2.1 Equations of State

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The equation of state can be written in different forms depending on the independent variables taken. Numerical algorithms should allow to calculate and optimize the axial turbine stages, both with an ideal and a real working fluid. It uses a single method of calculating the parameters of the state of the working fluid, in which as the independent variables are taken enthalpy i and pressure P:

For a perfect gas equation of state with P and i variables are very simple:

For the water steam approximation formula proposed in [7] is used, which established a procedure to calculate parameters of superheated and wet fluid. It is easy to verify that the knowledge of the value of the velocity coefficient Read More

1.5 The Practice of Numerical Methods Usage for Local Leveled Optimization Problems Solution

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To solve demanded by practice of axial turbines design multi-criteria problems, multi-parameter and multi-mode optimization of the multistage flow path further development and improvement of appropriate numerical methods and approaches required.

It should be noted some features of numerical solution of problems related to the optimization of design objects based on their modes of operation, multi-modal objective functions, as well as issues related to the multi-objective optimization problems.

Some aspects of the above problems solutions are given below.

1.5.1 Solution of the Multi-Criteria Optimization Problems

Set out in section 1.4 are the basic optimization techniques. However, depending on the formulation of the optimization problem, as well as the selected design object there are some features of numerical implementation of these methods and their applications.

It is known that the actual design object is usually characterized by a number of quality indicators and improvement in one of them leads to a deterioration in values of other quality criteria (Pareto principle). In such cases it is necessary to consider the optimization problem from many criteria.

The authors offer a well-established practice in solving multi-objective optimization problems – “convolution” of partial objective function weighted by u depending on the importance of a particular quality criteria in a comprehensive quality criteria based on the following:

Formula 1.37

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1.4 Optimization Methods

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1.4.1 General Information About the Extremal Problems

To solve problems with the single criterion of optimality rigorous mathematical methods are developed.

Direct methods of the calculus of variations – one of the branches of the theory of extreme problems for functional – reduce the problem of finding the functional extremum to the optimization of functions.

There are analytical and numerical methods for finding optimal solutions. As a rule, the real problems are solved numerically, and only in some cases it is possible to obtain an analytical solution.

Functions optimization using differentiation

Finding the extremum of the function of one or more variables possible by means of differential calculus methods. It’s said that the   point gives to function f (x) local maximum, if there is a number Ɛ>0 at which from the inequality | x-x̂| < Ɛ the inequality f (x) ≤ f (x̂) comes after.

The function is called one-extremal (unimodal) if it has a single extremum and multi-extremal (multimodal), if it has more than one extremum. The point at which the function has a maximum or minimum value of all local extrema, called a point of the global extremum.

A necessary condition for an extremum of a differentiable function of one variable gives the famous Fermat’s theorem: let f  (x) – function of one variable, differentiable at the point x̂. If x̂ – local extreme  point, then f’ (x̂) = 0.

The points at which this relationship is satisfied, called stationary. The stationary points are not necessarily the point of extreme. Sufficient conditions for the maximum and minimum functions of one variable – respectively f” (x̂) <0,  f” (x̂) > 0.

Before proceeding to the necessary and sufficient conditions for extrema of functions of several variables, we introduce some definitions. The gradient of function f  (x) is a vector

Formula Chapter 4

The real symmetric matrix H is called positive (negative) defined if XT = Hx>0(<0) for every set of real numbers x1 , x2, …. xn, not all of which are zero. Read More

1.3 Building Subsystems FMM

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1.3.1 FMM Basics

As noted, the FMM is an approximation of the original model, which means it can be obtained by statistical processing of the results of numerical experiment using OMM. The complexity of solving the equations of the original model forces minimize the number of sampling points, which is practically achieved by using methods of the theory of experiment design. Get the response function in the form (1.2) can, in particular, on the basis of three-level Box and Benken plans [1]. Special selection of sampling points on the boundary of the approximation:

Formula 1.11
Formula 1.11

and in its center possible in accordance with the least squares method to obtain the values of the coefficients according to (1.2), without resorting to the numerical solution of the normal equations. The number of sampling points is in the range from 13 at N = 3 to 385 at N = 16.

Similarly, relations (1.2) can also be obtained by using the three-level saturated plans by Rehtshafner [2]. In this case, the dimension of the observation vector will vary from 16 at N = 4 to 232 at N = 20. The feature of these plans is that it is the most economical plans that require a minimum number of calculations to generate a vector of observations, i.e. the number of calculations (experiments) equal to the number of the coefficients according to (1.2).
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1.2 Optimization of Complex Technical Devices

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1.2.1 Design Hierarchy

Block-hierarchical representation of the design process, implemented with the creation of complex technical devices, leads to a problem of such complexity that can be effectively resolved by means of modern computing, and the results of the decision – understood and analyzed by experts. Typically, the design hierarchy of tasks is formed along functional lines for turbine can have the form shown in Fig. 1.1.

Hierarchy of Turbine Design Problems
Figure 1.1 Hierarchy of turbine design problems
Nearby Hierarchy Levels of Optimization Problems
Figure 1.2 Nearby hierarchy levels of optimization problems.

The uniformity of mathematical models of the subsystems of the same level and local optimality criteria make it possible to organize the process of multi-level design, providing maximum global quality criterion of the whole system, in our case – the turbine. This process is based on the idea of so-called multilevel optimization approximation scheme that involves aggregation of mathematical models of the subsystems in the hierarchy when moving upward and disaggregation based on optimization results when moving downwards.

The problem of optimization the subsystem parameters described by OMM has the form (1.5). It can be solved by the methods of nonlinear programming and optimal control, depending on the form of the equations and the optimality criterion of the OMM.
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1.1 Mathematical Models and the Object Design Problem

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The methodology of a turbine optimal design as a complex multi-level engineering system should support the operation with diverse mathematical models, providing for each design problem communication between the neighboring subsystems levels.

One approach to turbine design with using of block-hierarchical representation consists in the transition from the original mathematical models for the subsystems and numerical methods of optimization to “all-purpose” mathematical model and general method of parameters optimization.

We will specify as original the mathematical model (OMM), which is a closed system of equations that describe the phenomena occurring in the designed object.

Regardless of the mathematical apparatus (algebraic, ordinary differential, integral, partial differential equations, etc.), OMM can be represented symbolically as follows:

Formula 1.1

where  X ⃗={x ⃗,u ⃗ };L(B ⃗,X ⃗) – the operator defining the model’s system of equations.

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Optimization of Axial Turbine Flow Paths: Preface

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The decades of the 1970s and 1980s of the last century were marked by the emergence and rapid development of a new scientific direction in turbine manufacturing – optimal design. A summary of the approaches, models, and optimization methods for axial turbine flow path is presented in the monographs [13–15 and 24].

It should be noted that work on the optimal design of the flow path of axial turbines and the results obtained not only have not lost their relevance, but are now widely developing. Evidence of this is the large number of publications on the topic and their steady growth. Optimization of the turbomachine flow path is a priority area of research and development of leading companies and universities.

Without the use of optimization, it is impossible nowadays to talk about progress made in the creation of high efficiency flow paths of turbomachines. It is worth noting that the widespread use in power engineering of modern achievements of hydro-aerodynamics, the theory of thermal processes, dynamics and strength of machines, materials science, and automatic control theory, is significantly expanding the range of tasks confronting the designer and greatly complicating them.

The proposed book comprehensively addresses the problem of turbomachine optimization, starting with the fundamentals of the optimization theory of the axial turbine flow paths, its development, and ending with specific examples of the optimal design of cylinder axial turbines. It should be noted that the mutual influence of designed objects of turbine
installations and the many design parameters of each object, which the product’s effectiveness depends on, is putting the task of multiparameter optimization on the agenda.

For turbines with extractions of working media for various needs, efficiency ceases to be the sole criterion of optimality. It is necessary to enable in the optimization process such important parameters as power supply. The task of optimal design of turbine has become multifaceted. It should also be stressed that often the turbo installation mode of operation is far from nominal. So taking into account the operating mode in the optimization can significantly improve the efficiency of the turbine.

In the book, along with the widely used methods of nonlinear programming, taking into account the complexity of the task and the many varied parameters, the use of the theory of planning the experiment coupled with the LP sequence to find the optimal solution is discussed. The first chapter of the book deals with general issues of the optimal design of complex technical systems and, in particular, the problem of optimization of turbomachines, using one of the approaches to the design of turbo installations – a block-hierarchical view of the design process. With this priority is given to flow path optimization of axial turbines. The task of object design and using mathematical models is formulated. A brief overview of optimization techniques, including the optimization method for turbines considering mode of operation is given.
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