5.4 Minimum Profile Loss Optimization

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A more rigorous formulation of creating an optimal cascade profile problem that provides design parameters of the flow at the exit and meet the requirements of strength and workability, is the problem of profiling, which objective function is the profile (or even better – integral) losses.

As mentioned above, the profile loss ratio can be presented as the sum of the friction loss coefficients of the profile ζfr and edge loss coefficient ζe.

Given that the ratio of the edge losses associated with the finite thickness of trailing edges, the value of which is predetermined and is practically independent of the profile configuration, the objective function can be assumed as [8].


In terms of flow profile, you must set a limit, excluding the boundary layer separation. Unseparated flow conditions according to Buri criterion can be written as [22]:


The constants B and m can be taken equal to: B = 0.013…0.020, m = 6.

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5.3 Optimization of Geometric Quality Criteria

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When used for the formation of the profile contour of polynomials of degree n (n > 5 for the convex part of the profile, and n > 3 for the concave part) the question arises about the correct choice of the missing n–5 (or n–3) boundary conditions which must be selected on the basis of the requirements of aerodynamic profile perfection.

One of the requirements of building the turbine profiles with good aerodynamic qualities is a gradually changing curvature along the outline of the profile [25]. Unfortunately, the question concerning the nature of the change of curvature along the profile’s surface, is currently not fully understood. Curvature along the profile’s surface, is currently not fully understood.

As a geometric criterion for smooth change of curvature in the lowest range of change in the absence of kinks on the profile, you can take the value of the  maximum curvature on the profile contour in the range [xc2,xc1] for the convex and for [xk2,xk1] the concave parts, by selecting the minimum of all possible values at the profile designs with the accepted parameters and restrictions. The requirement for the absence of curvature jumps in the description of the profile contour by power polynomials automatically fulfilled as all the derivatives of the polynomial are continuous functions. Agree to consider determined based on the geometric quality criterion, the missing boundary conditions in the form of derivatives of high orders in points C2, and K2 components of a vector Y ⃗ . For the concave part of the profile vector of
varied parameters Y ⃗ is as follows:


wherein k – the curvature of the profile, and the maximum is searched for in the range [xc2,xc1] on the convex portion of the profile and [xk2,xk1] – on the concave part of the profile using one of the one-dimensional search methods. Read More

5.2 Profiles Cascades Shaping Methods

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The resulting thermal calculations of optimal geometry and gas-dynamic parameters of the working fluid at the inlet and outlet of the blade row let you go to the next stage of optimization of the turbine flow path – the blade design. The solution of the latter problem, in turn, can be divided into two stages: the creation of planar profiles cascades and their reciprocal linkage also known as
stacking [25].

The optimal profiling problem formulated as follows: to design optimal from the standpoint of minimum aerodynamic losses profiles cascade with desired geometrical characteristics, provides necessary outlet flow parameters and satisfying the requirements of strength and processability.

To optimize the cascade’s profile shape profiling algorithm is needed, satisfying contradictory requirements of performance, reliability, clarity and high profiles quality.

Earlier, considerable effort has been expended to develop such algorithms [25]. Analyzing the results of these studies, the following conclusions may be done. First, great importance is the right choice of a class of basic curves, of which profiles build (which may be straight line segments and arcs, lemniscate, power polynomial, Bezier curves, etc.), which primarily determines the reliability and visibility of solutions. The quality of the obtained profiles associated with the favorable course of the curvature along the contours, the choice of which is carried out using the criteria of “dominant curvature”, minimum of maximum curvature, and other techniques.

First, consider the method of profiles constructing with power polynomials [15, 34]. The presentation will be carried out in relation to the rotor blade.

5.2.1 Turbine Profiles Building Using Power Polynomials

Initial data for the profile construction. Analysis of the thermal calculation results (entry β1 and exit β1 angles, values of flow velocities W1 and W2) and the requirements of durability and processability lead to the following initial profiling data (Fig. 5.1): β1g constructive entry angle; f – cross-sectional area; b – chord; t – cascade pitch. Optimal relative pitch of the cascade can be determined beforehand on the recommendations discussed in [25]; a – inter-blade chanel throat; ω1 – entry wedge angle; r1 – the radius of the leading edge rounding; r2 – the radius of the trailing edge rounding; ω2 – exit wedge angle; βs – profile stagger angle; β2g – constructive exit angle; δ – unguided turning angle.

Figure 5.1 The design parameters of the profile cascade.
Figure 5.1 The Design Parameters of the Profile Cascade

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5.1. The Cascade’s Basic Geometry Parameters Optimization

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Chapter 5 Introduction: Optimal Cascades Profiling

There are two different approaches to determining the optimal parameters of planar cascades of profiles for the designed axial turbine flow path.

The first one which is suitable for the early stages of design, does not takes into account the real profile shape, i.e. based on the involvement of empirical data on loss ratio, geometrical and strength characteristics depending on the most important dimensionless criteria (the relative height and pitch, geometric entry and exit angles, Mach and Reynolds numbers, relative roughness, etc.). The advantages of this approach are shown in the calculation of the optimal parameters of stages or groups of stages, as allow fairly quickly and accurately assess the mutual communication by various factors – aerodynamics, strength, technological and other, affecting the appearance of created design – and make an informed decision.

The second approach involves a rigorous solution of the profile contour optimal shape determining problem on the basis of a viscous compressible fluid flow modeling with varying impermeability boundary conditions of the profile walls. In practice, the task is divided into a number of sub-problems (building the profile of a certain class curve segments, the calculation of cascade fluid flow, the calculation of the boundary layer and the energy loss) solved repeatedly in accordance with the used optimization algorithm, designed to search for the profile configuration that provides an extremum of selected quality criteria (e.g., loss factor) with constraints related to strength, and other technological factors. Read More

4.4 The Effect of Tangential Lean on the Characteristics of Axial Turbine Stage

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One means of the flow control in the axial turbine stage is the use of blades with non-radial setting. In this case, there is a non-zero the blade surface lean angle.

Vortex equation for the case of flow in a rotating crown can be written as:


Turning to the new independent variable ψ – the stream function, we write (4.16) in final form:


Equation (4.17) for given geometrical parameters of the surface S’2 forms a closed system of ordinary differential equations in cross-section z = const together with the continuity equation:


Consider a three sections stage calculation which located on the entrance and exit edges of the guide vane and on the trailing edge of the impeller. Derivative:

is defined in terms of the flow of the working fluid in the free space (right side of the design section):


In the absence of lean (tgδ = 0 ) the equation (4.17) coincides with the previously obtained. Upheld algorithm for the stage calculation by sections and supplements it by specifying the lean angles of the guide and rotor blades output edges. Agreed δ(r) = const.

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4.3 The Axial Turbine Stage Optimization Along the Radius in View of Leakages

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The above numerical study results, confirmed experimentally, show, that leakages significantly affect the axial turbine stage crowns optimal twist laws. With a decrease in the length of the rotor blade (increase of Dm/ι ratio) this effect is amplified.

In this regard, the problem arises of determining the guide vanes and rotor optimal twist laws for a given stage geometry, inlet parameters, the rotor angular velocity, flow rate and heat drop. We restrict ourselves to the task of practically important case of the blades angles specification in the form (4.4). At the same time, while setting the flow and heat drop together, thermal calculation is performed by adjusting one of the angles α1m or β2m Described below optimization technique based on repeated conduct this kind of thermal calculations for the purpose of calculating the internal stage efficiency depending on one of α1m, β2m angles, and the exponents m1, m2 in the
expression (4.4).

Assume that the control variables are β2m, m1 and m2, whereby the back pressure at a predetermined flow rate must be specified by changing the angle α1 at the mean radius. The problem of the thermal stage calculation is written as

Figure 4.6

and its numerical solution is based on finding the roots of transcendental equations

Figure 4.7

After the solution of (4.7), which is conducted with the specification form of the stream lines, leakage values, velocity and flow rate coefficients, internal stage efficiency calculated as a function of three variables β2m, m1 and m2.

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4.2 The Impact of Leaks on the Axial Turbine Stages Crowns Twist Laws

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Significant impact on the stage efficiency have leakage of the working fluid through the seal gaps and discharge openings. The dependence of the leakage (and associated losses) of the stage bounding surfaces parameters can dramatically affect the distribution of the optimal parameters along the radii and, hence, the spatial structure of the flow therein. The latter, in turn, is determined by the shape and twist law of guide vane and impeller.

Development of algorithms for the axial turbine stages crowns twist laws optimization demanded the establishment of appropriate in the terms of computer time methods for calculating the quantities of leaks and losses on them, allowing the joint implementation of the procedure for calculating the spatial parameters of the flow in the stage.

The leakage calculation is necessary to conduct together with a spatial calculation step, as the results of which the parameters in the calculation sections are determined, including the meridian boundaries of the flow path. The flow capacity depends on the clearance (or leakages) values, in connection with which main stream flow calculation is made with the mass flow amplification at fixed the initial parameters and counter-pressure on the mean radius, or clarifying counter-pressure at fixed initial parameters and mass flow. The need for multiple stage spatial parameters calculation (in the optimization problem the number of direct spatial calculations increases many times) demanded a less time-consuming, but well reflecting the true picture of the flow,
methods of spatial stage calculation in the gaps described above (Fig. 2.3).

When calculating stage in view of leakage the continuity equation is convenient to take as [8]:


where μ – the mass transfer coefficient, which allows to take into account changes in the amount of fluid passing through the crowns, and at the same time to solve a system of ordinary differential equations in sections in front of and behind the impeller like with a constant flow rate.

The leakage mass transfer coefficients [13] is defined as follows:


In the case of wet steam flow with loss of moisture, crown overall mass transfer coefficient is given by


where ψm,i flow coefficient, is usually determined in function of the degree of humidity and pressure ratio [8].

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4.1 Formulation of the Problem

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Mathematical models of gas and steam turbines stages, discussed above, allow to put the task of their geometry and gas-dynamic parameters optimization. This optimization problem is solved by the direct problem of stage calculation. The reason for this are the following considerations:

  • – it is most naturally in optimizing to vary the geometry of the blades;
  • – in the streamlines form refinement it is convenient to use well-established methods for the solution of the direct problem in the general axisymmetric formulation;
  • – only a direct problem statement allows to optimize the stage, taking into account the off-design operation;
  • – for the stages to be optimized, assumed to be given:
  • – the distribution of the flow at the stage entrance;
  • – the form of the meridian contours;
  • – the number of revolutions of the rotor;
  • – mass flow of the working fluid;
  • – averaged integral heat drop.

In general, you want to determine the distribution along the certain axial sections of angles α1 and β2 to ensure maximum peripheral efficiency of the stage:


Here the inlet geometric angle of the rotor we assume equal to the angle of the inlet flow. Selection of the optimal angle β1g can be achieved solving an optimal profiling problem.

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3.2 Preliminary Design of the Multistage Axial Flow Turbine Method Description

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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.

Continue reading “3.2 Preliminary Design of the Multistage Axial Flow Turbine Method Description”

3.1 Analytical Solutions

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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 sections numbering in the turbine flow part section,

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