4. Varying boundary conditions

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 engine operating point.

### 4.1 Boundary temperatures

Oleynik has shown in his thesis [25] that the distribution of boundary temperatures 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 kT ¼ ð Þ Ti�THPC = T0 <sup>i</sup> �T0 ð Þ HPC is approximately constant and a current temperature at any section "i" of the HP surface can be expressed through this coefficient by:

$$\mathbf{T\_i = k\_T \cdot \left(T\_i^0 - T\_{\rm HPC}^0\right) + T\_{\rm HPC}} \tag{10}$$

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 func-

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

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

kjð Þ k<sup>α</sup> 1 � e

�τ Tj ð Þ kα

, <sup>j</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup> (13)

ust�u0 illustrated by Figure 5.

uð Þ¼ τ; k<sup>α</sup> ∑

by a weighted sum of two exponents and therefore can be presented by:

2 j¼1

influence of the coefficient k<sup>α</sup> is insignificant.

Advanced Nonlinear Modeling of Gas Turbine Dynamics

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

tion <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

Figure 5.

Figure 6.

163

Transient performance of a disk displacement.

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

The similarity coefficient is determined using the gas path temperatures computed by NDM at the reference and actual engine modes.

#### 4.2 Heat transfer coefficients

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:

$$\mathbf{k}\_{\mathbf{u}} = \frac{\mathbf{a}}{\mathbf{a}^{0}} = \left(\frac{\mathbf{n}}{\mathbf{n}^{0}} \frac{\mathbf{P}\_{\mathrm{HPC}}}{\mathbf{P}\_{\mathrm{HPC}}^{0}}\right)^{0.8} \left(\frac{\mathbf{T}\_{\mathrm{HPC}}}{\mathbf{T}\_{\mathrm{HPC}}^{0}}\right)^{-0.567} \tag{11}$$

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 the HP surface sections are determined by:

$$\alpha\_{\rm i} = \mathbf{k}\_a \alpha\_{\rm i}^0 \tag{12}$$

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> <sup>i</sup> can be simply determined through the NDM gas path variables, namely, HPT rotation speed n, high pressure compressor (HPC) discharge temperature THPC, and HPC discharge pressures PHPC.

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 heated parts.
