5. Varying thermal expansion displacements

The dynamics of the displacements caused by temperature loading is described below using a displacement transient performance. To determine this performance, Advanced Nonlinear Modeling of Gas Turbine Dynamics DOI: http://dx.doi.org/10.5772/intechopen.82015

4. Varying boundary conditions

engine operating point.

Aerospace Engineering

kT ¼ ð Þ Ti�THPC = T0

4.1 Boundary temperatures

<sup>i</sup> �T0 ð Þ HPC

4.2 Heat transfer coefficients

As mentioned above, the values of the boundary parameters Ti and α<sup>i</sup> in the sections of the HP surface (see Figure 3 for the case of the disk) are known only for the reference mode. To have the possibility to make the finite element calculation in ANSYS at any mode, we need to know how these parameters vary along with an

Oleynik has shown in his thesis [25] that the distribution of boundary tempera-

<sup>i</sup> � T0 HPC

The similarity coefficient is determined using the gas path temperatures com-

Paper [22] shows that the heat transfer coefficients α<sup>i</sup> change proportionally when an operating mode varies. Using known relations between different criteria of gas flow, this chapter derives the following equation for a similarity coefficient:

> PHPC P0 HPC

As the necessary actual and reference values of gas path variables are known from NDM, the similarity coefficient is simply calculated and the coefficients α<sup>i</sup> at

<sup>α</sup><sup>i</sup> <sup>¼</sup> <sup>k</sup>αα<sup>0</sup>

The next challenging problem was to create the relations for calculating the HP displacements, both temperature induced and force induced, at any engine dynamic operating point. Let us begin from the displacements due to thermal expansion of

The dynamics of the displacements caused by temperature loading is described below using a displacement transient performance. To determine this performance,

In this way, the distribution of the boundary variables T and <sup>α</sup><sup>i</sup> <sup>¼</sup> <sup>k</sup>αα<sup>0</sup>

simply determined through the NDM gas path variables, namely, HPT rotation speed n, high pressure compressor (HPC) discharge temperature THPC, and HPC

!<sup>0</sup>:<sup>8</sup>

is approximately constant and a current temperature at any

THPC T0 HPC

!�0:<sup>567</sup>

<sup>i</sup> (12)

(11)

<sup>i</sup> can be

� � <sup>þ</sup> THPC (10)

tures around HP at a current operating mode is similar to the distribution at a reference mode. The calculations made with NDM of the engine under analysis also confirm that gas path temperatures proportionally change from one operating point to another [22]. In this way, we can state that a temperature similarity coefficient

section "i" of the HP surface can be expressed through this coefficient by:

Ti <sup>¼</sup> kT � T0

puted by NDM at the reference and actual engine modes.

<sup>k</sup><sup>α</sup> <sup>¼</sup> <sup>α</sup>

5. Varying thermal expansion displacements

the HP surface sections are determined by:

discharge pressures PHPC.

the heated parts.

162

<sup>α</sup><sup>0</sup> <sup>¼</sup> <sup>n</sup> n0 the influence of a step change of boundary temperatures from 293 K (cold disk) to the distribution at the reference mode was simulated in ANSYS. It was found that, in addition to time τ, the displacement also depends on the heat transfer similarity coefficient kα, and the displacement performance was presented as a relative function <sup>u</sup>ð Þ¼ <sup>τ</sup>; <sup>k</sup><sup>α</sup> <sup>u</sup>ð Þ� <sup>τ</sup>;k<sup>α</sup> u0 ust�u0 illustrated by Figure 5.

Figure 6 shows the transient performances of the blade and casing absolute displacements obtained in ANSYS by the same mode. For these heated parts, the influence of the coefficient k<sup>α</sup> is insignificant.

Using the disk as an example, let us now show how to consider its displacement performance in a total process of the ENDM computing. Paper [22] demonstrates that, for each value kα, the corresponding curve in Figure 5 is accurately described by a weighted sum of two exponents and therefore can be presented by:

$$\overline{\mathbf{u}}(\mathbf{r}, \mathbf{k}\_a) = \sum\_{j=1}^{2} \mathbf{k}\_j(\mathbf{k}\_a) \left(\mathbf{1} - \mathbf{e}^{\frac{-\pi}{\mathbf{j}\_j(\mathbf{k}\_a)}}\right), \mathbf{j} = \mathbf{1}, \mathbf{2} \tag{13}$$

Figure 5. Transient performance of a disk displacement.

Figure 6. Transient performance of blade and casing displacements (a—blade, b—casing).

Figure 7.

Weighting coefficients and time constants vs. heat transfer similarity coefficient [21] (a—weighting coefficients, b—time constants).

where kj is a weighting coefficient and Tj is a time constant. For each value of kα, four parameters k1, k2, T1, and T2 were determined. Figure 7 illustrates their dependency on the coefficient kα.

The two mentioned exponents present analytical solutions of linear differential equations that for absolute displacements take a form:

$$\mathbf{T}\_{\mathbf{j}}(\mathbf{k}\_a) \frac{d\mathbf{u}\_{\mathbf{j}}}{d\tau} + \mathbf{u}\_{\mathbf{j}} = \mathbf{u}^0(\mathbf{k}\_a), \mathbf{j} = \mathbf{1}, 2 \tag{14}$$

elasticity change. To solve this problem, paper [22] proposes the concept of an

uF <sup>¼</sup> <sup>u</sup><sup>∘</sup>

uF <sup>¼</sup> <sup>u</sup><sup>∘</sup>

The equivalent temperature te is defined as a temperature of a uniformly heated engine part, which has load-induced displacements equal to the displacements of HP with an actual temperature state and the same mechanical loading. Using the temperature te, the displacement at an actual dynamic point is written for the disk

<sup>F</sup>ð Þ� te

<sup>F</sup>ð Þ� te

the constant mechanical load of the reference mode, but the HP heating conditions are varying and correspond to the actual engine operating point. A function u<sup>∘</sup>

was determined by simulating such hypothetical loading in ANSYS. Figure 8 illustrates the results of the disk displacement simulations. These results are approxi-

As follows from Eqs. (10) and (12), thermal loading on each heated part (disk, blade, and casing) depends on the temperature THPC (temperature of HPC air) and the similarity coefficients kT and kα. As described in Section 4, the radial displacement uF caused by the force depends on the temperature state of HP and, therefore, is related to the thermal loading. Thus, this relation can be written by a function

<sup>F</sup>ð Þ¼ te <sup>0</sup>:<sup>38557</sup> <sup>þ</sup> <sup>8</sup>:<sup>55627</sup> � <sup>10</sup>‐<sup>5</sup> � te <sup>þ</sup> <sup>1</sup>:<sup>83458</sup> � <sup>10</sup>‐<sup>8</sup> � <sup>t</sup>

n n0 � �<sup>2</sup>

> PHPC P0

<sup>F</sup> corresponds to a hypothetical situation when HP is under

HPC ! (18)

2

<sup>e</sup> (19)

(17)

<sup>F</sup>ð Þ te

equivalent temperature.

and blade by:

mated by:

Figure 8.

165

6.1 Equivalent temperature

Advanced Nonlinear Modeling of Gas Turbine Dynamics

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

and for the casing by:

The displacement u<sup>∘</sup>

u∘

6.2 Characteristic temperature

Disk displacement at the reference mode vs. equivalent disk temperature.

The following two equations:

$$\mathbf{u}\_{\mathbf{j}}(\tau, \mathbf{k}\_a) = \frac{\mathbf{T}\_{\mathbf{j}}(\mathbf{k}\_a)}{\mathbf{T}\_{\mathbf{j}}(\mathbf{k}\_a) + \Delta\tau} \mathbf{u}\_{\mathbf{j}}(\tau - \Delta\tau) + \frac{\Delta\tau}{\mathbf{T}\_{\mathbf{j}}(\mathbf{k}\_a) + \Delta\tau} \mathbf{u}^0(\mathbf{k}\_a), \mathbf{j} = \mathbf{1}, 2 \tag{15}$$

are numerical solutions of these equations. Their weighted sum:

$$\mathbf{u}(\boldsymbol{\pi}, \mathbf{k}\_a) = \sum\_{\mathbf{j=1}}^2 \mathbf{k}\_{\mathbf{j}}(\mathbf{k}\_a) \mathbf{u}\_{\mathbf{j}}(\boldsymbol{\pi}, \mathbf{k}\_a), \mathbf{j=1,2} \tag{16}$$

is a final expression to numerically compute the dynamic displacement caused by thermal disk expansion. Eqs. (15) and (16) present final steps in Block 1.3 of the enhanced nonlinear dynamic model (see Figure 1). Through the coefficient kα, the displacement calculation is adapted to an actual dynamic engine operating point. The blade and casing displacement (Blocks 1.3 and 2.3) are computed similarly.
