2. Enhanced nonlinear dynamic model

The structure and operation of the above-mentioned ENDM are illustrated by Figure 1. Modules 1–4 constitute SDCM of an engine component, compressor or turbine. Module 5 presents an initial engine model, which, in conjunction with SDCM, presents the enhanced nonlinear model. Only one component is presented on this scheme for simplicity. The software of ENDM includes SDCMs for all the components where the dynamic clearance effect is significant. The enhanced model has been developed for a low-bypass two-spool turbofan engine of a maneuverable aircraft. All simplified relations for the clearance model were obtained through exhaustive calculations in ANSYS with the solid models of engine stator and rotor hot parts, namely disk, blade, and casing. The below description of the clearance model is given for a high-pressure turbine (HPT) of this engine as a component example.

The initial nonlinear dynamic model includes the dynamics of two engine rotors. The corresponding differential equations are solved through their integration by an iterative procedure. In each step, the corrections to state variables (rotation speeds) are obtained and all engine variables are renewed. The enhanced engine model

Figure 1. Structure of the enhanced nonlinear dynamic model.

conserves this iterative character. At an actual step, the variables computed by NDM are used in SDCM to calculate a new tip clearance and corrections to component performances. The modified performances are employed at the next step. In this way, as with real engine dynamics, ENDM uses component performances that are dynamically changed.

Blocks 1.1–1.4 for the stator and Blocks 2.1–2.4 for the rotor illustrate what happens with heated parts (disk, blade, and casing) and tip clearances when engine operating mode dynamically changes. After the mode change, the gas path variables calculated by NMM (Block 5) begin to vary producing the change of heat exchange boundary conditions outside of the heated parts (Blocks 1.1 and 2.1). Because of heat accumulation or loss, the temperature state (distribution of metal temperature) of HPs begins to change (Blocks 1.2 and 2.2). Elevated temperatures of HP cause its thermal expansion. As the HP temperature state has a delay relative to the engine mode change, the corresponding displacement (Blocks 1.3 and 2.3) varies with a delay as well. The knowledge of the temperature state also allows us to correctly consider the action of forces on the HP radial displacements. As shown in [21], for a disk and blade, significant displacements are caused by a centrifugal force, while a pressure force is the most influencing for a casing. These forces are considered in Blocks 1.4 and 2.4 accordingly. When an engine operating point is changing, the force applied to HP changes as well, and the force-induced displacement reacts immediately. However, the engine mode variation also means the change of the heat exchange boundary conditions resulting in other temperature state, other metal elasticity, and an additional change of the displacement. Thus, the force-induced displacement has a static component that immediately reacts on the engine mode and a dynamic component that reacts with a delay. In this way, the total radial displacements of the surfaces that form a tip clearance have a complex dynamic behavior.

The calculation of a dynamic clearance change (see Module 3), which is necessary to evaluate the change in a component performance, is based on simple relations. An actual dynamic tip clearance δ that is a function of transient time can be expressed through a clearance δ<sup>0</sup> of a cold turbine and total displacements uC, uD, δ<sup>B</sup> of the casing, disk, and blade accordingly, resulting in

$$\delta = \delta\_0 + \mathbf{u}\_{\mathbb{C}} - (\mathbf{u}\_{\mathbb{D}} + \mathbf{u}\_{\mathbb{B}}) \tag{1}$$

where L is a blade length. Figure 2 illustrates this relation.

depending on the construction of a modeled engine and its component.

simulation of the HP displacements is described in the next section.

3. Finite element-based displacement simulation

these conditions and loads.

159

3.1 Thermal boundary conditions and mechanical loads

of a transient process according to an expression:

Advanced Nonlinear Modeling of Gas Turbine Dynamics

DOI: http://dx.doi.org/10.5772/intechopen.82015

Turbine efficiency losses vs. an increasing tip clearance.

Figure 2.

the effect of dynamically varying radial clearances.

Therefore, Module 4 computes the corrected component efficiency at each point

where ηst is the efficiency at an equivalent steady state and k is the coefficient

Since Module 5 employs the corrected efficiencies of all the engine components as input parameters, the engine variables simulated by NDM take into consideration

As can be seen from the above description, the equations of Modules 3 and 4 are

As shown in Figure 1 and described in Section 2, the displacements of the heated parts depend on thermal boundary conditions and mechanical loads applied. Shown in Figure 3, the design scheme of the disk, which is the most complex HP, illustrates

The boundary conditions are set by an external air and gas temperature T and heat transfer coefficient α at the disk surface. Since these conditions considerably vary at different parts of the surface, it is broken down into 24 sections with constant temperatures Ti and coefficients αi. At one engine steady state called a

simple and do not need additional explanations. A nonlinear dynamic model of Module 5 is not simple, but it does not need additional description because this type of gas turbine models is well described in literature. However, Modules 1 and 2 and their blocks were presented in this section in a schematic form necessary for a general understanding of the enhanced NDM. As described before, in these modules, the displacements of the heated parts are calculated. To know how these displacements depend on external factors, the displacements were simulated in ANSYS by creating the solid model of each HP and by applying the finite element method to determine the HP stress-stain state. By multiple numerical experiments of this type, the simplified relations between the displacement and external dynamic factors were formed and included in Modules 1 and 2. This ANSYS-based

<sup>η</sup> <sup>¼</sup> <sup>η</sup>st � <sup>k</sup>Δδ (5)

Since the HPT performance map used in NDM corresponds to the turbine parts completely warmed up at steady states, the necessary dynamic correction of the performance will depend on a difference Δδ between a dynamic clearance δ and a static clearance δst. Let us express the static clearance in the form of Eq. (1) but using static displacements of HPs. This yields:

$$\boldsymbol{\delta}^{\rm st} = \boldsymbol{\delta}\_0 + \mathbf{u}\_{\rm C}^{\rm st} - \left(\mathbf{u}\_{\rm D}^{\rm st} + \mathbf{u}\_{\rm B}^{\rm st}\right) \tag{2}$$

In this way, the dynamic clearance change is written by:

$$
\Delta \mathsf{S} = \left( \mathbf{u}\_{\mathrm{C}} - \mathbf{u}\_{\mathrm{C}}^{\mathrm{st}} \right) - \left( \mathbf{u}\_{\mathrm{D}} - \mathbf{u}\_{\mathrm{D}}^{\mathrm{st}} \right) - \left( \mathbf{u}\_{\mathrm{B}} - \mathbf{u}\_{\mathrm{B}}^{\mathrm{st}} \right) \tag{3}
$$

Using the clearance change Δδ as an input parameter, Module 4 corrects the component efficiency η because it is known that just this performance is affected by an increased clearance. Paper [23] shows that a turbine efficiency loss Δη is linearly dependent on a relative clearance change:

Δδ ¼ Δδ=<sup>L</sup> (4)

Advanced Nonlinear Modeling of Gas Turbine Dynamics DOI: http://dx.doi.org/10.5772/intechopen.82015

#### Figure 2.

conserves this iterative character. At an actual step, the variables computed by NDM are used in SDCM to calculate a new tip clearance and corrections to component performances. The modified performances are employed at the next step. In this way, as with real engine dynamics, ENDM uses component performances that

Blocks 1.1–1.4 for the stator and Blocks 2.1–2.4 for the rotor illustrate what happens with heated parts (disk, blade, and casing) and tip clearances when engine operating mode dynamically changes. After the mode change, the gas path variables calculated by NMM (Block 5) begin to vary producing the change of heat exchange boundary conditions outside of the heated parts (Blocks 1.1 and 2.1). Because of heat accumulation or loss, the temperature state (distribution of metal temperature) of HPs begins to change (Blocks 1.2 and 2.2). Elevated temperatures of HP cause its thermal expansion. As the HP temperature state has a delay relative to the engine mode change, the corresponding displacement (Blocks 1.3 and 2.3) varies with a delay as well. The knowledge of the temperature state also allows us to correctly consider the action of forces on the HP radial displacements. As shown in [21], for a disk and blade, significant displacements are caused by a centrifugal force, while a pressure force is the most influencing for a casing. These forces are considered in Blocks 1.4 and 2.4 accordingly. When an engine operating point is changing, the force applied to HP changes as well, and the force-induced displacement reacts immediately. However, the engine mode variation also means the change of the heat exchange boundary conditions resulting in other temperature state, other metal elasticity, and an additional change of the displacement. Thus, the force-induced displacement has a static component that immediately reacts on the engine mode and a dynamic component that reacts with a delay. In this way, the total radial displacements of the surfaces that form a tip clearance have a complex

The calculation of a dynamic clearance change (see Module 3), which is necessary to evaluate the change in a component performance, is based on simple relations. An actual dynamic tip clearance δ that is a function of transient time can be expressed through a clearance δ<sup>0</sup> of a cold turbine and total displacements uC, uD, δ<sup>B</sup>

Since the HPT performance map used in NDM corresponds to the turbine parts completely warmed up at steady states, the necessary dynamic correction of the performance will depend on a difference Δδ between a dynamic clearance δ and a static clearance δst. Let us express the static clearance in the form of Eq. (1) but

<sup>C</sup> � <sup>u</sup>st

Using the clearance change Δδ as an input parameter, Module 4 corrects the component efficiency η because it is known that just this performance is affected by an increased clearance. Paper [23] shows that a turbine efficiency loss Δη is linearly

<sup>D</sup> <sup>þ</sup> ust B

D � uB � <sup>u</sup>st

<sup>δ</sup>st <sup>¼</sup> <sup>δ</sup><sup>0</sup> <sup>þ</sup> <sup>u</sup>st

C � uD � <sup>u</sup>st

In this way, the dynamic clearance change is written by:

Δδ <sup>¼</sup> uC � <sup>u</sup>st

δ ¼ δ<sup>0</sup> þ uC � ð Þ uD þ uB (1)

(2)

(3)

B

Δδ ¼ Δδ=<sup>L</sup> (4)

of the casing, disk, and blade accordingly, resulting in

using static displacements of HPs. This yields:

dependent on a relative clearance change:

158

are dynamically changed.

Aerospace Engineering

dynamic behavior.

Turbine efficiency losses vs. an increasing tip clearance.

where L is a blade length. Figure 2 illustrates this relation.

Therefore, Module 4 computes the corrected component efficiency at each point of a transient process according to an expression:

$$
\boldsymbol{\eta} = \boldsymbol{\eta}^{\text{st}} - \mathbf{k} \Delta \overline{\boldsymbol{\delta}} \tag{5}
$$

where ηst is the efficiency at an equivalent steady state and k is the coefficient depending on the construction of a modeled engine and its component.

Since Module 5 employs the corrected efficiencies of all the engine components as input parameters, the engine variables simulated by NDM take into consideration the effect of dynamically varying radial clearances.

As can be seen from the above description, the equations of Modules 3 and 4 are simple and do not need additional explanations. A nonlinear dynamic model of Module 5 is not simple, but it does not need additional description because this type of gas turbine models is well described in literature. However, Modules 1 and 2 and their blocks were presented in this section in a schematic form necessary for a general understanding of the enhanced NDM. As described before, in these modules, the displacements of the heated parts are calculated. To know how these displacements depend on external factors, the displacements were simulated in ANSYS by creating the solid model of each HP and by applying the finite element method to determine the HP stress-stain state. By multiple numerical experiments of this type, the simplified relations between the displacement and external dynamic factors were formed and included in Modules 1 and 2. This ANSYS-based simulation of the HP displacements is described in the next section.
