6.1 Equivalent temperature

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 and blade by:

$$\mathbf{u}\_{\rm F} = \mathbf{u}\_{\rm F}^{\circ}(\mathbf{t}\_{\rm e}) \cdot \left(\frac{\mathbf{n}}{\mathbf{n}^{0}}\right)^{2} \tag{17}$$

and for the casing by:

where kj is a weighting coefficient and Tj is a time constant. For each value of kα,

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

The two mentioned exponents present analytical solutions of linear differential

<sup>þ</sup> uj <sup>¼</sup> <sup>u</sup><sup>0</sup>ð Þ <sup>k</sup><sup>α</sup> , <sup>j</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup> (14)

Tjð Þþ <sup>k</sup><sup>α</sup> Δτ <sup>u</sup><sup>0</sup>ð Þ <sup>k</sup><sup>α</sup> , <sup>j</sup> <sup>¼</sup> <sup>1</sup>, 2 (15)

kjð Þ k<sup>α</sup> ujð Þ τ; k<sup>α</sup> , j ¼ 1, 2 (16)

four parameters k1, k2, T1, and T2 were determined. Figure 7 illustrates their

dependency on the coefficient kα.

Figure 7.

b—time constants).

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The following two equations:

ujð Þ¼ τ; k<sup>α</sup>

n2

164

equations that for absolute displacements take a form:

Tjð Þ k<sup>α</sup>

6. Varying force-induced displacements

displacement will change along with the load change.

Tjð Þ k<sup>α</sup>

duj dτ

Tjð Þþ <sup>k</sup><sup>α</sup> Δτ ujð Þþ <sup>τ</sup> � Δτ Δτ

are numerical solutions of these equations. Their weighted sum:

2 j¼1

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.

The displacements induced in HPs by mechanical loads can be considered elastic and proportional to the load. For the disk and the blade, the main load is a centrifugal force and the displacements will be proportional to the rotation speed squared

. As the casing is mainly loaded by a pressure force, the displacement will linearly depend on the HPC pressure PHPC. The action of these forces has no delay and the

However, since the elasticity coefficient depends on the HP temperature, the HP

displacement should be simulated regarding this dependency. The temperature distribution within HP is nonuniform and dynamically changes during transient engine operation. For this reason, it will be difficult to directly simulate the

uð Þ¼ τ; k<sup>α</sup> ∑

$$\mathbf{u\_F} = \mathbf{u\_F^\circ(t\_e)} \cdot \left(\frac{\mathbf{P\_{HPC}}}{\mathbf{P\_{HPC}^0}}\right) \tag{18}$$

The displacement u<sup>∘</sup> <sup>F</sup> corresponds to a hypothetical situation when HP is under 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> <sup>F</sup>ð Þ te was determined by simulating such hypothetical loading in ANSYS. Figure 8 illustrates the results of the disk displacement simulations. These results are approximated by:

$$\mathbf{u}\_{\rm F}^{\circ}(\mathbf{t}\_{\rm e}) = \mathbf{0}.38557 + 8.55627 \times 10^{-5} \cdot \mathbf{t}\_{\rm e} + \mathbf{1.83458} \times 10^{-8} \cdot \mathbf{t}\_{\rm e}^{2} \tag{19}$$

#### 6.2 Characteristic temperature

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

Figure 8. Disk displacement at the reference mode vs. equivalent disk temperature.

uF ¼ f Tð Þ HPC; kT; k<sup>α</sup> . The three interrelated arguments make this function complex for realization. Paper [22] proposes the concept of a characteristic temperature to be used as the unique function argument. The characteristic temperature T is defined ~ as a weighted mean of a boundary temperature T.

$$\tilde{\mathbf{T}} = \frac{\int\_{A} \mathbf{T(A)} \alpha(\mathbf{A}) \mathbf{dA}}{\int\_{A} \alpha(\mathbf{A}) \mathbf{dA}} \tag{20}$$

where A is the surface of a heated part.

The characteristic temperature T has an important property that temperatures t ~ of a heated part tend to a value T when heat transfer approaches zero, i.e.: ~

$$\lim\_{\mathbf{k}\_a \to 0} \mathbf{t} = \mathbf{\tilde{T}} \tag{21}$$

simulations. Figure 9 presents the errors of both algorithms for different characteristic temperatures and consequently for different engine operating points. We can see that the original algorithm has significant errors (up to 9%), whereas for the enhanced algorithm, the errors are negligible (within 0.1%). So, the accuracy of the displacement simulation was drastically enhanced despite the simplicity of the

Errors of two force-induced displacement algorithms (scored line: algorithm that considers temperature-dependent elasticity; dashed line: algorithm that uses constant elasticity).

The equivalent temperature te determined in Eq. (24) as a function of T corre- ~ sponds to a completely warmed-up heated part and its final static displacement. Let

conditions have changed, the force-induced displacements will vary dynamically

tejð Þþ <sup>τ</sup> � Δτ Δτ

are also similar to displacement Eqs. (15) and (16) and the same parameters Tj and kj are employed. Using the dynamic value teð Þ τ; k<sup>α</sup> from Eq. (26) as an argu-

induced disk displacement uF from Eq. (17). The blade and casing force-induced

Tjð Þþ k<sup>α</sup> Δτ

t st

kjð Þ k<sup>α</sup> tejð Þ τ; k<sup>α</sup> , j ¼ 1, 2 (26)

<sup>F</sup>ð Þ te is determined from Eq. (19) and a total force-

displacement and the temperature te is practically linear (see Figure 8), their dynamic behavior will be similar. For this reason, the dynamics of te are described using the same displacement transient performances presented in Figures 5 and 6. For the disk, the algorithm to compute te is similar to that described in Section 5 for the thermal expansion displacements. The resulting equations to compute the

<sup>e</sup> <sup>T</sup><sup>~</sup>

. When the boundary

<sup>e</sup> . As the relation between the

<sup>e</sup> ð Þ k<sup>α</sup> , j ¼ 1, 2 (25)

proposed algorithm.

Figure 9.

equivalent temperature:

tejð Þ¼ τ; k<sup>α</sup>

ment, a dynamic displacement u<sup>∘</sup>

and

167

6.4 Dynamic force-induced displacement

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us call this temperature a static equivalent temperature tst

and the temperature te will dynamically approach tst

Tjð Þ k<sup>α</sup> Tjð Þþ k<sup>α</sup> Δτ

teð Þ¼ τ; k<sup>α</sup> ∑

2 j¼1

This property allows us to determine the characteristic temperature through ANSYS simulation of the heated part with an extremely low similarity coefficient kα. It is proven that such simulation yields low errors relatively a direct calculation of T according to Eq. (20). For example, given k ~ <sup>α</sup> ≈10�<sup>3</sup> , the error was 0.01 K.

The characteristic temperature was firstly computed at the reference mode and, with the known value T~ <sup>0</sup> , a temperature coefficient:

$$
\tilde{\Theta} = \frac{\tilde{\mathbf{T}}^0 - \mathbf{T}\_{\mathrm{HPC}}^0}{\mathbf{T}\_{\mathrm{g}}^0 - \mathbf{T}\_{\mathrm{HPC}}^0} \tag{22}
$$

was formed, where Tg denotes a HPT input temperature. Then, it was found that this coefficient does not depend on an operating mode and can be used to determine the characteristic temperature at any mode by a simple relation:

$$\mathbf{\tilde{T}} = \mathbf{\tilde{\Theta}} \cdot \left( \mathbf{T\_{\mathfrak{g}}} - \mathbf{T\_{\mathrm{HPC}}} \right) + \mathbf{T\_{\mathrm{HPC}}} \tag{23}$$

### 6.3 Static force-induced displacement

To determine the relation between the temperatures T and t ~ e, series of simulations in ANSYS have been conducted. For the disk under reference mechanical load, the thermal load parameters THPC, kt, and k<sup>α</sup> are varied and the displacement u0 <sup>F</sup> ¼ f THPC; kt ð Þ ; k<sup>α</sup> was determined for each combination of THPC, kt, and kα. The equivalent temperature te corresponding to each displacement was found from Eq. (19). The characteristic temperature T was calculated according to Eq. (23) ~ using a known value THPC and a gas temperature Tg computed by NDM. By doing so, multiple pairs of te and T values were found. With these data, the relation is ~ between te and T is described by: ~

$$\mathbf{t}\_{\mathbf{e}} = -16.631 + 1.0518 \cdot \mathbf{\bar{T}} - 3.9362 \times 10^{-5} \cdot \mathbf{\bar{T}}^2 \tag{24}$$

Thus, through a consecutive application of Eqs. (23), (24), (19), and (17), we can calculate a force-induced radial displacements of the disk as a function of the gas path variable THPC and Tg computed by NDM. The displacements of this enhanced algorithm as well as the original algorithm that consider constant disk elasticity were estimated by the comparison with the results of ANSYS-based

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#### Figure 9.

uF ¼ f Tð Þ HPC; kT; k<sup>α</sup> . The three interrelated arguments make this function complex for realization. Paper [22] proposes the concept of a characteristic temperature to be used as the unique function argument. The characteristic temperature T is defined ~

Ð

<sup>A</sup> T Að Þαð Þ A dA

The characteristic temperature T has an important property that temperatures t

<sup>A</sup> <sup>α</sup>ð Þ <sup>A</sup> dA (20)

<sup>t</sup> <sup>¼</sup> <sup>T</sup><sup>~</sup> (21)

� � <sup>þ</sup> THPC (23)

, the error was 0.01 K.

~ e, series of simula-

(22)

as a weighted mean of a boundary temperature T.

where A is the surface of a heated part.

<sup>T</sup><sup>~</sup> <sup>¼</sup> Ð

~ of a heated part tend to a value T when heat transfer approaches zero, i.e.:

> lim kα!0

, a temperature coefficient:

<sup>Θ</sup><sup>~</sup> <sup>¼</sup> <sup>T</sup><sup>~</sup> <sup>0</sup> � <sup>T</sup><sup>0</sup>

T0 <sup>g</sup> � T0 HPC

<sup>T</sup><sup>~</sup> <sup>¼</sup> <sup>Θ</sup><sup>~</sup> � Tg � THPC

This property allows us to determine the characteristic temperature through ANSYS simulation of the heated part with an extremely low similarity coefficient kα. It is proven that such simulation yields low errors relatively a direct calculation

The characteristic temperature was firstly computed at the reference mode and,

was formed, where Tg denotes a HPT input temperature. Then, it was found that this coefficient does not depend on an operating mode and can be used to determine

tions in ANSYS have been conducted. For the disk under reference mechanical load, the thermal load parameters THPC, kt, and k<sup>α</sup> are varied and the displacement

<sup>F</sup> ¼ f THPC; kt ð Þ ; k<sup>α</sup> was determined for each combination of THPC, kt, and kα. The equivalent temperature te corresponding to each displacement was found from Eq. (19). The characteristic temperature T was calculated according to Eq. (23) ~ using a known value THPC and a gas temperature Tg computed by NDM. By doing so, multiple pairs of te and T values were found. With these data, the relation is

Thus, through a consecutive application of Eqs. (23), (24), (19), and (17), we can calculate a force-induced radial displacements of the disk as a function of the gas path variable THPC and Tg computed by NDM. The displacements of this enhanced algorithm as well as the original algorithm that consider constant disk elasticity were estimated by the comparison with the results of ANSYS-based

te ¼ �16:<sup>631</sup> <sup>þ</sup> <sup>1</sup>:<sup>0518</sup> � <sup>T</sup><sup>~</sup> � <sup>3</sup>:<sup>9362</sup> � <sup>10</sup>�<sup>5</sup> � <sup>T</sup>~<sup>2</sup> (24)

HPC

~

of T according to Eq. (20). For example, given k ~ <sup>α</sup> ≈10�<sup>3</sup>

the characteristic temperature at any mode by a simple relation:

To determine the relation between the temperatures T and t

~

with the known value T~ <sup>0</sup>

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u0

166

6.3 Static force-induced displacement

between te and T is described by: ~

Errors of two force-induced displacement algorithms (scored line: algorithm that considers temperature-dependent elasticity; dashed line: algorithm that uses constant elasticity).

simulations. Figure 9 presents the errors of both algorithms for different characteristic temperatures and consequently for different engine operating points. We can see that the original algorithm has significant errors (up to 9%), whereas for the enhanced algorithm, the errors are negligible (within 0.1%). So, the accuracy of the displacement simulation was drastically enhanced despite the simplicity of the proposed algorithm.

#### 6.4 Dynamic force-induced displacement

The equivalent temperature te determined in Eq. (24) as a function of T corre- ~ sponds to a completely warmed-up heated part and its final static displacement. Let us call this temperature a static equivalent temperature tst <sup>e</sup> <sup>T</sup><sup>~</sup> . When the boundary conditions have changed, the force-induced displacements will vary dynamically and the temperature te will dynamically approach tst <sup>e</sup> . As the relation between the displacement and the temperature te is practically linear (see Figure 8), their dynamic behavior will be similar. For this reason, the dynamics of te are described using the same displacement transient performances presented in Figures 5 and 6. For the disk, the algorithm to compute te is similar to that described in Section 5 for the thermal expansion displacements. The resulting equations to compute the equivalent temperature:

$$\mathbf{t}\_{\rm ef}(\tau, \mathbf{k}\_a) = \frac{\mathbf{T}\_{\rm j}(\mathbf{k}\_a)}{\mathbf{T}\_{\rm j}(\mathbf{k}\_a) + \Delta\tau} \mathbf{t}\_{\rm ef}(\tau - \Delta\tau) + \frac{\Delta\tau}{\mathbf{T}\_{\rm j}(\mathbf{k}\_a) + \Delta\tau} \mathbf{t}\_{\rm e}^{\rm st}(\mathbf{k}\_a), \mathbf{j} = \mathbf{1}, 2 \tag{25}$$

and

$$\mathbf{t}\_{\mathbf{e}}(\pi, \mathbf{k}\_{\mathbf{a}}) = \sum\_{\mathbf{j=1}}^{2} \mathbf{k}\_{\mathbf{j}}(\mathbf{k}\_{\mathbf{a}}) \mathbf{t}\_{\mathbf{e}\mathbf{j}}(\pi, \mathbf{k}\_{\mathbf{a}}), \mathbf{j=1,2} \tag{26}$$

are also similar to displacement Eqs. (15) and (16) and the same parameters Tj and kj are employed. Using the dynamic value teð Þ τ; k<sup>α</sup> from Eq. (26) as an argument, a dynamic displacement u<sup>∘</sup> <sup>F</sup>ð Þ te is determined from Eq. (19) and a total forceinduced disk displacement uF from Eq. (17). The blade and casing force-induced

displacements are computed by similar algorithms. All these algorithms correspond to Blocks 1.4 and 2.4 of the engine ENDM presented in Figure 1.

a constant speed value 8100 rpm during the first 175 s, than a linear change to 12,400 rpm during 12 s, and the same constant value up to the transient end. Figure 11 illustrates the dynamics of the HPT radial clearance simulated by ENDM in comparison with the steady-state clearance simulation (completely warmed-up turbine parts). One can state that ENDM correctly reflects the physics of real warming-up. From the beginning of the engine acceleration, the clearance descends in 15 s because the blade is rapidly warmed up. Next, the clearance grows due to the casing warming up. Finally, the clearance descends once more as the disk

Advanced Nonlinear Modeling of Gas Turbine Dynamics

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Figures 12 and 13 present the results of the comparison of the initial and enhanced dynamic models between each other and with experimental data for the same test-case transient. The plots of a fuel consumption variable in Figure 12 clearly show that the ENDM and experimental curves practically coincide. Both show the same fuel consumption overshoot after the control parameter change, and this overshoot gradually decreases during 150 s for both curves. This elevated fuel

Dynamics of the HPT radial clearance (1—steady-state operating modes; 2—ENDM).

Fuel consumption dynamics (1—experimental data; 2—ENDM; 3—NDM).

begins to warm up.

Figure 11.

Figure 12.

169
