#### 3.2 Preliminary Design of the Multistage Axial Flow Turbine Method Description

In the early stages of the flow path (FP) design of the turbine, when determined the diameter, the blade heights, heat drops and other main characteristics of the stages, required to study alternatives with a view to the design solution, in the best sense of a quality criterion.

Most effectively, this problem is solved within the created turbine flow path CAD systems, because manage: to achieve a rational division of the designer, defining the strategy and computer, quickly and accurately perform complex calculations and presents the results in human readable numeric or graphical form; to take into account many different factors influencing the efficiency,
reliability, manufacturability, cost and other indicators of the quality of the design being created; organize dialogue or fully automatic determination of optimal parameters, etc [29].

Most methods of the multi-stage turbine parameters optimization is designed to select the number of gas-dynamic and geometric parameters on the basis of the known prototype, the characteristics of which are taken as the initial approximation.

When using complex mathematical models, a large number of variables and constraints, the solution of such problems requires considerable computer time and for the purposes of CAD that require quick response of the system is often unacceptable.

It is desirable to have a method of design that combines simplicity, reliability and speed of obtaining results with an accuracy of the mathematical model, a large number of factors taken into account and optimized, the depth of finding the optimal variant. This inevitably certain assumptions, the most important of which are: the synthesis parameters of “good”, competitive structure without attracting accurate calculation models; in-depth analysis and refinement of the parameters are not taken into account at the first stage; optimization of the basic parameters by repeatedly performing the steps of the synthesis and analysis.

Design of the FP in such a formulation will be called preliminary (PD). PD does not claim to such a detailed optimization of parameters, as in the above-mentioned methods of optimal design. Its goal – to offer a workable, effective enough design, the characteristics of which, if necessary, can be selected as the initial approximation for more accurate calculations.

Major challenges in creating a PD method are:

• – a rational approach to the problem of the preliminary design, the selection of the quality criteria and the constraints system;
• – development of a method for the multi-stage flow path basic parameters selection;
• – formation of requirements for a mathematical models complex describing different aspects of turbines and their efficient numerical implementation;
• – selection of the appropriate algorithm for finding the optimal solution;
• – a flexible software creation for a dialog based solution of the design problems in various statements and visual representation of the results.

#### 3.1 Analytical Solutions

An important objective in the design of a multi-stage axial turbine is to determine the optimal number of stages in the module and the distribution of heat drop between stages.

Typically, a given quantity is the module’s heat drop, and should vary the number of stages and the rotational speed (diameter). It should be understood that the circumferential velocity reduction, and hence the diameters of the stages, reduces the disc friction losses, increase height of the blades (and therefore reduce the proportion of end losses), decrease the flow path leakage. At the
same time it leads to an increase in the optimal number of stages, which causes an increase in losses due to discs friction and an additional amount of the turbine rotor elongation. Immediately aggravated questions of reliability and durability (the critical number of revolutions), materials consumption, increase cost of turbine production and power plant construction.

A special place in the problem of the number of stages optimization is the correct assessment of the flow path shape influence, keeping its meridional disclosure in assessing losses in stages. As you know, the issue is most relevant for the powerful steam turbines LPC. It is therefore advisable for the problem of determining the optimal number of stages to be able to fix the form of the flow path for the LPC and at the same time to determine its optimal shape in the HPC and IPC.

It should also be noted that the choice of the degree of reaction at the stages mean radius (the amount of heat drop also associated with it) must be carried out with a view to ensuring a positive value thereof at the root. Formulated in this section methods and algorithms:

• – May serve as a basis for further improvement of the mathematical model and complexity of the problem with the accumulation of experience, methods and computer programs used in the algorithm to optimize the flow of the axial turbine;
• – Allow the analysis of the influence of various factors on the optimal characteristics of the module, which gives reason for their widespread use in teaching purposes, the calculations for the understanding of the processes taking place in stages, to evaluate the impact of the various losses components on a stage operation;
• – Allow to perform heat drop distribution between stages and to determine the optimal number of stages in a module within the modernization of the turbine, i.e. at fixed rotational speeds (diameters) and a given flow path shape or at the specified law or the axial velocity component change along the cylinder under consideration.

A possible variant of the form setting of n stages group of the flow path can be carried out by taking the known axial and circumferential velocity components in all cross-sections, which the numbering will be carried out as shown in Fig. 3.1.

The axial velocity components we refer to the axial velocity at the entrance to the stages group: Read More

#### 2.5 Thermal Cycles Modelling

Imagine the process of analyzing the thermal cycle in the example of gas turbine unit (GTU) (Fig. 2.18) in the following sequence:

1. the structure diagram presentation as a set of standard elements and connections between them;
2. entering the input data on the elements;
3. generation of computer code in the internal programming language based on the chosen problem statement;
4. processing;
5. post-processing and analysis of results.

This sequence of actions combines a high degree of automation of routine operations (input-output and storage of data, programming, presentation of the results of calculations, and so on) with the possibility of human intervention in the process of calculations at any stage (editing of data, changing the program code in the domestic language, writing additional custom code for non-standard calculations performing, etc.).

#### 2.4 Flow Path Elements Macro Modelling

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

##### 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

##### 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

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

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

##### 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

##### 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).