#### 2.4 Flow Path Elements Macro Modelling

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Macromodels are dependencies of the “black box” type with a reduced number of internal relations. This is most convenient to create such dependence in the form of power polynomials. Obtaining formal macromodels (FMM) as a power polynomial based on the analysis of the results of numerical experiments conducted with the help of the original mathematical models (OMM).

Therefore, the problem of formal macro modelling includes two subtasks:

1. The FMM structure determining.
2. The numerical values of the FMM parameters (polynomial coefficients) finding.

As is known, the accuracy of the polynomial and the region of its adequacy greatly depend on its structure and order. At the same time, obtaining polynomials of high degrees requires analysis of many variants of the investigated flow path elements, which leads to significant computer resources cost and complicates the process of calculating the coefficients of the polynomial.

#### 2.3 Geometric and Strength Model

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##### 2.3.1 Statistical Evaluation of Geometric Characteristics of the Cascade Profiles

For accurate estimates of the size of the blades, which takes into account not only their aerodynamic properties and conditions of safe operation, it is required to calculate the set of dependent geometric characteristics of the profiles (DGCP) as a function of a number of parameters that determine the shape of the profile. When the shape of the profiles is not yet known, to assess DGCP should use statistical relations. From the literature are known attempts to solve a similar problem [25, 26] on the basis of the regression analysis.

The DGCP include: f – area; Ie and In minimum and maximum moments of inertia;Iu – moment of inertia about an axis passing through the center of gravity of the cross section parallel to the axis of rotation u; φ the angle between the central axis of the minimum moment of inertia and the axis u; Χgcgc the coordinates of the center of gravity;βi – stagger angle;lss – the distance from the outermost points of the edges and suction side to the axis Ε; linlout – the distance from the outermost points of the edges to the axis Ν; We, Wss, Win, Wout, – moments of profile resistance.

The listed DGCP values most essentially dependent on the following independent parameters (IGCP) β1g – geometric entry angle; β2eff – effective exit angle; – chord; t/b  – relative pitch; r1, r2 – edges radii; ω1, ω2 – wedges angles.

Formal macromodelling techniques usage tends to reduce the IGCP number, taking into account only meaningful and independent parameters. In this case, you can exclude from consideration the magnitude of  r1, r2, ω2 taking them equal r1 =0.03b; r2=0.01b ; ω2=0.014Kωω1/(0.2 +ω1) , Kω = 1…3 , depending on the type of profile [26].

We obtained basic statistical DGCP relationships using profiles class, designed on the basis of geometric quality criteria – a minimum of maximum curvature of high order power polynomials [15] involving the formal macromodelling technique. Approximation relations or formal macromodel (FMM) are obtained in the form of a complete quadratic polynomial of the form (1.2):

The response function y(q ⃗’) values (DGCP) corresponding to the points of a formal macromodelling method, calculated by the mathematical model of cascades profiling using geometric quality criteria.

Analysis of profiles used in turbine building reveals, that two of remaining four IGCP β1g and t/b highly correlated.

#### 2.2 Aerodynamic Models

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##### 2.2.1 Axisymmetric Flow in the Axial Turbine Stage

Assume that in the flow path of the turbine:

• The flow is steady relatively to the impeller, rotating at a constant angular velocity ω about the z-axis or stationary guide vanes.
• The fluid is compressible, non-viscous and not thermally conductive, and the effect of viscous forces is taken into account in the form of heat recovery in the energy and the process equations, i.e., friction losses are accounted energetically.
• If the working fluid is real (wet steam) it is considered the equilibrium process of expansion.
• the flow is axisymmetric, i.e., its parameters are independent of the circumferential coordinate.

Under these assumptions the system of equations describing the steady axisymmetric compressible flow motion, includes:

1. The equation of motion in the relative coordinate system in the Crocco form

2. Continuity equation

3. The equation of the process or system of equations describing the process

4. The equations of state

5. The equation of the flow surface

where n ⃗’ – normal to the S2 surface (Fig. 2.1).

6. The equation of blade force orthogonality to the flow surface

Projections of the vortex in the relative motion rot W ⃗’ = ∇ * W ⃗’ to be determined by the formulas:

Taking into account (2.12), projection of the equation of motion (2.6) on the axes of cylindrical coordinate system can be written as follows: Read More

#### 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:

#### 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

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:

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).

#### 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.

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.

#### 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:

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

#### [:en]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.