � <sup>Δ</sup>NIP <sup>þ</sup> c22 � <sup>Δ</sup>NHP <sup>þ</sup> <sup>с</sup><sup>23</sup> � <sup>Δ</sup>NLP <sup>þ</sup> d21 � <sup>Δ</sup>Wf � TTC dTTC dt

dt and TTC are determined from experimental data.

The resultant time constant of thermocouple as a function of corrected rotation

Starting simulation algorithm implements the structure, which is presented in

1. Initial values of parameters are set as NHP cor ¼ NIP cor ¼ NLP cor ¼ 0, Wf ¼ 0,

2.Knowing NHP cor, P*in*, T*in* (in first step, the initial values are used) and applying the static model (Wf st cor ¼ f Nð Þ HP cor , NIP st cor ¼ f Nð Þ HP cor , NLP st cor ¼ f Nð Þ HP cor , P2 st cor ¼ f Nð Þ HP cor , T2 st cor ¼ f Nð Þ HP cor , Wair st cor ¼ f Nð Þ HP cor , Т4 st cor ¼ f Nð Þ HP cor ), static values of the simulated parameters are calculated:

3.Physical values of the parameters Wf st, NIP st, NLP st, P2 st, T2 st, Wair st, Т4 st

4.Knowing NHP cor, P*in*, T*in* and using the relations a11 cor ¼ f Nð Þ HP cor , b11 cor ¼ f Nð Þ HP cor , b21 cor ¼ f Nð Þ HP cor , a31 cor ¼ f Nð Þ HP cor , a33 cor ¼ f Nð Þ HP cor , b31 cor ¼ f Nð Þ HP cor , d11 cor ¼ f Nð Þ HP cor , d21 cor ¼ f Nð Þ HP cor , the coefficients of the linear

Wf st cor, NIP st cor, NLP st cor, P2 st cor, T2 st cor, Wair st cor, Т4 st cor.

time constant is generally unknown.

*Turbine Engine Starting Simulation*

*DOI: http://dx.doi.org/10.5772/intechopen.90803*

*Estimated time constant of thermocouple as a function of rotation speed.*

system (27):

**Figure 8.**

where dTTC

**25**

speed is shown in **Figure 8**.

**7. Starting simulation algorithm**

**Figure 1**, as the following sequence of stages:

P2 ¼ P*in*, T4 ¼ Т<sup>2</sup> ¼ TTC ¼ T*in*, Wair cor ¼ 0.

are calculated using equations of correction.

As an example, **Figure 7** represents the results of a11 approximation on the engine operational mode. They correspond to the obtained dependence

$$\mathbf{a}\_{11\text{ cor}} = -4.0276 \cdot 10^{-9} \text{N}^2\_{\text{HP cor}} - 3.0863 \cdot 10^{-5} \text{N}\_{\text{HP cor}} + 0.0716.\tag{36}$$

It is known that the coefficient smoothly increases when the rotational speed goes down; therefore, the results of its identification are reliable.

For a more adequate model, the coefficient is refined according to the following relationship:

$$\mathbf{b}\_{11} = \frac{\dot{\mathbf{N}}\_{\text{IP}} \,\exp\left(-\mathbf{a}\_{11} \cdot \left(\mathbf{N}\_{\text{IP}} \,\exp\left(-\mathbf{N}\_{\text{IP}} \,\mathbf{s}\right)\right)\right)}{\mathbf{W}\_{\text{f}} \,\exp\left(-\mathbf{W}\_{\text{f}} \,\mathbf{s}\right)}.\tag{37}$$

#### **6. Model of thermocouple**

Measuring the temperature at the sub-idle modes is challenging because the thermocouple has an extremely high time constant. Estimating the real temperature requires complementing the linear dynamic model with an additional equation

$$
\varepsilon \frac{\mathbf{d} \mathbf{T}\_{\rm TC}}{\mathbf{d} \mathbf{t}} + \mathbf{T}\_{\rm TC} = \mathbf{T}\_{\rm 4},\tag{38}
$$

where ε is the time constant of the thermocouple.

The time constant experimental observation requires mounting two thermocouples at the same station: a regular thermocouple and a quick-response thermocouple. Then, the stored data are analyzed to find the temperature delay in time.

*Turbine Engine Starting Simulation DOI: http://dx.doi.org/10.5772/intechopen.90803*

#### **Figure 8.**

Thus, the problem of the evaluation of coefficients a11, b11 reduces to the problem of determining the coefficients of polynomials approximating the change of

> HP cor j þ q1NHP cor j þ q0 � �ΔWf cor j

> > ��<sup>2</sup>

*:* (35)

*:* (37)

NHP cor j þ 0*:*0716*:* (36)

these coefficients in the sub-idle region:

*Relation of coefficient a11 on operation mode.*

<sup>¼</sup> <sup>X</sup> N

j¼1

<sup>þ</sup> p2N<sup>2</sup>

a11 cor ¼ �4*:*<sup>0276</sup> � <sup>10</sup>�9N2

**6. Model of thermocouple**

¼ J q2, q1, q0, p2, p1, p0 � �

�

*Modeling of Turbomachines for Control and Diagnostic Applications*

HP cor j þ p1NHP cor j þ p0 � �ΔNIP cor j

engine operational mode. They correspond to the obtained dependence

goes down; therefore, the results of its identification are reliable.

ε dTTC

where ε is the time constant of the thermocouple.

As an example, **Figure 7** represents the results of a11 approximation on the

HP cor j � <sup>3</sup>*:*<sup>0863</sup> � <sup>10</sup>�<sup>5</sup>

It is known that the coefficient smoothly increases when the rotational speed

b11 <sup>¼</sup> <sup>N</sup>\_ IP exp � a11 � NIP exp � NIP st

Measuring the temperature at the sub-idle modes is challenging because the thermocouple has an extremely high time constant. Estimating the real

temperature requires complementing the linear dynamic model with an additional

The time constant experimental observation requires mounting two thermocouples at the same station: a regular thermocouple and a quick-response thermocouple. Then, the stored data are analyzed to find the temperature delay in time.

For a more adequate model, the coefficient is refined according to the following

Wf exp � Wf st

� �

dt <sup>þ</sup> TTC <sup>¼</sup> T4, (38)

N\_ IP cor j � � q2N<sup>2</sup>

J θ � �!

**Figure 7.**

relationship:

equation

**24**

*Estimated time constant of thermocouple as a function of rotation speed.*

Unfortunately, the performance of this experiment is notably rare. Hence, the time constant is generally unknown.

Fortunately, this constant can be estimated using the one-time special analysis of the experimental data. The time constant can be identified for the sub-idle modes by substituting TT in Eq. (38) using the last equation of the system (27):

$$\varepsilon = \frac{\mathbf{c}\_{21} \cdot \Delta \mathbf{N}\_{\mathrm{IP}} + \mathbf{c}\_{22} \cdot \Delta \mathbf{N}\_{\mathrm{HP}} + \mathbf{c}\_{23} \cdot \Delta \mathbf{N}\_{\mathrm{LP}} + \mathbf{d}\_{21} \cdot \Delta \mathbf{W}\_{\mathrm{f}} - \mathbf{T}\_{\mathrm{TC}}}{\frac{d \mathbf{T}\_{\mathrm{TC}}}{d \mathbf{t}}},\tag{39}$$

where dTTC dt and TTC are determined from experimental data.

The resultant time constant of thermocouple as a function of corrected rotation speed is shown in **Figure 8**.

## **7. Starting simulation algorithm**

Starting simulation algorithm implements the structure, which is presented in **Figure 1**, as the following sequence of stages:


dynamic model are determined for a current step of the solution procedure: a11 cor, b11 cor, b21 cor, a31 cor, a33 cor, b31 cor, d11 cor, d21 cor.


<sup>Δ</sup><sup>U</sup> <sup>¼</sup> <sup>Δ</sup>Wf <sup>¼</sup> Wf‐Wf st; <sup>Δ</sup>x1 <sup>¼</sup> <sup>Δ</sup>NIP <sup>¼</sup> NIP‐NIP st; <sup>Δ</sup>x2 <sup>¼</sup> <sup>Δ</sup>NHP <sup>¼</sup> 0;Δx3 <sup>¼</sup> <sup>Δ</sup>NLP <sup>¼</sup> NLP‐NLP st.


$$\begin{cases} \mathbf{N\_{IP}} \mathbf{i} + \mathbf{1} = \mathbf{N\_{IP}} \cdot \mathbf{i} + \mathbf{t\_{step}} \cdot (\mathbf{a\_{11}} \cdot \Delta \mathbf{N\_{IP}} + \mathbf{b\_{11}} \cdot \Delta \mathbf{W\_{f}}); \\\\ \mathbf{N\_{HP}} \mathbf{i} + \mathbf{1} = \mathbf{N\_{HP}} \cdot \mathbf{i} + \mathbf{t\_{step}} \cdot (\mathbf{b\_{21}} \cdot \Delta \mathbf{W\_{f}} + \mathbf{b\_{24}} \cdot \mathbf{M\_{start}}); \\\\ \mathbf{N\_{LP}} \mathbf{i} + \mathbf{1} = \mathbf{N\_{LP}} \mathbf{i} + \mathbf{t\_{step}} \cdot (\mathbf{a\_{31}} \cdot \Delta \mathbf{N\_{LP}} + \mathbf{a\_{33}} \cdot \Delta \mathbf{N\_{LP}} + \mathbf{b\_{31}} \cdot \Delta \mathbf{W\_{f}}); \\\\ \mathbf{P\_{2i+1}} = \mathbf{P\_{2st}} \mathbf{i} + \mathbf{1} + \mathbf{d\_{11}} \cdot \Delta \mathbf{W\_{f}}; \\\\ \mathbf{T\_{4i+1}} = \mathbf{T\_{4t}} \mathbf{i} + \mathbf{1} + \mathbf{d\_{21}} \cdot \Delta \mathbf{W\_{f}}; \\\\ \mathbf{T\_{TC}} \mathbf{i+1} = \mathbf{T\_{TC}} \mathbf{i+1} + \frac{\mathbf{t\_{step}}}{\mathbf{e}} (\mathbf{T\_{4i+1}} \cdot \mathbf{T\_{TC}}). \end{cases} \tag{40}$$

9.NHP cor is determined on known NHP.

Points 2–9 of this algorithm are repeated up to the end of the starting sequence diagram.

> fact that the real combustion chamber ignition was some earlier than it is declared in the technical requirements to ACS, and the model implements these

The section discussed a new method for the synthesis of the mathematical model of engine operation in the sub-idle region. As shown, this model belongs to the class of dynamic models. To ensure the unity of the structure with the model of operating modes, a simplified static model is introduced into the dynamic model, which takes into account the synthesized characteristics of the engine components in the format of direct dependencies between the parameter-argument and the parameters used

requirements.

*Turbine Engine Starting Simulation*

*DOI: http://dx.doi.org/10.5772/intechopen.90803*

**Figure 9.** *Starting diagrams.*

**27**

**9. Conclusion**

for the synthesis of ACS.

### **8. Starting model verification**

For adequacy of the model and its quality checking, the simulation results were compared with experimental data. For this purpose, the starting process was chosen, which was not used for the model identification and differs from other experiments by a law of the fuel supplying and by ambient conditions. The experimental fuel flow (see **Figure 9a**) was inputted to the model. The time moments of the combustion ignition and the starter switching off are the same as at experiments, which were used for the model identification. The results are shown in **Figure 9b–f**.

Comparison of the simulation results with experimental data shows that the model meets all demands to models, which are used for the ACS development. Visible difference between experimental and simulated data is explained by the

## *Turbine Engine Starting Simulation DOI: http://dx.doi.org/10.5772/intechopen.90803*

dynamic model are determined for a current step of the solution procedure:

5.Using the equations of correction, physical values of the LM coefficients are

<sup>Δ</sup>x1 <sup>¼</sup> <sup>Δ</sup>NIP <sup>¼</sup> NIP‐NIP st; <sup>Δ</sup>x2 <sup>¼</sup> <sup>Δ</sup>NHP <sup>¼</sup> 0;Δx3 <sup>¼</sup> <sup>Δ</sup>NLP <sup>¼</sup> NLP‐NLP st.

7.The starter torsion torque Mstart is determined. The model of starter (see **Figure 2**) contains characteristics of the starter, which determines the starter torsion torque as a function of the starter rotation speed Nstart, pressure and temperature of the air at the starter inlet Pair, Tair, and the air pressure at the

starter nozzle vanes PNB. The moment of the starter switching off is

NLP iþ<sup>1</sup> ¼ NLP i þ t*step* � ð Þ a31 � ΔNLP þ a33 � ΔNLP þ b31 � ΔWf ;

<sup>ε</sup> ð Þ T4 iþ<sup>1</sup>‐T<sup>Т</sup>C i *:*

Points 2–9 of this algorithm are repeated up to the end of the starting sequence

For adequacy of the model and its quality checking, the simulation results were compared with experimental data. For this purpose, the starting process was chosen, which was not used for the model identification and differs from other experiments by a law of the fuel supplying and by ambient conditions. The experimental fuel flow (see **Figure 9a**) was inputted to the model. The time moments of the combustion ignition and the starter switching off are the same as at experiments, which were used for the model identification. The results are shown in **Figure 9b–f**. Comparison of the simulation results with experimental data shows that the model meets all demands to models, which are used for the ACS development. Visible difference between experimental and simulated data is explained by the

determined according to the starting sequence diagram. After this moment,

8.Using the quasi-linear dynamic model, new values of the simulated parameters

(see Eq. (27)) for the current step of the solution procedure

(40)

a11 cor, b11 cor, b21 cor, a31 cor, a33 cor, b31 cor, d11 cor, d21 cor.

*Modeling of Turbomachines for Control and Diagnostic Applications*

determined: a11, b11, b21, a31, a33, b31, d11, d21.

!

NHP, NIP, NLP, P2, T4, TTC are calculated:

P2 iþ<sup>1</sup> ¼ P2 st iþ<sup>1</sup> þ d11 � ΔWf; Т4 iþ<sup>1</sup> ¼ Т4 st iþ<sup>1</sup> þ d21 � ΔWf;

9.NHP cor is determined on known NHP.

T<sup>Т</sup>C iþ<sup>1</sup> ¼ T<sup>Т</sup>C iþ<sup>1</sup> þ

**8. Starting model verification**

NIP iþ<sup>1</sup> ¼ NIP i þ t*step* � ð Þ a11 � ΔNIP þ b11 � ΔWf ;

t*step*

NHP iþ<sup>1</sup> ¼ NHP i þ t*step* � ð Þ b21 � ΔWf þ b24 � М*start* ;

6.Deviations ΔU, ΔX

<sup>Δ</sup><sup>U</sup> <sup>¼</sup> <sup>Δ</sup>Wf <sup>¼</sup> Wf‐Wf st;

Mstart is set to be zero.

8

>>>>>>>>>>>>>><

>>>>>>>>>>>>>>:

diagram.

**26**

are calculated:

fact that the real combustion chamber ignition was some earlier than it is declared in the technical requirements to ACS, and the model implements these requirements.

## **9. Conclusion**

The section discussed a new method for the synthesis of the mathematical model of engine operation in the sub-idle region. As shown, this model belongs to the class of dynamic models. To ensure the unity of the structure with the model of operating modes, a simplified static model is introduced into the dynamic model, which takes into account the synthesized characteristics of the engine components in the format of direct dependencies between the parameter-argument and the parameters used for the synthesis of ACS.

A mathematical model of the thermocouple was also presented and it was proved to become an integral part of the starting model, due to the large inertia of temperature sensors in the sub-idle.

The starting model can be used in the development of control laws for the starting, dead cranking, and in-flight starting.

## **Acknowledgements**

This work has been carried out with the support of the Ministry of Education and Science of Ukraine (research project no. D203-3/2019-П).

**Author details**

**29**

Sergiy Yepifanov\* and Feliks Sirenko

*Turbine Engine Starting Simulation*

*DOI: http://dx.doi.org/10.5772/intechopen.90803*

provided the original work is properly cited.

\*Address all correspondence to: aedlab@gmail.com

National Aerospace University "Kharkiv Aviation Institute", Kharkiv, Ukraine

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

## **Nomenclature**


## **Indexes**


*Turbine Engine Starting Simulation DOI: http://dx.doi.org/10.5772/intechopen.90803*

A mathematical model of the thermocouple was also presented and it was proved to become an integral part of the starting model, due to the large inertia of

The starting model can be used in the development of control laws for the

This work has been carried out with the support of the Ministry of Education

and Science of Ukraine (research project no. D203-3/2019-П).

*Modeling of Turbomachines for Control and Diagnostic Applications*

cp specific heat capacity at constant pressure condition

temperature sensors in the sub-idle.

**Acknowledgements**

**Nomenclature**

J moment of inertia L specific work M torque

ε thermocouple time constant

N rotation speed P pressure Pow power T temperature W air (gas) flow rate Δ absolute deviation

η efficiency π pressure ratio τ engine time constant

C compressor

cor corrected parameter

HP high-pressure cascade

LP low-pressure cascade

IP intermediate-pressure cascade

**Indexes**

**28**

a air

f fuel

in inlet

res resistance st static T turbine TC thermocouple

starting, dead cranking, and in-flight starting.

## **Author details**

Sergiy Yepifanov\* and Feliks Sirenko National Aerospace University "Kharkiv Aviation Institute", Kharkiv, Ukraine

\*Address all correspondence to: aedlab@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Gaudet SR, Gauthier JED. A simple sub-idle component map extrapolation method. In: ASME Paper No. GT2007– 27193. 2007. 9 p

[2] Sexton WR. A method to control turbofan engine starting by varying compressor surge valve bleed [thesis]. Virginia: Virginia Polytechnic Institute and State University; 2001

[3] Bahlawan H, Morini M, Pinelli M, Ruggero Spina P, Venturini M. Development of reliable NARX models of gas turbine cold, warm and hot startup. In: ASME paper GT2017–63332. 2017. 14 p

[4] Kurzke J. Correlations hidden in compressor maps. In: ASME Paper No. GT2011–45519. 2011. 11 p

[5] Walsh PP, Fletcher P. Gas Turbine Performance. Oxford, UK: Blackwell Science; 1998 646 p

[6] Benzer WA, Johnsen IA, Bullock RO. Compressor operation with one or more blade rows stalled. In: NASA Aerodynamic Design of Axial-Flow Compressors. Vol. SP-36. 1965. pp. 341-346

[7] Kim S, Ellis S, Challener M. Real-time engine modeling of a three shafts turbofan engine: from sub-idle to max power rate. In: ASME Paper No. GT2006–90656. 2006. 7 p

[8] Agrawal RK, Yunis M. A generalized mathematical model to estimate gas turbine starting characteristics. Journal of Propulsion and Power. 1982;**104**: 194-201

[9] Kong C, Ki J, Kang M. A new scaling method for component maps of gas turbine using system identification. Journal of Engineering for Gas Turbines and Power. 2003;**125**(4):979-985

[10] Bretschneider S, Reed J. Modeling of start-up from engine-off conditions using high fidelity turbofan engine simulations. In: ASME Paper GT2015– 43528. 2015. 8 p

**Chapter 3**

**Abstract**

frequencies.

**1. Introduction**

**31**

Capacities

Gas Turbine Simulation Taking

into Account Dynamics of Gas

The chapter considers one of the main dynamic factors of the turbine engine the dynamics of gas capacities. Typically, the most influencing capacities in the turbine engine are combustion chamber, afterburner, mixing chamber, secondary duct of turbofan, and jet nozzle. Simulation of high-frequency transients in turbine engines needs taking into account this factor. For the needs of automatic control and parametric diagnostics, the equations of capacities must be combined with the equations of rotor dynamics and, sometimes, with the equations of a measurement system and actuators. The model complexity consists in two features. The first feature is in how many segments are used to simulate the capacity. The second feature is in what of three basic laws are taken into account at the gas motion description: the mass conservation law, the energy conservation law, and the momentum conservation law. This chapter includes the analysis of models of different complexity followed by the conclusions about their applicability. In the last part of the chapter, the real case of the engine dynamics analysis is considered when the designer does not need the simulation of the capacities' dynamics in time, but needs estimating of the capacities' ability to oscillate and in their natural oscillation

*Sergiy Yepifanov and Roman Zelenskyi*

**Keywords:** turbine engine dynamics simulation, gas capacities,

Engine development is known to include numerous stages and, among them, control systems and engine health management systems development. The development of these systems, however, includes conducting much experimental work, which is not a good choice, keeping in mind the cost of tests and time expenses. An alternative choice to be made is the involvement of the mathematical modeling into a development process [1]. One of the topics to be discussed in this chapter deals with the problem of considerable pneumatic volumes of a gas turbine: main and afterburning combustion chambers, bypass, exhaust nozzle, transition ducts, etc. Stationary gas turbines have extra volumes that must be considered when building up the mathematical model: intake with an air purification system and stack with a noise suppression device. In the models of a gas turbine-driven natural gas pumping

differential equations, linearization, Eigen frequency

[11] Ferrand A, Bellenoue M, Bertin Y, Cirligeanu R, Marconi P, Mercier-Calvairac F. High fidelity modeling of the acceleration of a turboshaft engine during a restart. In: ASME Paper GT2018–76654. 2018. 11 p

[12] Jones G, Pilidis P, Curnock B. Extrapolation of compressor characteristics to the low-speed region for sub-idle performance modelling. In: ASME Turbo Expo 2002: Power for Land, Sea, and Air, American Society of Mechanical Engineers. 2002. pp. 861-867

[13] Pourfarzaneh H, Hajilouy-Benisi A, Farshchi M. An analytical model of a gas turbine components performance and its experimental validation. In: ASME Paper No. GT2010–23369. 2010. 6 p

[14] Sukhovii SI, Yepifanov SV, Pavluk EV, Kulik TV. Designexperiment method of GTE starting process linear dynamic simulator forming. In: Aerospace Technique and Technology. Vol. 9/56. Kharkiv: National Aerospace University; 2008. pp. 147-151

## **Chapter 3**

**References**

27193. 2007. 9 p

2017. 14 p

[1] Gaudet SR, Gauthier JED. A simple sub-idle component map extrapolation method. In: ASME Paper No. GT2007–

*Modeling of Turbomachines for Control and Diagnostic Applications*

[10] Bretschneider S, Reed J. Modeling of start-up from engine-off conditions using high fidelity turbofan engine simulations. In: ASME Paper GT2015–

[11] Ferrand A, Bellenoue M, Bertin Y, Cirligeanu R, Marconi P, Mercier-Calvairac F. High fidelity modeling of the acceleration of a turboshaft engine during a restart. In: ASME Paper GT2018–76654. 2018. 11 p

[12] Jones G, Pilidis P, Curnock B. Extrapolation of compressor

Mechanical Engineers. 2002.

[14] Sukhovii SI, Yepifanov SV, Pavluk EV, Kulik TV. Designexperiment method of GTE starting process linear dynamic simulator forming. In: Aerospace Technique and Technology. Vol. 9/56. Kharkiv: National Aerospace University; 2008.

pp. 861-867

pp. 147-151

characteristics to the low-speed region for sub-idle performance modelling. In: ASME Turbo Expo 2002: Power for Land, Sea, and Air, American Society of

[13] Pourfarzaneh H, Hajilouy-Benisi A, Farshchi M. An analytical model of a gas turbine components performance and its experimental validation. In: ASME Paper No. GT2010–23369. 2010. 6 p

43528. 2015. 8 p

[2] Sexton WR. A method to control turbofan engine starting by varying compressor surge valve bleed [thesis]. Virginia: Virginia Polytechnic Institute

[3] Bahlawan H, Morini M, Pinelli M, Ruggero Spina P, Venturini M.

Development of reliable NARX models of gas turbine cold, warm and hot startup. In: ASME paper GT2017–63332.

[4] Kurzke J. Correlations hidden in compressor maps. In: ASME Paper No.

[5] Walsh PP, Fletcher P. Gas Turbine Performance. Oxford, UK: Blackwell

[6] Benzer WA, Johnsen IA, Bullock RO. Compressor operation with one or more

[7] Kim S, Ellis S, Challener M. Real-time engine modeling of a three shafts turbofan engine: from sub-idle to max power rate. In: ASME Paper No. GT2006–90656. 2006. 7 p

[8] Agrawal RK, Yunis M. A generalized mathematical model to estimate gas turbine starting characteristics. Journal of Propulsion and Power. 1982;**104**:

[9] Kong C, Ki J, Kang M. A new scaling method for component maps of gas turbine using system identification. Journal of Engineering for Gas Turbines and Power. 2003;**125**(4):979-985

GT2011–45519. 2011. 11 p

blade rows stalled. In: NASA Aerodynamic Design of Axial-Flow Compressors. Vol. SP-36. 1965.

Science; 1998 646 p

pp. 341-346

194-201

**30**

and State University; 2001
