*:* (52)

, (53)

dt2 <sup>þ</sup> <sup>δ</sup><sup>P</sup> <sup>¼</sup> <sup>1</sup>

<sup>δ</sup>P tðÞ¼ <sup>1</sup> 2

ð Þ 2Pinl 0 � P0 ΔPinl � Pinl 0ΔP 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pinl 0ð Þ Pinl 0 � P0

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Pinl 0ð Þ Pinl 0 � P0 ξRT

The system, which behaves like this, is oscillatory. The solution of this equation in the case of inlet or outlet pressure step is changed by δP ¼ AP must

$$\mathbf{P}\_0 = \sqrt{\mathbf{P}\_{\text{in l 0}}^2 + \left(\frac{\mathbf{P}\_{\text{in l 0}} - \mathbf{P}\_{\text{out 0}}}{2}\right)^2} - \frac{\mathbf{P}\_{\text{in l 0}} - \mathbf{P}\_{\text{out 0}}}{2}. \tag{54}$$

Then, when switching to the relative deviations, we get

$$
\tau\_\mathrm{P} \frac{\mathrm{d}(\delta \mathrm{P})}{\mathrm{d}\mathrm{t}} + \delta \mathrm{P} = \mathrm{K}\_{\mathrm{in}}^\mathrm{P} \delta \mathrm{P}\_{\mathrm{in}} + \mathrm{K}\_{\mathrm{out}}^\mathrm{P} \delta \mathrm{P}\_{\mathrm{out}},\tag{55}
$$

where <sup>τ</sup><sup>P</sup> <sup>¼</sup> P0 Pinl 0þ2P0�Pout 0 ξτ0M0 is a time constant; M0 <sup>¼</sup> c0 <sup>a</sup> is a Mach number; and KP inl <sup>¼</sup> 2Pinl 0�P0 Pinl 0þ2P0�Pout 0 Pinl 0 P0 and KP out <sup>¼</sup> Pout 0 Pinl 0þ2P0�Pout 0 are gain coefficients.

The solution of this equation in the case of inlet pressure perturbation δPinl ¼ AP can be presented by

$$\delta \mathbf{P}(\mathbf{t}) = \mathbf{A}\_{\mathrm{P}} \mathbf{K}\_{\mathrm{inl}}^{\mathrm{P}} \left( \mathbf{1} - \mathbf{e}^{-\frac{\mathbf{t}}{\tau\_{\mathrm{p}}}} \right). \tag{56}$$

#### **A.1.3 Model 1.3**

Let us linearize Eqs. (6), (9) and (10):

$$\frac{\text{LA}}{\gamma \text{RT}} \frac{\text{d}(\Delta \text{P})}{\text{dt}} = \Delta \text{W}\_{\text{inl}} - \Delta \text{W}\_{\text{out}};\tag{57}$$

$$\frac{\mathbf{d}(\Delta \mathbf{W}\_{\rm in})}{\mathbf{d}t} = \frac{\mathbf{2A}}{\mathbf{L}} \left[ \Delta \mathbf{P}\_{\rm inl} - \Delta \mathbf{P} - \frac{\xi \mathbf{R} \mathbf{T}}{2\mathbf{A}^2} \left( \frac{2\mathbf{W}\_0}{\mathbf{P}\_{\rm inl}} \Delta \mathbf{W}\_{\rm inl} - \frac{\mathbf{W}\_0^2}{\mathbf{P}\_{\rm inl}^2} \Delta \mathbf{P}\_{\rm inl} \right) \right];\tag{58}$$

$$\frac{\mathbf{d}(\Delta \mathbf{W}\_{\rm out})}{\mathbf{d}t} = \frac{2\mathbf{A}}{\mathbf{L}} \left[ \Delta \mathbf{P} - \Delta \mathbf{P}\_{\rm out} - \frac{\xi \mathbf{R} \mathbf{T}}{2\mathbf{A}^2} \left( \frac{2\mathbf{W}\_0}{\mathbf{P}\_0} \Delta \mathbf{W}\_{\rm out} - \frac{\mathbf{W}\_0^2}{\mathbf{P}\_0^2} \Delta \mathbf{P} \right) \right]. \tag{59}$$

Next, we differentiate the equation for the pressure and substitute the derivatives of airflow. Then, considering P0 Pinl 0 <sup>Δ</sup>Winl � <sup>Δ</sup>Wout <sup>≈</sup> LA γRT dð Þ ΔP dt , we get

$$\begin{split} \frac{\mathbf{L}^2}{2\gamma \mathbf{R} \mathbf{T}} \frac{\mathbf{d}^2(\Delta \mathbf{P})}{\mathbf{d}\mathbf{t}^2} &= \left(\frac{1}{2} + \frac{\xi \mathbf{R} \mathbf{T} \mathbf{W}\_0^2}{4\mathbf{A}^2 \mathbf{P}\_{\mathrm{in}\,1}^2 \mathbf{0}}\right) \Delta \mathbf{P}\_{\mathrm{in}\,1} + \frac{1}{2} \Delta \mathbf{P}\_{\mathrm{out}} - \left(\mathbf{1} + \frac{\xi \mathbf{R} \mathbf{T} \mathbf{W}\_0^2}{4\mathbf{A}^2 \mathbf{P}\_0^2}\right) \Delta \mathbf{P} \\ &- \frac{\xi \mathbf{R} \mathbf{T} \mathbf{W}\_0}{2\mathbf{A}^2 \mathbf{p}\_0} \frac{\mathbf{L} \mathbf{A}}{2\gamma \mathbf{R} \mathbf{T}} \frac{\mathbf{d}(\Delta \mathbf{P})}{\mathbf{d}\mathbf{t}}. \end{split} \tag{60}$$

In a relative deviations format

$$\begin{split} & \frac{\mathsf{\tau}\_{0}^{2}}{2 + \mathsf{0}.5\mathsf{\xi}\mathsf{\gamma}\mathsf{M}\_{0}^{2}} \frac{\mathsf{d}^{2}(\mathsf{\delta P})}{\mathsf{d}\mathsf{t}^{2}} + \frac{\mathsf{\tau}\_{\mathsf{P}}}{\mathsf{1} + \mathsf{0}.25\mathsf{\xi}\mathsf{\gamma}\mathsf{M}\_{0}^{2}} \frac{\mathsf{d}(\mathsf{\delta P})}{\mathsf{d}\mathsf{t}} + \mathsf{\delta P} \\ &= \frac{2 + \mathsf{\xi}\mathsf{\gamma}\mathsf{M}\_{0}^{2}}{\mathsf{4} + \mathsf{\xi}\mathsf{\gamma}\mathsf{M}\_{0}^{2}} \mathsf{\delta P}\_{\mathsf{in}} + \frac{2}{\mathsf{4} + \mathsf{\xi}\mathsf{\gamma}\mathsf{M}\_{0}^{2}} \frac{\mathsf{P}\_{\text{out}} \; \mathsf{0}}{\mathsf{P}\_{0}} \mathsf{\delta P}\_{\text{out}}. \end{split} \tag{61}$$

The transient process, which is initiated by the inlet pressure perturbation δPinl ¼ AP, is expressed as

$$\mathbf{P(t)} = \mathbf{A}\_{\mathbf{P}} \left[ \mathbf{1} - \mathbf{e}^{-\frac{\xi}{2\pi}} \left( \cos \alpha t + \frac{\xi}{2\sqrt{2}} \sin \alpha t \right) \right],\tag{62}$$

where <sup>ω</sup> <sup>≈</sup> ffiffi 2 p τ0 .

## **A.1.4 Model 2.1**

The linearization of Eq. (17) outputs the equation in the absolute deviations

$$\frac{\text{PLA}}{\gamma \text{RG}} \frac{\text{d}(\Delta \text{T})}{\text{dt}} = -\text{T}\_{\text{inl }0} \Delta \text{T} + \text{T}\_{0} \Delta \text{T}\_{\text{inl}},\tag{63}$$

which in the relative deviations has the following form:

$$
\sigma\_\mathrm{T} \frac{\mathrm{d}(\mathrm{\delta T})}{\mathrm{d}\mathrm{t}} + \mathrm{\delta T} = \mathrm{\delta T}\_{\mathrm{inl}}, \tag{64}
$$

where <sup>τ</sup><sup>T</sup> <sup>¼</sup> PAL <sup>γ</sup>RG0T0 <sup>¼</sup> <sup>τ</sup> γ is a time constant.

The transient process, which is initiated by the inlet temperature perturbation δTinl ¼ AT, is described as

$$\delta \mathbf{T}(\mathbf{t}) = \mathbf{A}\_{\mathbf{T}} \left( \mathbf{1} - \mathbf{e}^{-\frac{t}{T}} \right). \tag{65}$$

(67)

ττ<sup>p</sup> γ d2 ð Þ δP dt2 <sup>þ</sup>

> ττ<sup>P</sup> γ

<sup>a</sup> <sup>¼</sup> <sup>1</sup>

<sup>b</sup> <sup>¼</sup> <sup>1</sup> ττ<sup>P</sup>

d2 ð Þ δT dt2 <sup>þ</sup>

where A <sup>¼</sup> <sup>a</sup>þ<sup>d</sup>

where A1 <sup>¼</sup> <sup>a</sup>

δTinl ¼ AT, is described as

where A2 <sup>¼</sup> <sup>a</sup>þd1

and the temperature:

**51**

**A.1.6 Model 3.1**

ττ<sup>P</sup> �2τ � ð Þ γ þ 1 τ<sup>P</sup> �

�2τ � ð Þ γ þ 1 τ<sup>P</sup> þ

τ γ

> τ γ

<sup>þ</sup> <sup>γ</sup> <sup>þ</sup> <sup>1</sup> 2γ τP � � <sup>d</sup>ð Þ <sup>δ</sup><sup>P</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.90490*

<sup>þ</sup> <sup>γ</sup> <sup>þ</sup> <sup>1</sup> 2γ τP � � <sup>d</sup>ð Þ <sup>δ</sup><sup>T</sup>

a að Þ �<sup>b</sup> ; B <sup>¼</sup> <sup>b</sup>þ<sup>d</sup>

a að Þ �<sup>b</sup> ; B1 <sup>¼</sup> <sup>b</sup>

a að Þ �<sup>b</sup> ; B1 <sup>¼</sup> <sup>b</sup>þd1

τ γ

dð Þ δT

τ γ

> 1 2

dð Þ δP

dt <sup>þ</sup> <sup>δ</sup><sup>P</sup> <sup>¼</sup> <sup>τ</sup>

*Gas Turbine Simulation Taking into Account Dynamics of Gas Capacities*

2γ

� τ <sup>4</sup> <sup>ξ</sup>M2 0

> 2γ τP

Aeat <sup>þ</sup> Bebt <sup>þ</sup> <sup>τ</sup><sup>P</sup>

<sup>2</sup> � 16kττ<sup>P</sup>

<sup>2</sup> � <sup>16</sup>γττ<sup>P</sup>

� <sup>γ</sup> � <sup>1</sup> γ

dt <sup>þ</sup> <sup>δ</sup><sup>T</sup> <sup>¼</sup> <sup>γ</sup> <sup>þ</sup> <sup>1</sup>

The transient that is initiated by the pressure perturbation AP is described as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>δ</sup>P tðÞ¼ AP

b bð Þ �<sup>a</sup> ; d <sup>¼</sup> <sup>γ</sup>

� � q

� � q

<sup>δ</sup>T tðÞ¼ APð Þ <sup>γ</sup> � <sup>1</sup> ξγτM2 0

<sup>δ</sup>P tðÞ¼� ATξγM2

b bð Þ �<sup>a</sup> ; d1 <sup>¼</sup> <sup>2</sup><sup>γ</sup>

dt ¼ �δ<sup>T</sup> <sup>þ</sup> <sup>δ</sup>Tinl <sup>þ</sup> <sup>γ</sup> � <sup>1</sup>

b bð Þ �<sup>a</sup> .

<sup>δ</sup>T tðÞ¼ ATð Þ <sup>γ</sup> <sup>þ</sup> <sup>1</sup> 2τ

2τ<sup>P</sup>

τ ;

ð Þ 2τ þ ð Þ γ þ 1 τ<sup>P</sup>

ð Þ 2τ þ ð Þ γ þ 1 τ<sup>P</sup>

The transient state, which is initiated by the temperature perturbation

0 4τ<sup>P</sup>

<sup>τ</sup>Pð Þ <sup>γ</sup>þ<sup>1</sup> .

The linearized Eqs. (6), (7), (20), and (21) in the relative deviations format are

Let us transform Eqs. (76)–(78) to get the differential equations for the pressure

A2eat <sup>þ</sup> B2ebt <sup>þ</sup>

dδPinl dt þ

dδPout dt � � <sup>þ</sup>

dt <sup>þ</sup> <sup>δ</sup>Tinl

� �, (72)

;

;

A1eat <sup>þ</sup> B1ebt � �, (73)

A1eat <sup>þ</sup> B1ebt � �; (74)

<sup>γ</sup> ð Þ <sup>δ</sup>Winl � <sup>δ</sup>Wout ; (77)

2τ γ þ 1 � �, (75)

dt <sup>¼</sup> <sup>δ</sup>Winl � <sup>δ</sup>Wout <sup>þ</sup> <sup>δ</sup>Tinl � <sup>δ</sup>T; (76)

γτ0M0ðδWinl � δWoutÞ ¼ δPinl þ δPout � 2δP*:* (78)

dð Þ δPinl dt þ

dð Þ δTinl dt

dδTinl

τP ξγM2 0 1 2

ð Þ δPinl þ δPout

dð Þ δPout

dt � �*:*

(70)

(71)

#### **A.1.5 Model 2.2**

Let us transform Eq. (18) and linearize it:

$$\frac{\mathbf{L}}{\gamma} \sqrt{\frac{\xi}{2\mathbf{R}}} \frac{\mathbf{dP}}{\mathbf{dt}} = \mathbf{T} \left[ \sqrt{\frac{\mathbf{P}\_{\text{inl}} (\mathbf{P}\_{\text{inl}} - \mathbf{P})}{\mathbf{T}\_{\text{inl}}}} - \sqrt{\frac{\mathbf{P} (\mathbf{P} - \mathbf{P}\_{\text{out}})}{\mathbf{T}}} \right];\tag{66}$$

$$\begin{split} \frac{\mathbf{L}}{\gamma} \sqrt{\frac{\xi}{2\mathbf{R}}} \frac{\mathbf{d}(\Delta \mathbf{P})}{\mathbf{dt}} &= \frac{\mathbf{W}\_{0}}{2\mathbf{A}} \sqrt{\frac{\xi \mathbf{R}}{2}} (\Delta \mathbf{T}\_{\text{inl}} - \Delta \mathbf{T}) \\ &+ \frac{\mathbf{A}}{2\mathbf{W}\_{0}} \sqrt{\frac{2}{\xi \mathbf{R}}} [(2\mathbf{P}\_{\text{inl}} \mathbf{0} - \mathbf{P}\_{0}) \Delta \mathbf{P}\_{\text{inl}} + \mathbf{P}\_{0} \Delta \mathbf{P}\_{\text{out}} - (\mathbf{P}\_{\text{inl}} \mathbf{0} + 2\mathbf{P}\_{0} - \mathbf{P}\_{\text{out}}) \Delta \mathbf{P}\_{\text{inl}}]. \end{split}$$

Next, we transform the coefficients and change the equation to the relative deviations:

$$\mathbf{r}'\_{\rm P} \frac{\mathbf{d}(\delta \mathbf{P})}{\mathbf{d}t} + \delta \mathbf{P} = \mathbf{K}\_{\rm T}^{\rm P}(\delta \mathbf{T} - \delta \mathbf{T}\_{\rm inl}) + \mathbf{K}\_{\rm inl}^{\rm P} \delta \mathbf{P}\_{\rm inl} + \mathbf{K}\_{\rm out}^{\rm P} \delta \mathbf{P}\_{\rm out},\tag{68}$$

where τ<sup>0</sup> <sup>P</sup> ¼ 2τ<sup>P</sup> P0 Pinl 0þ2P0�Pout 0; K<sup>P</sup> <sup>T</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> ξγM2 0 P0 Pinl 0þ2P0�Pout 0. In a similar manner we transform Eq. (19):

$$\frac{2\pi}{\gamma+1}\frac{\mathbf{d}(\delta\mathbf{T})}{\delta\mathbf{t}} + \delta\mathbf{T} = \delta\mathbf{T}\_{\mathrm{inl}} + \mathbf{K}\_{\mathrm{inl}}^{\mathrm{T}}\delta\mathbf{P}\_{\mathrm{inl}} + \mathbf{K}\_{\mathrm{out}}^{\mathrm{T}}\delta\mathbf{P}\_{\mathrm{out}} - \mathbf{K}\_{\mathrm{p}}^{\mathrm{T}}\delta\mathbf{P},\tag{69}$$

where K<sup>T</sup> inl <sup>¼</sup> <sup>2</sup>ð Þ <sup>γ</sup>�<sup>1</sup> γ γð Þ <sup>þ</sup><sup>1</sup> <sup>ξ</sup>M<sup>2</sup> 0 2Pinl 0�P0 P0 Pinl 0 P0 ; K<sup>T</sup> out <sup>¼</sup> <sup>2</sup>ð Þ <sup>γ</sup>�<sup>1</sup> γ γð Þ <sup>þ</sup><sup>1</sup> <sup>ξ</sup>M<sup>2</sup> 0 Pout 0 P0 ;

$$\mathbf{K}\_{\mathbf{P}}^{\mathrm{T}} = \frac{\mathbf{2}(\boldsymbol{\gamma} - \mathbf{1})}{\boldsymbol{\gamma}(\boldsymbol{\gamma} + \mathbf{1})\mathbf{\tilde{\mathsf{E}}}\mathbf{M}\_{0}^{2}} \frac{\mathbf{P}\_{\mathrm{inl\ 0}} + \mathbf{2}\mathbf{P}\_{0} - \mathbf{P}\_{\mathrm{out\ 0}}}{\mathbf{P}\_{0}}.$$

Having combined (68) and (69), we will get the differential equations for pressure and temperature:

*Gas Turbine Simulation Taking into Account Dynamics of Gas Capacities DOI: http://dx.doi.org/10.5772/intechopen.90490*

$$\begin{aligned} \frac{\pi \tau\_{\rm p}}{\gamma} \frac{\mathrm{d}^2(\delta \mathbf{P})}{\mathrm{d}\mathbf{t}^2} + \left(\frac{\tau}{\gamma} + \frac{\gamma + 1}{2\gamma} \tau\_{\rm p}\right) \frac{\mathrm{d}(\delta \mathbf{P})}{\mathrm{d}\mathbf{t}} + \delta \mathbf{P} &= \frac{\tau}{2\gamma} \left(\frac{\mathrm{d}\delta \mathbf{P}\_{\mathrm{inl}}}{\mathrm{d}\mathbf{t}} + \frac{\mathrm{d}\delta \mathbf{P}\_{\mathrm{out}}}{\mathrm{d}\mathbf{t}}\right) + \frac{1}{2} (\delta \mathbf{P}\_{\mathrm{inl}} + \delta \mathbf{P}\_{\mathrm{out}}) \\ & \qquad - \frac{\tau}{4} \xi \mathsf{M}\_0^2 \frac{\mathrm{d}(\delta \mathbf{T}\_{\mathrm{inl}})}{\mathrm{d}\mathbf{t}} \end{aligned} \tag{70}$$

ττ<sup>P</sup> γ d2 ð Þ δT dt2 <sup>þ</sup> τ γ <sup>þ</sup> <sup>γ</sup> <sup>þ</sup> <sup>1</sup> 2γ τP � � <sup>d</sup>ð Þ <sup>δ</sup><sup>T</sup> dt <sup>þ</sup> <sup>δ</sup><sup>T</sup> <sup>¼</sup> <sup>γ</sup> <sup>þ</sup> <sup>1</sup> 2γ τP dδTinl dt <sup>þ</sup> <sup>δ</sup>Tinl � <sup>γ</sup> � <sup>1</sup> γ τP ξγM2 0 dð Þ δPinl dt þ dð Þ δPout dt � �*:* (71)

The transient that is initiated by the pressure perturbation AP is described as

$$\delta \mathbf{P}(\mathbf{t}) = \frac{\mathbf{A}\_{\mathrm{P}}}{2\pi\_{\mathrm{P}}} \left( \mathbf{A} \mathbf{e}^{\mathrm{at}} + \mathbf{B} \mathbf{e}^{\mathrm{bt}} + \pi\_{\mathrm{P}} \right), \tag{72}$$

$$\begin{aligned} \text{where } \mathbf{A} &= \frac{\mathbf{a} + \mathbf{d}}{\mathbf{a}(\mathbf{a} - \mathbf{b})}; \mathbf{B} = \frac{\mathbf{b} + \mathbf{d}}{\mathbf{b}(\mathbf{b} - \mathbf{a})}; \mathbf{d} = \frac{\mathbf{y}}{\mathbf{z}};\\ \mathbf{a} &= \frac{1}{\tau \mathbf{r}\_{\rm P}} \left[ -2\pi - (\mathbf{y} + \mathbf{1})\tau\_{\rm P} - \sqrt{\left(2\pi + (\mathbf{y} + \mathbf{1})\tau\_{\rm P}\right)^{2} - \mathbf{1}\mathbf{f}\mathbf{k}\tau\tau\_{\rm P}} \right];\\ \mathbf{b} &= \frac{\mathbf{1}}{\tau \tau\_{\rm P}} \left[ -2\pi - (\mathbf{y} + \mathbf{1})\tau\_{\rm P} + \sqrt{\left(2\pi + (\mathbf{y} + \mathbf{1})\tau\_{\rm P}\right)^{2} - \mathbf{1}\mathbf{f}\mathbf{y}\tau\tau\_{\rm P}} \right];\\ \mathbf{\delta T(t)} &= \frac{\mathbf{A}\_{\rm P}(\mathbf{y} - \mathbf{1})}{\xi\_{\rm P}\tau\mathbf{x}\mathbf{M}\_{0}^{2}} \left( \mathbf{A}\_{\rm P}\mathbf{e}^{\mathbf{at}} + \mathbf{B}\_{\rm P}\mathbf{e}^{\mathbf{b}t} \right), \end{aligned} \tag{73}$$

where A1 <sup>¼</sup> <sup>a</sup> a að Þ �<sup>b</sup> ; B1 <sup>¼</sup> <sup>b</sup> b bð Þ �<sup>a</sup> .

The transient state, which is initiated by the temperature perturbation δTinl ¼ AT, is described as

$$\delta \mathbf{P}(\mathbf{t}) = -\frac{\mathbf{A}\_{\mathrm{T}} \mathbf{\tilde{\xi}} \gamma \mathbf{M}\_{0}^{2}}{4\pi\_{\mathrm{P}}} \left(\mathbf{A}\_{\mathrm{1}} \mathbf{e^{\mathrm{at}}} + \mathbf{B}\_{\mathrm{1}} \mathbf{e^{\mathrm{bt}}}\right);\tag{74}$$

$$\text{ST}(\mathbf{t}) = \frac{\mathbf{A}\_{\text{T}}(\boldsymbol{\gamma} + \mathbf{1})}{2\pi} \left( \mathbf{A}\_{2}\mathbf{e}^{\text{at}} + \mathbf{B}\_{2}\mathbf{e}^{\text{bt}} + \frac{2\pi}{\boldsymbol{\gamma} + \mathbf{1}} \right), \tag{75}$$

where A2 <sup>¼</sup> <sup>a</sup>þd1 a að Þ �<sup>b</sup> ; B1 <sup>¼</sup> <sup>b</sup>þd1 b bð Þ �<sup>a</sup> ; d1 <sup>¼</sup> <sup>2</sup><sup>γ</sup> <sup>τ</sup>Pð Þ <sup>γ</sup>þ<sup>1</sup> .

#### **A.1.6 Model 3.1**

**A.1.4 Model 2.1**

where <sup>τ</sup><sup>T</sup> <sup>¼</sup> PAL

**A.1.5 Model 2.2**

L γ

ffiffiffiffiffiffi ξ 2R <sup>r</sup> <sup>d</sup>ð Þ <sup>Δ</sup><sup>P</sup>

deviations:

where τ<sup>0</sup>

where K<sup>T</sup>

**50**

δTinl ¼ AT, is described as

<sup>γ</sup>RG0T0 <sup>¼</sup> <sup>τ</sup> γ

Let us transform Eq. (18) and linearize it:

dt <sup>¼</sup> <sup>T</sup>

ffiffiffiffiffi 2 ξR

dt <sup>þ</sup> <sup>δ</sup><sup>P</sup> <sup>¼</sup> <sup>K</sup><sup>P</sup>

P0 Pinl 0þ2P0�Pout 0; K<sup>P</sup>

In a similar manner we transform Eq. (19):

dð Þ δT

KT

ffiffiffiffiffiffi ξ 2R r dP

> ffiffiffiffiffi ξR 2

s

r

L γ

þ A 2W0

τ0 P dð Þ δP

<sup>P</sup> ¼ 2τ<sup>P</sup>

2τ γ þ 1

inl <sup>¼</sup> <sup>2</sup>ð Þ <sup>γ</sup>�<sup>1</sup> γ γð Þ <sup>þ</sup><sup>1</sup> <sup>ξ</sup>M<sup>2</sup> 0

pressure and temperature:

dt <sup>¼</sup> W0 2A

The linearization of Eq. (17) outputs the equation in the absolute deviations

The transient process, which is initiated by the inlet temperature perturbation

� t τT � �

�

" # r

δT tðÞ¼ AT 1 � e

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pinlð Þ Pinl � P Tinl

Next, we transform the coefficients and change the equation to the relative

<sup>T</sup>ð Þþ <sup>δ</sup><sup>T</sup> � <sup>δ</sup>Tinl KP

<sup>2</sup> ξγM2 0

<sup>T</sup> <sup>¼</sup> <sup>1</sup>

dt <sup>þ</sup> <sup>δ</sup><sup>T</sup> <sup>¼</sup> <sup>δ</sup>Tinl <sup>þ</sup> KT

<sup>P</sup> <sup>¼</sup> <sup>2</sup>ð Þ <sup>γ</sup> � <sup>1</sup> γ γð Þ <sup>þ</sup> <sup>1</sup> <sup>ξ</sup>M<sup>2</sup>

Pinl 0 P0 ; K<sup>T</sup>

2Pinl 0�P0 P0

dt ¼ �Tinl 0Δ<sup>T</sup> <sup>þ</sup> T0ΔTinl, (63)

dt <sup>þ</sup> <sup>δ</sup><sup>T</sup> <sup>¼</sup> <sup>δ</sup>Tinl, (64)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P Pð Þ � Pout T

ð Þ 2Pinl 0 � P0 ΔPinl þ P0ΔPout ½ �ð Þ Pinl 0 þ 2P0 � Pout 0 ΔP�*:*

inlδPinl <sup>þ</sup> <sup>K</sup><sup>P</sup>

outδPout � <sup>K</sup><sup>T</sup>

*:*

P0 Pinl 0þ2P0�Pout 0.

inlδPinl <sup>þ</sup> KT

Pinl 0 þ 2P0 � Pout 0 P0

out <sup>¼</sup> <sup>2</sup>ð Þ <sup>γ</sup>�<sup>1</sup> γ γð Þ <sup>þ</sup><sup>1</sup> <sup>ξ</sup>M<sup>2</sup> 0 Pout 0 P0 ;

0

Having combined (68) and (69), we will get the differential equations for

*:* (65)

; (66)

outδPout, (68)

<sup>p</sup> δP, (69)

(67)

PLA γRG dð Þ ΔT

*Modeling of Turbomachines for Control and Diagnostic Applications*

which in the relative deviations has the following form:

τT dð Þ δT

is a time constant.

s

ð Þ ΔTinl � ΔT

The linearized Eqs. (6), (7), (20), and (21) in the relative deviations format are

$$\frac{\pi}{\chi} \frac{\mathbf{d}(\delta \mathbf{P})}{\mathbf{d}\mathbf{t}} = \delta \mathbf{W}\_{\text{inl}} - \delta \mathbf{W}\_{\text{out}} + \delta \mathbf{T}\_{\text{inl}} - \delta \mathbf{T};\tag{76}$$

$$\frac{\pi}{\gamma} \frac{\mathbf{d}(\delta \mathbf{T})}{\mathbf{d}\mathbf{t}} = -\delta \mathbf{T} + \delta \mathbf{T}\_{\text{inl}} + \frac{\gamma - \mathbf{1}}{\gamma} (\delta \mathbf{W}\_{\text{inl}} - \delta \mathbf{W}\_{\text{out}});\tag{77}$$

$$\frac{1}{2}\gamma\mathbf{\tau}\_0\mathbf{M}\_0(\delta\mathbf{W}\_{\text{inl}}-\delta\mathbf{W}\_{\text{out}}) = \delta\mathbf{P}\_{\text{inl}} + \delta\mathbf{P}\_{\text{out}} - 2\delta\mathbf{P}.\tag{78}$$

Let us transform Eqs. (76)–(78) to get the differential equations for the pressure and the temperature:

1 4 τ2 τ0M0 d3 ð Þ δP dt3 <sup>þ</sup> 1 <sup>4</sup> γττ0M0 d2 ð Þ δP dt<sup>2</sup> <sup>þ</sup> <sup>τ</sup> dð Þ δP dt <sup>þ</sup> <sup>δ</sup><sup>P</sup> <sup>¼</sup> ¼ 1 <sup>4</sup> γττ0M0 d2 Tinl dt2 <sup>þ</sup> 1 2 τ dð Þ δPinl dt þ dð Þ δPout dt � � <sup>þ</sup> 1 2 ð Þ δPinl þ δPout *:* (79) 1 4 τ2 τ0M0 d3 ð Þ δT dt3 <sup>þ</sup> 1 <sup>4</sup> γττ0M0 d2 ð Þ δT dt2 <sup>þ</sup> þ τ dð Þ δT dt <sup>þ</sup> <sup>δ</sup><sup>T</sup> <sup>¼</sup> <sup>1</sup> <sup>4</sup> γττ0M0 d2 Tinl dt<sup>2</sup> <sup>þ</sup> <sup>δ</sup>Tinl <sup>þ</sup> <sup>γ</sup> � <sup>1</sup> 2γ τ dð Þ δPinl dt þ dð Þ δPout dt � �*:* (80)

The transient that is initiated by the temperature perturbation δTinl= AT is described as

$$\delta \mathbf{T}(\mathbf{t}) = \mathbf{A}\_{\Gamma} \frac{\gamma}{\pi} (\mathbf{A}\_{3} \mathbf{e}^{\mathrm{at}} \sin \left( \alpha \mathbf{t} + \beta\_{1} \right) + \mathbf{B}\_{3} \mathbf{e}^{\mathrm{s\_{1}t}} + \mathbf{K}), \tag{81}$$

dWinl

dt � dWout

dt <sup>¼</sup> <sup>2</sup>

dWinl

Let us determine <sup>d</sup>ð Þ <sup>δ</sup><sup>P</sup>

the temperature in the volume:

<sup>¼</sup> γτ<sup>2</sup> 0 2C d2 ð Þ δTinl

<sup>þ</sup> τ γð Þ � <sup>1</sup> C

0.

get from Eqs. (87) and (88). The final result is

<sup>C</sup> ð Þ <sup>0</sup>*:*5<sup>γ</sup> <sup>þ</sup> <sup>2</sup><sup>ξ</sup> <sup>d</sup><sup>2</sup>

<sup>þ</sup> ξγM2 0 � �δPinl <sup>þ</sup>

dð Þ δTinl dt þ

<sup>δ</sup>Pout � �*:*

τ2 0

τ0ξγM0 C

magnitude of the step is AT) is described as

γ τ 0

BB@

þ B6

Pinl 0 P0

ττ<sup>2</sup> 0 2C d3 ð Þ δT dt3 <sup>þ</sup>

where C <sup>¼</sup> <sup>2</sup> <sup>þ</sup> ξγM2

and WT

<sup>¼</sup> γτ<sup>2</sup> 0 2C d2 ð Þ δTinl dt<sup>2</sup> <sup>þ</sup>

**53**

ττ<sup>2</sup> 0 2C d3 ð Þ δP dt3 <sup>þ</sup>

> þ 1 C

δT tðÞ¼ AT

γτ0M0

*DOI: http://dx.doi.org/10.5772/intechopen.90490*

�ξγM<sup>2</sup>

On the other hand, from Eq. (77) we get

dt � dWout

dð Þ δP dt <sup>¼</sup> <sup>γ</sup>

> τ2 0

<sup>f</sup> Pinl 0 P0

*Gas Turbine Simulation Taking into Account Dynamics of Gas Capacities*

dt <sup>¼</sup> <sup>γ</sup>

γ � 1

dt from (76) and (78):

γ � 1

<sup>C</sup> ð Þ <sup>0</sup>*:*5<sup>γ</sup> <sup>þ</sup> <sup>2</sup><sup>ξ</sup> <sup>d</sup><sup>2</sup>

Pinl 0 P0

Let us use the Laplace transform and transfer functions W<sup>p</sup>

dt<sup>2</sup> <sup>þ</sup> ð Þ <sup>γ</sup> <sup>þ</sup> <sup>1</sup> ξτ0M0 C

> <sup>þ</sup> ξγM2 0 � � <sup>d</sup>ð Þ <sup>δ</sup>Pinl

> > ð Þ δP dt<sup>2</sup> <sup>þ</sup>

τ C

A6 � 1 tð Þ� ð Þ A6 <sup>þ</sup> B6 <sup>e</sup>s1t

ω

K3 B6 � α � �<sup>2</sup>

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>þ</sup> ξγM2 0 P0 Pinl 0 � �δPinl <sup>þ</sup>

<sup>0</sup>½2ðδWinl � δWoutÞ þ δTinl � δT�g*:*

τ γ d2 ð Þ δT dt2 <sup>þ</sup>

dð Þ δT dt þ

Having equalized the right sides of Eqs. (85) and (86), derived the obtained equation, and substituted the derivative (87) in it, we get a differential equation for

> ð Þ δT dt2 <sup>þ</sup>

1 τ

τ

dð Þ δTinl

dt þ

Tinlð Þs to obtain the equation for the pressure. The transfer functions we will

τ

Pinl 0 P0

The transient state, which is initiated by the temperature perturbation δTinl, (the

þ ω<sup>2</sup>

Pout 0 P0

Pout 0 P0

dð Þ δT

dt � <sup>d</sup>ð Þ <sup>δ</sup>Tinl dt !*:* (86)

ð Þ <sup>δ</sup><sup>T</sup> � <sup>δ</sup>Tinl � �*:* (87)

<sup>C</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>2</sup><sup>γ</sup> <sup>þ</sup> <sup>1</sup> <sup>ξ</sup>M<sup>2</sup>

Pout 0 P0

dt <sup>þ</sup> <sup>δ</sup>Tinl

dt � � (88)

<sup>C</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>2</sup><sup>γ</sup> <sup>þ</sup> <sup>1</sup> <sup>ξ</sup>M2

<sup>þ</sup> ξγM<sup>2</sup> 0 � � <sup>d</sup>ð Þ <sup>δ</sup>Pinl

þ

<sup>e</sup><sup>α</sup><sup>t</sup> � sin <sup>ω</sup><sup>t</sup> <sup>þ</sup> <sup>β</sup><sup>3</sup> ð Þ

0

dð Þ δPout

<sup>T</sup>ð Þ<sup>s</sup> , WT

0

dt � �<sup>þ</sup>

dt þ

1

CCA

, (90)

� � dð Þ δP

dt <sup>þ</sup> <sup>δ</sup><sup>T</sup>

pinlð Þ<sup>s</sup> , WT

dt <sup>þ</sup> <sup>δ</sup><sup>P</sup> <sup>¼</sup>

dð Þ δPout

(89)

Pout 0 P0

outð Þs ,

� � dð Þ δT

<sup>δ</sup>Pout � <sup>2</sup> <sup>þ</sup> ξγM<sup>2</sup>

0 � � <sup>þ</sup> <sup>δ</sup><sup>P</sup>

(85)

where <sup>α</sup> and <sup>ω</sup> are expressed by (38) and (39); A3 <sup>¼</sup> <sup>1</sup> ω ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>α</sup>2�ω2<sup>þ</sup> <sup>4</sup> γτ2 0 � �<sup>2</sup> þð Þ <sup>2</sup>αω <sup>2</sup> <sup>α</sup>2þω<sup>2</sup> ð Þ ð Þ <sup>α</sup>�s1 <sup>2</sup> <sup>þ</sup>ω<sup>2</sup> ½ � vuuut ;

$$\mathbf{B}\_{3} = \frac{\mathbf{s}\_{1}^{4} + \mathbf{s}\_{0}^{4}}{\mathbf{s}\_{1} \left[ \left( \mathbf{s}\_{1} - \alpha \right)^{2} + \alpha^{2} \right]}; \mathbf{K} = -\frac{4}{\mathcal{T}\_{0}^{2} \mathbf{s}\_{1} \left( \mathbf{a}^{2} + \alpha^{2} \right)}; \boldsymbol{\varnothing} = \mathbf{ar} \mathbf{c} \mathbf{g} \frac{2 \mathbf{m}}{a^{2} - a^{2} + \frac{4}{r\_{0}^{4}}} - \mathbf{ar} \mathbf{c} \mathbf{g} \frac{a}{a - s\_{1}} - \mathbf{ar} \mathbf{c} \mathbf{g} \frac{a}{a};$$

$$\boldsymbol{\mathfrak{S}} \mathbf{P}(\mathbf{t}) = \mathbf{A}\_{\mathbf{T}} \frac{\mathbf{y}}{\mathbf{c}} (\mathbf{A}\_{\mathbf{t}} \mathbf{e}^{\mathbf{at}} \sin \left( \alpha \mathbf{t} + \boldsymbol{\beta} \right) + \mathbf{B}\_{\mathbf{t}} \mathbf{e}^{\mathbf{at}}),\tag{82}$$

where A4 <sup>¼</sup> <sup>1</sup> ω ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>α</sup>2�ω<sup>2</sup> ð Þ<sup>2</sup> þð Þ <sup>2</sup>αω <sup>2</sup> <sup>α</sup>2þω<sup>2</sup> ð Þ ð Þ <sup>α</sup>�s1 <sup>2</sup> <sup>þ</sup>ω<sup>2</sup> ½ � <sup>r</sup> ; B4 <sup>¼</sup> s2 1 s1 ð Þ s1�<sup>α</sup> <sup>2</sup> <sup>þ</sup>ω<sup>2</sup> ½ �.

The transient that is initiated by the pressure perturbation AP is described as

$$\text{S7(t)} = \mathbf{A}\_{\text{T}} \frac{\mathbf{2(\gamma - 1)} }{\gamma \mathbf{r}\_0^2} (\mathbf{A}\_{\text{5}} \mathbf{e^{at}} \sin \left( \alpha \mathbf{t} + \beta\_1 \right) + \mathbf{B}\_{\text{5}} \mathbf{e^{s\_{\text{if}}}}), \tag{83}$$

where A5 <sup>¼</sup> <sup>1</sup> ω ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> ð Þ <sup>α</sup>�s1 <sup>2</sup> þω<sup>2</sup> <sup>q</sup> ; B5 <sup>¼</sup> <sup>1</sup> ð Þ s1�<sup>α</sup> <sup>2</sup> <sup>þ</sup>ω<sup>2</sup>; <sup>β</sup><sup>1</sup> ¼ �arctg <sup>ω</sup> α�s1 ;

$$\delta \mathbf{P}(\mathbf{t}) = \mathbf{A}\_{\rm T} \frac{2}{\pi\_0^2} (\mathbf{A}\_{\rm 6} \mathbf{e}^{\rm at} \sin \left( \alpha \mathbf{t} + \mathfrak{R}\_2 \right) + \mathbf{B}\_{\rm 6} \mathbf{e}^{\rm s\_{\rm 1} \rm t} + \mathbf{K}\_2), \tag{84}$$

$$\begin{array}{l} \textbf{where } \textbf{A}\_{6} = \frac{1}{\alpha} \sqrt{\frac{\left(\frac{\tau\_{0}^{2}}{\pi} + a\right)^{2} + \alpha^{2}}{\left(\alpha^{2} + \alpha^{2}\right)\left[\left(\alpha - \mathbf{s}\_{1}\right)^{2} + \alpha^{2}\right]}}; \textbf{B}\_{6} = \frac{\textbf{s}\_{1} + \frac{\tau\_{0}^{2}}{2\pi}}{\textbf{s}\_{1}\left[\left(\mathbf{s}\_{1} - \mathbf{u}\right)^{2} + \alpha^{2}\right]}; \textbf{K}\_{2} = -\frac{\tau\_{0}^{2}}{2\pi \mathbf{s}\_{1}\left(\mathbf{a}^{2} + \mathbf{a}^{2}\right)}; \textbf{K}\_{2} = -\frac{\textbf{s}\_{0}^{2}}{2\pi \mathbf{s}\_{1}\left(\mathbf{a}^{2} + \mathbf{a}^{2}\right)}; \textbf{K}\_{2} = \frac{\textbf{s}\_{1} + \frac{\tau\_{0}^{2}}{2\pi}}{\textbf{s}\_{1}\left(\mathbf{a}^{2} + \mathbf{a}^{2}\right)}. \end{array}$$

#### **A.1.7 Model 3.2**

The model consists of Eqs. (6), (7), (9) and (10). Linearized Eqs. (6) and (7) are of the format presented in Eqs. (76) and (78). As a result of the linearization, we get the missing difference dWinl dt � dWout dt :

*Gas Turbine Simulation Taking into Account Dynamics of Gas Capacities DOI: http://dx.doi.org/10.5772/intechopen.90490*

$$\begin{aligned} \frac{\text{dW}\_{\text{inl}}}{\text{dt}} - \frac{\text{dW}\_{\text{out}}}{\text{dt}} &= \frac{2}{\gamma \text{r}\_{0} \text{M}\_{0}} \{ \left( \frac{\text{P}\_{\text{inl}} \, \text{0}}{\text{P}\_{0}} + \text{\text{\textdegree} \, \text{M}\_{0}^{2}} \frac{\text{P}\_{0}}{\text{P}\_{\text{inl}} \, \text{0}} \right) \text{\textdegree P}\_{\text{inl}} + \frac{\text{P}\_{\text{out}} \, \text{0}}{\text{P}\_{0}} \ $ \text{P}\_{\text{out}} - \left( \text{Z} + \text{\textdegree} \, \text{M}\_{0}^{2} \right) + \$  \text{P}\_{\text{inl}} \\ &- \text{\textdegree} \, \text{M}\_{0}^{2} \{ \text{Z} (\ $ \text{W}\_{\text{inl}} - \$  \text{W}\_{\text{out}}) + \ $ \text{T}\_{\text{inl}} - \$  \text{T} \} ). \end{aligned} \tag{85}$$

On the other hand, from Eq. (77) we get

1 4 τ2 τ0M0 d3 ð Þ δP dt3 <sup>þ</sup>

<sup>4</sup> γττ0M0

dt <sup>þ</sup> <sup>δ</sup><sup>T</sup> <sup>¼</sup> <sup>1</sup>

d2 Tinl dt2 <sup>þ</sup>

> 1 4 τ2 τ0M0 d3 ð Þ δT dt3 <sup>þ</sup>

¼ 1

dð Þ δT

s2 <sup>1</sup><sup>þ</sup> <sup>4</sup> γτ2 0 s1 ð Þ s1�<sup>α</sup> <sup>2</sup>

where A4 <sup>¼</sup> <sup>1</sup>

where A5 <sup>¼</sup> <sup>1</sup>

where A6 <sup>¼</sup> <sup>1</sup>

β<sup>2</sup> ¼ �arctg <sup>ω</sup>

**A.1.7 Model 3.2**

**52**

<sup>þ</sup>ω<sup>2</sup> ½ �; K ¼ � <sup>4</sup>

ω

ω

ω

αþ τ2 0 2τ

the missing difference dWinl

r

þ τ

B3 ¼

described as

1 <sup>4</sup> γττ0M0

*Modeling of Turbomachines for Control and Diagnostic Applications*

d2 Tinl

γ τ

where <sup>α</sup> and <sup>ω</sup> are expressed by (38) and (39); A3 <sup>¼</sup> <sup>1</sup>

γ τ

2ð Þ γ � 1 γτ<sup>2</sup> 0

ð Þ s1�<sup>α</sup> <sup>2</sup>

α.

dð Þ δPinl dt þ

1 2 τ

<sup>4</sup> γττ0M0

δT tðÞ¼ AT

γτ<sup>2</sup>

δP tðÞ¼ AT

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>α</sup>2�ω<sup>2</sup> ð Þ<sup>2</sup> þð Þ <sup>2</sup>αω <sup>2</sup>

<sup>þ</sup>ω<sup>2</sup> ½ �

2 τ2 0

þω<sup>2</sup>

<sup>þ</sup>ω<sup>2</sup> ½ � vuut ; B6 <sup>¼</sup> s1<sup>þ</sup>

<sup>α</sup>�s1 � arctg <sup>ω</sup>

dt � dWout dt :

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>τ</sup><sup>2</sup> 0 <sup>2</sup>τþα � �<sup>2</sup>

<sup>α</sup>2þω<sup>2</sup> ð Þ ð Þ <sup>α</sup>�s1 <sup>2</sup>

� arctg <sup>ω</sup>

<sup>α</sup>2þω<sup>2</sup> ð Þ ð Þ <sup>α</sup>�s1 <sup>2</sup>

δT tðÞ¼ AT

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> ð Þ <sup>α</sup>�s1 <sup>2</sup> þω<sup>2</sup> <sup>q</sup> ; B5 <sup>¼</sup> <sup>1</sup>

δP tðÞ¼ AT

d2 ð Þ δP dt<sup>2</sup> <sup>þ</sup> <sup>τ</sup>

� �

1 <sup>4</sup> γττ0M0

dt<sup>2</sup> <sup>þ</sup> <sup>δ</sup>Tinl <sup>þ</sup> <sup>γ</sup> � <sup>1</sup>

The transient that is initiated by the temperature perturbation δTinl= AT is

<sup>0</sup>s1 <sup>α</sup>2þω<sup>2</sup> ð Þ; <sup>β</sup> <sup>¼</sup> arctg <sup>2</sup>αω

; B4 <sup>¼</sup> s2

The transient that is initiated by the pressure perturbation AP is described as

dð Þ δPout dt

dð Þ δP

þ 1 2

d2 ð Þ δT dt2 <sup>þ</sup>

2γ τ

<sup>α</sup>2�ω2<sup>þ</sup> <sup>4</sup> γτ2 0

1 s1 ð Þ s1�<sup>α</sup> <sup>2</sup> <sup>þ</sup>ω<sup>2</sup> ½ �.

<sup>þ</sup>ω<sup>2</sup>; <sup>β</sup><sup>1</sup> ¼ �arctg <sup>ω</sup>

τ2 0 2τ s1 ð Þ s1�<sup>α</sup> <sup>2</sup>

The model consists of Eqs. (6), (7), (9) and (10). Linearized Eqs. (6) and (7) are of the format presented in Eqs. (76) and (78). As a result of the linearization, we get

dt <sup>þ</sup> <sup>δ</sup><sup>P</sup> <sup>¼</sup>

dð Þ δPinl dt þ

A3e<sup>α</sup><sup>t</sup> sin <sup>ω</sup><sup>t</sup> <sup>þ</sup> <sup>β</sup><sup>1</sup> ð Þþ B3es1t ð Þ <sup>þ</sup> <sup>K</sup> , (81)

ω

� arctg <sup>ω</sup>

A4e<sup>α</sup><sup>t</sup> sin ð Þþ <sup>ω</sup><sup>t</sup> <sup>þ</sup> <sup>β</sup> B4es1t ð Þ, (82)

A5e<sup>α</sup><sup>t</sup> sin <sup>ω</sup><sup>t</sup> <sup>þ</sup> <sup>β</sup><sup>1</sup> ð Þþ B5es1t ð Þ, (83)

α�s1 ;

A6e<sup>α</sup><sup>t</sup> sin <sup>ω</sup><sup>t</sup> <sup>þ</sup> <sup>β</sup><sup>2</sup> ð Þþ B6es1t ð Þ <sup>þ</sup> K2 , (84)

<sup>þ</sup>ω<sup>2</sup> ½ �; K2 ¼ � <sup>τ</sup><sup>2</sup>

0 <sup>2</sup>τs1 <sup>α</sup>2þω<sup>2</sup> ð Þ;

ð Þ δPinl þ δPout *:*

� �

dð Þ δPout dt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

vuuut ;

<sup>α</sup>2�ω2<sup>þ</sup> <sup>4</sup> γτ2 0

� �<sup>2</sup>

<sup>α</sup>2þω<sup>2</sup> ð Þ ð Þ <sup>α</sup>�s1 <sup>2</sup>

<sup>α</sup>�s1 � arctg <sup>ω</sup>

*:*

þð Þ <sup>2</sup>αω <sup>2</sup>

<sup>þ</sup>ω<sup>2</sup> ½ �

α;

(79)

(80)

$$\frac{\text{d}\mathbf{W}\_{\text{inl}}}{\text{d}\mathbf{t}} - \frac{\text{d}\mathbf{W}\_{\text{out}}}{\text{d}\mathbf{t}} = \frac{\gamma}{\gamma - 1} \left( \frac{\text{\tau}}{\gamma} \frac{\text{d}^2(\text{\delta T})}{\text{d}\mathbf{t}^2} + \frac{\text{d}(\text{\delta T})}{\text{d}\mathbf{t}} - \frac{\text{d}(\text{\delta T}\_{\text{inl}})}{\text{d}\mathbf{t}} \right). \tag{86}$$

Let us determine <sup>d</sup>ð Þ <sup>δ</sup><sup>P</sup> dt from (76) and (78):

$$\frac{\mathbf{d}(\delta \mathbf{P})}{\mathbf{d}\mathbf{t}} = \frac{\gamma}{\gamma - 1} \left[ \frac{\mathbf{d}(\delta \mathbf{T})}{\mathbf{d}\mathbf{t}} + \frac{\mathbf{1}}{\pi} (\delta \mathbf{T} - \delta \mathbf{T}\_{\mathrm{inl}}) \right]. \tag{87}$$

Having equalized the right sides of Eqs. (85) and (86), derived the obtained equation, and substituted the derivative (87) in it, we get a differential equation for the temperature in the volume:

$$\begin{split} \frac{\pi \tau\_{0}^{2}}{2\mathbf{C}} \frac{\mathbf{d}^{3}(\delta \mathbf{T})}{\mathbf{d}^{3}} + \frac{\tau\_{0}^{2}}{\mathbf{C}} (0.5\boldsymbol{\gamma} + 2\boldsymbol{\xi}) \frac{\mathbf{d}^{2}(\delta \mathbf{T})}{\mathbf{d}\mathbf{t}^{2}} + \frac{\tau}{\mathbf{C}} \left[ 2 + (2\boldsymbol{\gamma} + 1)\xi \mathbf{M}\_{0}^{2} \right] \frac{\mathbf{d}(\delta \mathbf{T})}{\mathbf{d}\mathbf{t}} + \delta \mathbf{T} \\ = \frac{\mathbf{\gamma} \tau\_{0}^{2}}{2\mathbf{C}} \frac{\mathbf{d}^{2}(\delta \mathbf{T}\_{\text{inl}})}{\mathbf{d}\mathbf{t}^{2}} + \frac{(\boldsymbol{\gamma} + 1)\xi \tau\_{0} \mathbf{M}\_{0}}{\mathbf{C}} \frac{\mathbf{d}(\delta \mathbf{T}\_{\text{inl}})}{\mathbf{d}\mathbf{t}} + \delta \mathbf{T}\_{\text{inl}} \\ + \frac{\tau(\boldsymbol{\gamma} - 1)}{\mathbf{C}} \left[ \left( \frac{\mathbf{P}\_{\text{inl}} \, \mathrm{o}}{\mathrm{P}\_{0}} + \xi \gamma \mathbf{M}\_{0}^{2} \right) \frac{\mathbf{d}(\delta \mathbf{S}\_{\text{inl}})}{\mathbf{d}\mathbf{t}} + \frac{\mathbf{P}\_{\text{out}} \, \mathrm{d}(\delta \mathbf{S}\_{\text{out}})}{\mathbf{P}\_{0}} \right] \end{split} \tag{88}$$

where C <sup>¼</sup> <sup>2</sup> <sup>þ</sup> ξγM2 0.

Let us use the Laplace transform and transfer functions W<sup>p</sup> <sup>T</sup>ð Þ<sup>s</sup> , WT pinlð Þ<sup>s</sup> , WT outð Þs , and WT Tinlð Þs to obtain the equation for the pressure. The transfer functions we will get from Eqs. (87) and (88). The final result is

$$\begin{aligned} &\frac{\pi \tau\_0^2}{2\mathbf{C}} \frac{\mathbf{d}^3(\mathbf{S} \mathbf{P})}{\mathbf{d}^3} + \frac{\tau\_0^2}{\mathbf{C}} (\mathbf{0}.5\mathbf{\gamma} + 2\xi) \frac{\mathbf{d}^2(\delta \mathbf{P})}{\mathbf{d}\mathbf{t}^2} + \frac{\tau}{\mathbf{C}} \left[2 + (2\mathbf{\gamma} + 1)\xi \mathbf{M}\_0^2\right] \frac{\mathbf{d}(\delta \mathbf{P})}{\mathbf{d}\mathbf{t}} + \delta \mathbf{P} = \\ &\frac{\gamma \tau\_0^2}{2\mathbf{C}} \frac{\mathbf{d}^2(\delta \mathbf{T}\_{\text{ind}})}{\mathbf{d}^2} + \frac{\tau\_0 \xi \mathbf{y} \mathbf{M}\_0}{\mathbf{C}} \frac{\mathbf{d}(\delta \mathbf{T}\_{\text{ind}})}{\mathbf{d}\mathbf{t}} + \frac{\tau}{\mathbf{C}} \left[\left(\frac{\mathbf{P}\_{\text{ind}} \, \mathbf{0}}{\mathbf{P}\_0} + \xi \mathbf{y} \mathbf{M}\_0^2\right) \frac{\mathbf{d}(\delta \mathbf{P}\_{\text{ind}})}{\mathbf{d}\mathbf{t}} + \frac{\mathbf{P}\_{\text{out 0}} \, \mathbf{d}(\delta \mathbf{S}\_{\text{out}})}{\mathbf{P}\_0}\right] + \\ &+ \frac{\mathbf{1}}{\mathbf{C}} \left[\left(\frac{\mathbf{P}\_{\text{ind}} \, \mathbf{0}}{\mathbf{P}\_0} + \xi \mathbf{y} \mathbf{M}\_0^2\right) \delta \mathbf{P}\_{\text{ind}} + \frac{\mathbf{P}\_{\text{out 0}} \, \mathbf{0}}{\mathbf{P}\_0} \delta \mathbf{P}\_{\text{out}}\right]. \end{aligned} \tag{89}$$

The transient state, which is initiated by the temperature perturbation δTinl, (the magnitude of the step is AT) is described as

$$\delta \mathbf{T}(\mathbf{t}) = \mathbf{A}\_{\rm T} \frac{\eta}{\pi} \begin{pmatrix} \mathbf{A}\_{\rm 6} \cdot \mathbf{1}(\mathbf{t}) - (\mathbf{A}\_{\rm 6} + \mathbf{B}\_{\rm 6}) \mathbf{e}^{s\_{\rm 1} \rm t} \\\\ + \frac{\mathbf{B}\_{\rm 6} \sqrt{\left(\frac{\mathbf{K}\_{\rm 6}}{\rm B\_{\rm 6}} - \mathbf{a}\right)^{2} + \alpha^{2}}}{\alpha} \mathbf{e}^{\rm at} \cdot \sin\left(\alpha \mathbf{t} + \boldsymbol{\beta}\_{3}\right) \end{pmatrix},\tag{90}$$

where <sup>α</sup> and <sup>ω</sup> are expressed by (48) and (39): A6 ¼ � b1 s1 <sup>α</sup>2þω<sup>2</sup> ð Þ; B6 <sup>¼</sup> <sup>2</sup>αs1�s<sup>2</sup> ð Þ<sup>1</sup> A6�s1�a1 <sup>α</sup>2þω2�s1ð Þ <sup>2</sup>α�s1 ; K3 <sup>¼</sup> <sup>1</sup> <sup>þ</sup> s1A6 � ð Þ <sup>2</sup><sup>α</sup> � s1 B6; <sup>β</sup><sup>3</sup> <sup>¼</sup> arctg <sup>ω</sup> K3 B6 �α ; a1 <sup>¼</sup> <sup>2</sup>ð Þ <sup>γ</sup>þ<sup>1</sup> <sup>ξ</sup>M0 γτ<sup>0</sup> ; b1 <sup>¼</sup> 2C γτ<sup>2</sup> 0 ;

$$\delta \mathbf{P}(\mathbf{t}) = \mathbf{A}\_{\Gamma} \frac{\gamma}{\pi} \left( \mathbf{A}\_{\mathcal{T}} \mathbf{e}^{\mathbf{s} \mathbf{t}} + \frac{\mathbf{B}\_{\mathcal{T}} \sqrt{\left(\frac{\mathbf{K}\_{4}}{\mathcal{B}\_{\mathcal{T}}} - \alpha\right)^{2} + \alpha^{2}}}{\alpha} \mathbf{e}^{\mathbf{at}} \cdot \sin\left(\alpha \mathbf{t} + \beta\_{4}\right) \right), \tag{91}$$

where A7 ¼ �B7; B7 ¼ � a2þs1 <sup>α</sup>2þω2�s1ð Þ <sup>2</sup>α�s1 ; K4 <sup>¼</sup> <sup>α</sup>2þω<sup>2</sup> ð ÞA7�a2 s1 ; <sup>β</sup><sup>4</sup> <sup>¼</sup> arctg <sup>ω</sup> D2 C2 �α ; a2 <sup>¼</sup> <sup>2</sup>ξγM0 <sup>τ</sup>0<sup>γ</sup> .

The transient that is caused by the pressure perturbation AP is described as

$$\delta \mathbf{T}(\mathbf{t}) = \mathbf{A}\_{\mathsf{P}} \frac{2(\mathsf{y} - \mathsf{I})}{\pi\_{\mathsf{0}}^{2}} \left( \frac{\mathsf{P}\_{\text{in1}} \mathsf{o}}{\mathsf{P}\_{\text{0}}} + \mathsf{f} \mathsf{y} \mathsf{M}\_{\text{0}}^{2} \right) \left( \mathsf{A}\_{\mathsf{S}} \mathsf{e}^{\mathsf{s}\_{\mathsf{t}} \mathsf{t}} + \frac{\mathsf{B}\_{\mathsf{S}} \sqrt{\left( \frac{\mathsf{K}\_{\mathsf{S}}}{\mathsf{B}\_{\mathsf{s}}} - \mathsf{a} \right)^{2} + \mathsf{a}^{2}}}{\mathsf{a} \mathsf{o}} \mathsf{e}^{\mathsf{at}} \cdot \sin \left( \mathsf{ot} + \mathsf{f}\_{\mathsf{S}} \right) \right), \tag{92}$$

where A8 ¼ �B8; B8 ¼ � <sup>1</sup> <sup>α</sup>2þω2�s1ð Þ <sup>2</sup>α�s1 ; K5 <sup>¼</sup> <sup>α</sup>2þω<sup>2</sup> ð ÞA8�<sup>1</sup> s1 ; <sup>β</sup><sup>1</sup> <sup>¼</sup> arctg <sup>ω</sup> K5 B8 �α .

$$\delta \mathbf{P}(\mathbf{t}) = \mathbf{A}\_{\mathsf{P}} \frac{2}{\pi\_{0}^{2}} \left( \frac{\mathbf{P}\_{\text{in}1}}{\mathbf{P}\_{0}} + \xi\_{7} \mathsf{M}\_{0}^{2} \right) \left( \mathbf{A}\_{\mathsf{P}} \cdot \mathbf{1}(\mathbf{t}) - \left( \mathbf{A}\_{\mathsf{P}} + \mathsf{B}\_{\mathsf{P}} \right) \mathsf{e}^{\mathsf{a}\_{1} \mathsf{t}} + \frac{\mathsf{B} \sqrt{\left( \frac{\mathbf{K}\_{\mathsf{d}}}{\mathbf{B}\_{\mathsf{p}}} - \mathsf{a} \right)^{2} + \mathsf{a}^{2}}}{\mathsf{a} \mathsf{o}} \mathsf{e}^{\mathsf{a} \mathsf{t}} \cdot \sin \left( \mathsf{ot} + \mathsf{\mathcal{J}}\_{6} \right) \right), \tag{93}$$

**Author details**

**55**

Sergiy Yepifanov and Roman Zelenskyi\*

provided the original work is properly cited.

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

*Gas Turbine Simulation Taking into Account Dynamics of Gas Capacities*

*DOI: http://dx.doi.org/10.5772/intechopen.90490*

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

© 2019 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,

$$\begin{aligned} \text{where } \mathbf{A}\_{\theta} &= -\frac{\mathbf{b}\_{3}}{s\_{1}(\mathbf{a}^{2} + \mathbf{a}^{2})}; \mathbf{B}\_{\theta} = \frac{\left(2\mathbf{a}\mathbf{s}\_{1} - \mathbf{s}\_{1}^{2}\right)\mathbf{A}\_{\theta} - 1}{\mathbf{a}^{2} + \mathbf{a}^{2} - s\_{1}(2\mathbf{a} - \mathbf{s}\_{1})}; \mathbf{K}\_{6} = \mathbf{s}\_{1}\mathbf{A}\_{\theta} - (2\mathbf{a} - \mathbf{s}\_{1})\mathbf{B}\_{\theta};\\ \mathbf{B}\_{6} = \mathbf{a}\mathbf{c}\mathbf{t}\mathbf{g}\frac{\mathbf{a}}{\mathbf{k}\_{\theta} - \mathbf{a}}; \mathbf{b}\_{3} = \frac{1}{\tau}. \end{aligned}$$

*Gas Turbine Simulation Taking into Account Dynamics of Gas Capacities DOI: http://dx.doi.org/10.5772/intechopen.90490*

## **Author details**

where <sup>α</sup> and <sup>ω</sup> are expressed by (48) and (39): A6 ¼ � b1

*Modeling of Turbomachines for Control and Diagnostic Applications*

B7

r

<sup>α</sup>2þω2�s1ð Þ <sup>2</sup>α�s1 ; K3 <sup>¼</sup> <sup>1</sup> <sup>þ</sup> s1A6 � ð Þ <sup>2</sup><sup>α</sup> � s1 B6; <sup>β</sup><sup>3</sup> <sup>¼</sup> arctg <sup>ω</sup> K3

K4 B7 � α � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>α</sup>2þω2�s1ð Þ <sup>2</sup>α�s1 ; K4 <sup>¼</sup> <sup>α</sup>2þω<sup>2</sup> ð ÞA7�a2

B8

r

ω

The transient that is caused by the pressure perturbation AP is described as

A8es1t <sup>þ</sup>

<sup>α</sup>2þω2�s1ð Þ <sup>2</sup>α�s1 ; K5 <sup>¼</sup> <sup>α</sup>2þω<sup>2</sup> ð ÞA8�<sup>1</sup>

0

BB@

A9 � 1 tð Þ� ð Þ A9 <sup>þ</sup> B9 <sup>e</sup>s1t <sup>þ</sup>

þ ω<sup>2</sup>

B6 <sup>¼</sup> <sup>2</sup>αs1�s<sup>2</sup> ð Þ<sup>1</sup> A6�s1�a1

δP tðÞ¼ AT

2ð Þ γ � 1 τ2 0

γ τ 0

BB@

where A7 ¼ �B7; B7 ¼ � a2þs1

Pinl 0 P0

where A8 ¼ �B8; B8 ¼ � <sup>1</sup>

<sup>þ</sup> ξγM2 0

� �

; b3 <sup>¼</sup> <sup>1</sup> τ . <sup>þ</sup> ξγM<sup>2</sup> 0

� �

0 BB@

s1 <sup>α</sup>2þω<sup>2</sup> ð Þ; B9 <sup>¼</sup> <sup>2</sup>αs1�s<sup>2</sup> ð Þ<sup>1</sup> A9�<sup>1</sup>

A7es1t <sup>þ</sup>

b1 <sup>¼</sup> 2C γτ<sup>2</sup> 0 ;

a2 <sup>¼</sup> <sup>2</sup>ξγM0 <sup>τ</sup>0<sup>γ</sup> .

δT tðÞ¼ AP

δP tðÞ¼ AP

β<sup>6</sup> ¼ arctg <sup>ω</sup> K6

**54**

2 τ2 0

where A9 ¼ � b3

B9 �α

Pinl 0 P0

s1 <sup>α</sup>2þω<sup>2</sup> ð Þ;

; a1 <sup>¼</sup> <sup>2</sup>ð Þ <sup>γ</sup>þ<sup>1</sup> <sup>ξ</sup>M0 γτ<sup>0</sup> ;

, (91)

C2 �α ;

<sup>e</sup><sup>α</sup><sup>t</sup> � sin <sup>ω</sup><sup>t</sup> <sup>þ</sup> <sup>β</sup><sup>5</sup> ð Þ

B8 �α .

<sup>ω</sup> <sup>e</sup><sup>α</sup><sup>t</sup> � sin <sup>ω</sup><sup>t</sup> <sup>þ</sup> <sup>β</sup><sup>6</sup> ð Þ

1

CCA,

1 CCA,

(93)

(92)

1

CCA

B6 �α

s1 ; <sup>β</sup><sup>4</sup> <sup>¼</sup> arctg <sup>ω</sup> D2

þ ω<sup>2</sup>

s1 ; <sup>β</sup><sup>1</sup> <sup>¼</sup> arctg <sup>ω</sup> K5

þ ω<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K6 B9 � α � �<sup>2</sup>

<sup>e</sup><sup>α</sup><sup>t</sup> � sin <sup>ω</sup><sup>t</sup> <sup>þ</sup> <sup>β</sup><sup>4</sup> ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ω

K5 B8 � α � �<sup>2</sup>

B9

<sup>α</sup>2þω2�s1ð Þ <sup>2</sup>α�s1 ; K6 <sup>¼</sup> s1A9 � ð Þ <sup>2</sup><sup>α</sup> � s1 B9;

r

Sergiy Yepifanov and Roman Zelenskyi\* National Aerospace University "Kharkiv Aviation Institute", Kharkiv, Ukraine

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

© 2019 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**

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[2] Fawke AJ, Saravanamuttoo HIH. Digital Computer Methods for Prediction of Gas Turbine Dynamic Response. Report No. 710550. England: Society of Automotive Engineers, Station, Newcastle; 1971

[3] Fawke AJ. Digital computer simulation of gas turbine dynamic behavior [thesis]. Bristol, England: University of Bristol; 1970

[4] Glikman BF. Mathematical Models of Pneumatic Hydraulic Systems. Russia, Nauka: Moscow; 1986. 368p

[5] Gas Turb. Available from: http:// www.gasturb.de

[6] Zhao YS, Hu J, Wu TY, Chen JJ. Steady-state and transient performances simulation of large civil aircraft auxiliary power unit. ISABE Paper ISABE-2011-1323; 2011. 10p

[7] Ghigliazza F, Traverso A, Pascenti M, Massardo AF. Micro gas turbine realtime modeling: Test rig verification. ASME Paper GT2009-59124; 2009. 8p

[8] Davison CR, Birk AM. Comparison of transient modeling techniques for a micro turbine engine. ASME Paper GT2006-91088; 2006. 10p

[9] Koçer G, Uzol O, Yavrucuk İ. Simulation of the transient response of a helicopter turboshaft engine to hot-gas ingestion. ASME Paper GT2008-51164; 2008. 6p

[10] Gang Y, Jianguo S, Xianghua H, Wei-Sheng S. A Non-Iterative Method of Aero-Engine Modeling Using Complementary Variables. Journal of

Aerospace Power. Beijing, China: Chinese Society of Aeronautics and Astronautics; 2003

with a gas turbine performance model. ASME Paper GT2017-63881; 2017. 11p

*DOI: http://dx.doi.org/10.5772/intechopen.90490*

*Gas Turbine Simulation Taking into Account Dynamics of Gas Capacities*

[19] Mohajer A, Abbasi E. Development of compression system dynamic simulation code for testing and

designing of anti-surge control system. ASME Paper GT2017-63212; 2017. 8p

[20] Shevyakov AA. Automatic Control Systems of the Air-Breathing Power

[21] Dobryansky GV, Martyanova ТS. Dynamics of Aircraft Engines.

Mechanical Engineering: Moscow; 1989.

[22] Kurosaki M, Sasamoto M, Asaka K, Nakamura K, Kakiuchi D. An efficient transient simulation method for a gas turbine volume dynamics model. ASME

Paper GT2018-75353; 2018. 11p

[23] Gurevich O. Aircraft Engines Automatic Control Systems.

Press; 2011. 208p

University Press; 1994

**57**

Encyclopedic Guide. Moscow: Torus

[24] Chen CT. Introduction to Linear System Theory. Philadelphia: Holt, Rinehart and Winston; 1970

[25] Bryson AE. Control of a Spacecraft and Aircraft. Princeton, NJ: Princeton

[26] Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. Definitions, theorems and Formulas for Reference and Review. New York/ Toronto/London: McGraw Hill Book Company, Inc.; 1961. pp. 47-48

Plants. Moscow: Mechanical Engineering; 1992. 432p

240p

[11] Kong X, Wang X, Tan D, He A. A non-iterative aero engine model based on volume effect. AIAA Paper AIAA2011-6623; 2011

[12] Yepifanov SV, Zelenskyi RL. The simulation of the pneumatic volume dynamics as a part of transients in the gas path of gas turbine engines. Aerospace Technique and Technology. 2007;**10**(46):49-54

[13] Shi Y, Tu Q, Jiang P, Zheng H, Cai Y. Investigation of the compressibility effects on engine transient performance. ASME Paper GT2015-42889; 2015. 8p

[14] Culmone MV, Garcia-Rosa N, Carbonneau X. Sensitivity analysis and experimental validation of transient performance predictions for a shortrange turbofan. ASME Paper GT2016-57257; 2016. 10p

[15] Wang C, Li Y-G, Yang B-Y. Transient performance simulation of aircraft engine integrated with fuel and control systems. ISABE Paper ISABE-2015-20103; 2015. 11p

[16] Kopasakis G, Connolly JW, Paxson DE, Ma P. Volume dynamics propulsion system modeling for supersonics vehicle research. ASME Paper GT2008-50524; 2008. 10p

[17] Henke M, Monz T, Aigner M. Introduction of a new numerical simulation tool to analyze micro gas turbine cycle dynamics. ASME Paper GT2016-56335; 2016. 11p

[18] Wang C, Li YG. Hydraulic fuel system simulation using Newton-Raphson method and its integration *Gas Turbine Simulation Taking into Account Dynamics of Gas Capacities DOI: http://dx.doi.org/10.5772/intechopen.90490*

with a gas turbine performance model. ASME Paper GT2017-63881; 2017. 11p

**References**

[1] Jaw LC, Mattingly JD. Aircraft Engine Controls: Design, System Analysis, and Health Monitoring. Reston, USA: AIAA; 2009. 294p

*Modeling of Turbomachines for Control and Diagnostic Applications*

Aerospace Power. Beijing, China: Chinese Society of Aeronautics and

on volume effect. AIAA Paper

[11] Kong X, Wang X, Tan D, He A. A non-iterative aero engine model based

[12] Yepifanov SV, Zelenskyi RL. The simulation of the pneumatic volume dynamics as a part of transients in the gas path of gas turbine engines. Aerospace Technique and Technology.

[13] Shi Y, Tu Q, Jiang P, Zheng H,

[14] Culmone MV, Garcia-Rosa N, Carbonneau X. Sensitivity analysis and experimental validation of transient performance predictions for a short-

range turbofan. ASME Paper GT2016-57257; 2016. 10p

[15] Wang C, Li Y-G, Yang B-Y. Transient performance simulation of aircraft engine integrated with fuel and control systems. ISABE Paper ISABE-

[16] Kopasakis G, Connolly JW, Paxson DE, Ma P. Volume dynamics propulsion system modeling for supersonics vehicle research. ASME Paper GT2008-50524; 2008. 10p

[17] Henke M, Monz T, Aigner M. Introduction of a new numerical simulation tool to analyze micro gas turbine cycle dynamics. ASME Paper

[18] Wang C, Li YG. Hydraulic fuel system simulation using Newton-Raphson method and its integration

GT2016-56335; 2016. 11p

2015-20103; 2015. 11p

Cai Y. Investigation of the compressibility effects on engine transient performance. ASME Paper

GT2015-42889; 2015. 8p

Astronautics; 2003

AIAA2011-6623; 2011

2007;**10**(46):49-54

[2] Fawke AJ, Saravanamuttoo HIH. Digital Computer Methods for Prediction of Gas Turbine Dynamic Response. Report No. 710550. England: Society of Automotive Engineers,

Station, Newcastle; 1971

University of Bristol; 1970

Nauka: Moscow; 1986. 368p

www.gasturb.de

[3] Fawke AJ. Digital computer simulation of gas turbine dynamic behavior [thesis]. Bristol, England:

[4] Glikman BF. Mathematical Models of Pneumatic Hydraulic Systems. Russia,

[5] Gas Turb. Available from: http://

[6] Zhao YS, Hu J, Wu TY, Chen JJ. Steady-state and transient performances

[7] Ghigliazza F, Traverso A, Pascenti M, Massardo AF. Micro gas turbine realtime modeling: Test rig verification. ASME Paper GT2009-59124; 2009. 8p

[8] Davison CR, Birk AM. Comparison of transient modeling techniques for a micro turbine engine. ASME Paper

Simulation of the transient response of a helicopter turboshaft engine to hot-gas ingestion. ASME Paper GT2008-51164;

[10] Gang Y, Jianguo S, Xianghua H, Wei-Sheng S. A Non-Iterative Method of Aero-Engine Modeling Using Complementary Variables. Journal of

GT2006-91088; 2006. 10p

2008. 6p

**56**

[9] Koçer G, Uzol O, Yavrucuk İ.

simulation of large civil aircraft auxiliary power unit. ISABE Paper ISABE-2011-1323; 2011. 10p

[19] Mohajer A, Abbasi E. Development of compression system dynamic simulation code for testing and designing of anti-surge control system. ASME Paper GT2017-63212; 2017. 8p

[20] Shevyakov AA. Automatic Control Systems of the Air-Breathing Power Plants. Moscow: Mechanical Engineering; 1992. 432p

[21] Dobryansky GV, Martyanova ТS. Dynamics of Aircraft Engines. Mechanical Engineering: Moscow; 1989. 240p

[22] Kurosaki M, Sasamoto M, Asaka K, Nakamura K, Kakiuchi D. An efficient transient simulation method for a gas turbine volume dynamics model. ASME Paper GT2018-75353; 2018. 11p

[23] Gurevich O. Aircraft Engines Automatic Control Systems. Encyclopedic Guide. Moscow: Torus Press; 2011. 208p

[24] Chen CT. Introduction to Linear System Theory. Philadelphia: Holt, Rinehart and Winston; 1970

[25] Bryson AE. Control of a Spacecraft and Aircraft. Princeton, NJ: Princeton University Press; 1994

[26] Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. Definitions, theorems and Formulas for Reference and Review. New York/ Toronto/London: McGraw Hill Book Company, Inc.; 1961. pp. 47-48

**Chapter 4**

System

**Abstract**

A New Approach for Model

Unmeasured Parameters in an

Monitoring systems to predict the remaining lifetime of gas turbine engines are a major field of investigation, in particular, the monitoring systems that allow an online prediction. This chapter introduces and analyzes a new approach to develop mathematical models to estimate unmeasured parameters in an engine lifetime monitoring system; these models in contrast to previously developed models allow an on-line monitoring of unmeasured parameters, which are necessary to perform an on-line lifetime prediction. The blade of a high-pressure turbine (HPT) of a twospool free turbine power plant is the test case. Several candidate models are developed for each unmeasured parameter; the best models are selected by their accuracy and robustness using the instrumental and truncation error as criteria. Ten faulty engine conditions are considered to analyze the model robustness. Two methods for model developing are compared; the first method uses physics-based models (proposed in this chapter), and the second method develops the models using the similarity concept (reference methodology). The results of the comparison show that the physics-based models are more robust to engine faults and overall they

Engine Lifetime Monitoring

deliver a significantly more accurate prediction of the engine lifetime.

relations, unmeasured parameters

lifetime and improvement of the engine's reliability.

**1. Introduction**

**59**

**Keywords:** gas turbine, lifetime prediction, model developing, thermodynamic

Lifetime monitoring systems are an effective way to perform condition base maintenance of gas turbine engines [1–3]; this allows a better use of the available

Several approaches exist to predict the remaining lifetime, such as neural networks [4–7], finite element analysis (FEA) [8–10], and statistical methods [11]; however, in order to significantly enhance the accuracy of the lifetime prediction, it is necessary to estimate the lifetime in real time (on-line prediction) using actual conditions [12, 13]. All of the previously cited approaches require a large amount of computing resources, making them not suitable for an on-line application; another

Developing to Estimate

*Cristhian Maravilla and Sergiy Yepifanov*

## **Chapter 4**
