1.2 Thermodynamic model

The above reasoning explains why a component-based nonlinear gas turbine model is considered as principle for the design of the engine itself and for developing its control and monitoring systems. It is a highly complex thermodynamic model based on the aerothermal calculations of a gas path and the description of engine's components (compressor, combustion chamber, turbine, etc.) by nonlinear performance maps. Foundations of the thermodynamic models can be found in [2, 4].

The thermodynamic model comprises two interrelated parts, namely nonlinear static model and nonlinear dynamic model (NDM). The nonlinear static model allows investigating steady-state performances of the engine before its final creation. This model may include more than 100 algebraic and transcendent mathematical relations and, in general, presents a system of nonlinear equations (see [2]). These equations reflect the mass and energy balance between engine components during stationary operation, and the number of such equations typically varies from 5 to 15.

Once the static model has been created, the detailed nonlinear dynamic model can be developed with fewer efforts because it is similar to the static model. The principal difference is that the mentioned algebraic equations of mass and energy balance at steady states are now written in the form of differential equations at transients. The number of such equations corresponds to the number of mass and energy accumulators simulated. Since NDM is a complex and relatively slow procedure, many simplified models are constructed on its basis to be used for the aims of engine control and diagnosis. Nevertheless, along with increasing processor and developing the methods of execution time minimization, direct use of the NDM in real time is becoming possible [2].

have many other applications, such as design of a control system, condition monitoring, fault diagnosis, and system identification. The latter, for example, enables simulating the performances of a particular engine by model fitting to experimental data collected in test beds or at field conditions. In this way, control and diagnostic algorithms can be improved due to a more accurate individual engine model used

The models of technical systems, in particular GT models, can be divided, on the one hand, into linear and nonlinear and, on the other hand, data-driven (also wellknown as black-box models) and physics-based (also called white-box models). In spite of wide application of simplified linear modeling and simulation of GTs, the behavior of these machines is usually nonlinear, and precise nonlinear models are

The data-driven models do not need detailed knowledge about the system to model. Instead, they use available empiric information and are determined by optimization methods or, in the case of artificial neural networks, through machine learning. Because of their simplicity, such models are widely used in GT design. A detailed description of different gas turbine data-driven models can be found, for

Physics-based modeling relies on physical laws of the functioning of turbomachines and therefore allows realistic simulation of their behavior. These models are more complex and less used. However, they contain the information difficult to draw from empiric data and are frequently employed as a basis to create simpler data-driven models. Thus, physics-based modeling may be considered as a main gas

The above reasoning explains why a component-based nonlinear gas turbine model is considered as principle for the design of the engine itself and for developing its control and monitoring systems. It is a highly complex thermodynamic model based on the aerothermal calculations of a gas path and the description of engine's components (compressor, combustion chamber, turbine, etc.) by nonlinear performance maps. Foundations of the thermodynamic models can be found in

The thermodynamic model comprises two interrelated parts, namely nonlinear static model and nonlinear dynamic model (NDM). The nonlinear static model allows investigating steady-state performances of the engine before its final creation. This model may include more than 100 algebraic and transcendent mathematical relations and, in general, presents a system of nonlinear equations (see [2]). These equations reflect the mass and energy balance between engine components during stationary operation, and the number of such equations typically varies from

Once the static model has been created, the detailed nonlinear dynamic model can be developed with fewer efforts because it is similar to the static model. The principal difference is that the mentioned algebraic equations of mass and energy balance at steady states are now written in the form of differential equations at transients. The number of such equations corresponds to the number of mass and energy accumulators simulated. Since NDM is a complex and relatively slow procedure, many simplified models are constructed on its basis to be used for the aims of engine control and diagnosis. Nevertheless, along with increasing processor and developing the methods of execution time minimization, direct use of the NDM in

instead of a general model [2].

unavoidable [3].

Aerospace Engineering

example, in book [3].

turbine mathematical modeling type.

real time is becoming possible [2].

1.2 Thermodynamic model

[2, 4].

5 to 15.

154

The development and use of the thermodynamic models have started in the 1970s, in many respects, by the studies of Saravanamuttoo et al. (for example, [5]). Since that time, many improvements related to higher accuracy and more detailed engine's component description were introduced in this model; some of them are mentioned below.

Stamatis et al. proposed in [6] the scheme of adaptive simulation by a nonlinear system identification technique that later was used for multipoint gas turbine diagnosis [7]. Ellipsoid functions were introduced in [8] for more accurate description of the components' maps and better identification of a whole engine at steady states and transients. The authors of paper [9] developed a stage-based compressor model to be used in the thermodynamic model instead of a compressor performance map. This modified thermodynamic model allows the localization of the faulty stages of a multistage compressor and identification of the three compressor degradation mechanisms: fouling, tip clearance increase, and erosion of aerofoils. Thus, gas turbine diagnostics become more profound. Since the early 1990s, Joachim Kurzke has developed the universal program GasTurb for nonlinear physics-based gas turbine simulation [10, 11]. This commercial software allows simulating different types of engines and helps to solve various design and analysis problems. The program GasTurb has special tools to analyze, correct, and enhance the component maps contributing in this way to the accuracy of final engine simulation. Another way to improve the simulation accuracy is proposed by Volponi et al. [12]. As an engine measurement system has individual systematic measurement errors, the authors propose to compensate them by an additional data-driven model on the basis of artificial neural networks. The introduced hybrid model is constructed from a traditional thermodynamic model and this data-driven model. It is shown that the hybrid model can more accurately simulate the performance of a particular engine than the thermodynamic model itself.

The above improvements are related to a static part of the thermodynamic model or both static and dynamic parts. However, the description of engine transients has specific problems to solve, and their solution can additionally improve the dynamic part, namely, detailed nonlinear dynamic model. For gas turbine control and monitoring systems as a whole and, more importantly, for the systems of aircraft gas turbine engines, accurate and fast NDMs are in high demand [2, 13]. These detailed nonlinear models will be useful for the implementation of model predictive control and more effective diagnosis at transients where simplified Kalman filter-based techniques have often been used to date [14]. Since here, this chapter will deal only with such models.

Modern NDMs generally take into considerations three "accumulators":


The volume dynamics is very fast and it is important for controller design. The rotor dynamics lasts for aircraft engines about 10–15 s and has the largest influence on engine performance. The heat exchange dynamics may last many minutes but its direct influence on gas path variables is small because the heat interchange between gas flow and engine-heated parts (HPs) is by far smaller than total energy of the gas. These reasons explain why the models that simulate the rotor dynamics only are still used in diagnostics.
