Chapter 7 Introduction: Experience and Examples of Optimization of Axial Turbines Flow Paths
In this chapter, as an example of practical use of the developed theory of optimal design of axial turbines flow paths, the results of the studies, related to the optimization of parameters of flow path of the high pressure cylinders (HPC) of 220, 330 and 540 MW capacities turbines, operating at nominal mode, as well as examples of optimization turbo-expander and low pressure turbine of gas turbine unit, taking into account the mode of its operation, are presented. The entire complex of calculation research was conducted using mathematical models of flow path (FP) of axial turbines, described in Chapter 2.
In addition, in the studies variants of mathematical models of FP “with the specified profiles”  were also used, which allowed with more accuracy determine geometric characteristics of turbine cascades, in particular, the inlet geometric angles of working and nozzle cascades, that are changing with the changing of stagger angles of the profiles. The latter had a significant impact on the amount of additional losses related to the incidence angle of inlet flow of working fluid. Read More
Review of research on the application of the complex tangential lean and its optimization, as well as conducted computational research has shown that using of complex lean gives the possibility to increase aerodynamic efficiency of turbine cascades. However, as previously noted, research on optimization of complex tangential lean with preserving mass flow rate through the cascade with high precision, currently we do not have. Using developed optimization approach it is possible to preserve in optimal cascade mass flow rate at the level of the initial cascade with a high accuracy.
Complex tangential lean reduces integral losses by reducing secondary losses. It is known, that with increasing l/b there is a reducing in the part of the secondary losses in integral losses and, accordingly, the benefit from optimization has to diminish.
Relative height criterion was taken not l/b, but the cascade’s characteristic relation a/l, by analogy with the flows in the swivel tubes of rectangular cross-section.
Optimization problem is solved using two methods of stacking line parameterization. Research of the efficiency of the algorithm consists in attempts of optimization of turbine cascade at different a/l = 0,16; 0,23; 044 by changing of the blade height. It should be noted that for the blades with a/l ≤ 0.16 optimization, using both methods of stacking parameterization, no longer led tot he reducing losses compared to the cascade without lean.
The size of the throat varies slightly due to the changing of stagger angle of the profile, which is associated with the preserving of the mass flow rate.
Special attention was given to the FMM accuracy, since it determines the validity of the results obtained with used optimization approach. Criterion of the accuracy is deviation of the values of the target function and the constraint function, which we obtain in FMM and in checking CFD calculation.
6.7.1 Optimization with Various a/l Using Method 1
The results of the optimization for a/l = 0.44
Taking into account the experience of previous studies, in Table 6.3 the ranges of parameters variation have shown. The correctness of their choice is confirmed by the fact that the optimal combination of varied parameters falls in this range already at the first step of the optimization.
Then, a plan is created in accordance with the algorithm and relevant CFD calculations are produced (Table 6.4). The objective function – integral losses ζ, restriction function – mass flow rate through the calculation channel G Read More
Object of study and boundary conditions are identical to the turbine cascade, that described in the previous section, with the exception of the relative height of the blade, which in this case amounted to l/b = 0.714. Complex lean was carried out according to 2-nd method without changing the stagger angle of the profile.
Using proposed algorithm (section 6.4) the optimal blade’s shape of the specified turbine guide blade was found on the sixth step of the variation parameters range refinement.
All 56 configurations of turbine blades shape were counted.
To solve the same problem using genetic algorithm, probably, hundreds of calculations would have required. The Table 6.1 shows the best value of the varied parameters and the best value of target function for each of the optimization stages.
At the 1–5 stages optimization of minimum objective function fall on the border range of variation parameters, at that on 4-th phase the function is minimal on the right edge of the border of variation parameters range, while on 5-th phase the function is minimal on the left edge. As a result, after the 6-th phase the values of the optimal parameters became Ys = 0.77 and Yh = 0.80.
Fig. 6.14 shows isolines of the objective function in the space of parameters Ys and Yh.
It is known that the lean along the flow leads to increasing secondary flow losses on the periphery and to reducing them at the root. Lean against the stream leads, correspondingly, to the opposite result.
The lean to the opposite flow direction allows to alter the distribution of flow parameters along height, so that the leakages in the axial gap is reduced on the periphery, that positively affects stage efficiency. Read More
Sources of geometric information related to turbines blades are quite varied. These can be drawings in the paper or electronic form, the results of measurement of coordinates of the dots multitude, using mechanical or laser devices, coordinates of cross sections by flat or conical surfaces.
When the surface of the blade is represented by sets of dots its conical (cylindrical for axial machines) cross sections, it is assumed that the program, taking this information, will build the splines on cross sections and then will stretch the spline on surface. In this sense, this description is procedural.
In particular, the BladeGen preprocessor (Ansys CFX) offers two format of procedural form of the blades storing – RTZT and CURVE. Because the information in the CURVE file is not enough for permanent storage of data of the blade wheel, we have developed its extension – CUR format. It additionally includes the number of blades in the crown, the number of the cross sections
and the number of the profile sectors in the cross sections, the number of dots in the sectors, etc.
The number of sectors on the contour of the cross section can be 1, 2 or 4. In the first case, the surface of the blade is formed by one spline. Accordingly, in the second and third cases, the blades is formed by two (suction and pressure sides) or four (suction side, leading edge, pressure side and trailing edge) spline segments.
The order and type of splines (for example, interpolation or approximation)are not stored in a file, because these options are implementation-dependent. They must be specified in the reading procedure. A fragment of the blade wheel, described using the CUR format with four sectors in each of the five initial cross sections of the blade, is shown at Fig. 6.1. The cross sections, shown at
Fig. 6.1, came out as a result of spline-approximation of original dots (dots on the sites have different colors).
The process of drawing up the blades from known cross sections (flat or cylindrical) is called stacking. To do this, specific point of each cross section, which coincides with the stacking line of this cross section, must be selected. Often, for convenience, the centers of the edges or the centers of gravity of the cross sections are chosen as stacking points (Fig. 6.2). In general, this selection can significantly change the shape of the blades.
Any deviation of the stacking line from radial location will be named lean. With a simple lean stacking line remains straight and is characterized by a single parameter –the angle of incline. When we have a complex lean it can take
For easy use, it makes sense to limit parameterization of stacking line by Bezier curves.
6.2.3 Forming the Lateral Surfaces of the Blades
The surface of the blade is described by the parametric functions-interpolation or approximation B-splines based on two parameters: u-along the contour of each cross section and v-along the direction of stacking. Interpolation spline passes exactly through all dots of cross-section of the blade, and approximation spline – in accordance with supporting polygon, which is build using the dots of cross section or by least squares method . All dots of the surface can be found, when u and v parameters taking the values from 0 to 1. In some cases, it is required to allow extrapolation towards the staking and then the v parameter may become a bit less than 0 or greater than 1.
The blade can be described either by one surface or by several. In our implementation (as already noted) 2 or 4 surfaces, that can be useful for some applications, in particular, when constructing grids, are allowed. Since there is no joining on surface boundary, the junction’s error could be managed only changing the spline order along u and v directions. It is usually within the range 2…5.
6.2.4 Three-Dimensional the Turbine Blade Parametric Model
One of the key elements of the 3D aerodynamic optimization of the turbine cascades algorithm is the turbine blade model parameterization, which consists in the possibility of changing blade shape (curvature) by variation of the limited number of numeric parameters that describes the stacking line.
The Bezier curve of 3-d (method I) and 4-th order (method 2) it seems convenient to use as the binding line. The second method allows creating a stacking line with practically straight-line the middle segment. (Fig. 6.4, 6.5). The number of independent variables in both cases can be reduced to two ( Yh and Ys ).
Parametric model of turbine blades must provide an opportunity to check the mass flow through the cascade. This requires incorporation in the model parameter, which allows controlling the mass flow during optimization. Most eventually the last gives the possibility to ensure equality of the mass flow between initial and optimized cascades with the same flow parameters before
and after cascades. The changing of stagger angle of the blade, relative to the original, can be taken as such parameter.
In addition to complex lean, the simple lean was implemented in methodical aim, which consists in turning of the turbine blades profiles relatively of the axis of rotation of the turbine on the specified angle. In general, developed parametric model of turbine blade allows producing its curvature in the circumferential (tangential) direction as well as in axial direction, simultaneously or separately.
6.2.5 The Grids Construction
As it is known, the results of the CFD calculations may depend on the type of calculation grids. One of the tasks, needing to be addressed, is to build up a three-dimensional parametric calculation grids, satisfying the form of parameterized blades.
Fast and reliable building of the parametric calculation grids is an integral part of the optimization studies as it implies the calculation of a large number of variants of the geometry of the turbine cascade. Since the developed algorithm of optimization should not be tied up with solitary CFD-solver, a specialized grids builder has been developed.
We will describe in details the work with H-grids, which represent a convenient compromise between complexity of the grids creation and quality of the obtained solutions when flows computation in turbomachine cascades occur. H-grid topologically is equivalent to the cube. Therefore, a data structure is simple enough for description. It intentionally is made redundant to accelerate frequently meeting operations. This, of course, slightly reduces the maximum size of the grid when a limited amount of RAM available to the computer, but it is not critical to the solving problem.
The structured calculation grid for channel between the blades is obtained because of deformation in the direction of each of the coordinate axes of a rectangular parallelepiped (in space) or rectangle (flat case).
Inter-blade channel is formed by concave and convex sides of the two adjacent blades (or profiles in the planar case). For selecting the high pressure and low pressure sides of the blade, the blade is made up by two splines, connecting in the points of minimum and maximum x-coordinate sections. Parametric lines v = const of these splines give the calculated coordinates of D
sections of the grid in the radial direction. Next inter-blade channels are supplemented by input and output sections of the specified length, representing segments of the rings (for circumferential blade cascade) or parallelepiped (for flat cascade).The resulting area in each section is split up to cells of grids in the direction of x-coordinates dimension L. On the inlet and outlet sections is
usually taken by L/4 cells. Other cells are located on the profile and coordinates of the nodes are calculated by interpolation spline via dots of the splines of the high and low pressure sides.
Finally, channels are split on H sections along the directions x = const, that completes the structured grid formation. In the process of grid building a primitives numbering (nodes, edges, verges, and cells), topological ties formation and geometrical data calculation are made. All information is entered into a data structure.
As such, the calculated grids are not yet suitable for conducting reliable calculations of viscous flows in the blades cascades. They should be improved in order to fit the peculiarities of the flow near the walls of the channel.
Thickening structured grid is performed independently for each of the coordinate directions. Law of deformation of the grid can be different and should reflect the physical characteristics of the flow in the area of thickening. For example, near the wall polynomial law for changing the grid can be used, which corresponds to the rate of changing the velocity in the boundary layer. In
the area of input and output edges the deformation may be exponential in nature that is less aggressive. In either case, a number of parameters controlling the thickening as for the rate of deformation as well as the ratio of the sizes of areas of the channel subject to or not to distortion should be entered.
In general three-dimensional case, the grid, suitable for calculations of viscous flows, is presented at Fig. 6.6.
6.2.6 File Format for Grids Storage
The diversity of formats creates some difficulty in reading these files by different CFD-applications. CGNS-standard for CFD calculations data storage is positioned as a “common, portable and extensible”. Software implementation of the standard is an open, cross-platform and well documented that, in principle, precludes differences of various applications.
Data in CGNS format are stored in binary form and access to it is implemented through a set of functions for reading, writing, and modifying of the contents of the files which can be called from application in different programming languages. In general case CGNS file can contain data which is associated with viscous compressible fluid flow, but suitable for solutions of the Euler equations and potential flows.
The standard includes the following data types: structured, unstructured hybrid grids; data of the CFD calculations; information on the sub-grids docking or overlapping; boundary conditions; descriptions of equations of state, turbulence models etc.; nonstationary solutions, including deformation of calculation grids in time; dimension of variables; variables reference points;
history of calculations; user’s and other data.
For the purpose of specific tasks solution there is no need to implement in full all the functionality, supported by the CGNS (this is not currently doing even such advanced products like CFX). It is enough, for example, organize saving of the structured grids and setting of the boundary conditions, satisfying the terms of the calculation task. This significantly speeds up the preparation of data for CFD calculations. Analysis of output information, perhaps, you might need to implement by means of post-processors of used packages, since not all of them conserve the results of calculations in CGNS format.
The rapid development of computational aerodynamics methods not only puts on the agenda introduction of the spatial calculations into the turbines design practice, but also raises the need to develop the blades shape and other turbine flow path elements optimization methods taking into account 3D flow .
Formulations of the blades spatial optimization problems, which essentially cannot be solved by using one-dimensional and two-dimensional models, for minimization of the secondary flows loses, arising at the tip and the hub of the blades, are of the greatest interest .
Analyzing the results of the research, three main reasons for formation of the secondary flows in the turbine cascades could be singled out:
Turning of the flow. In channels with flow turning (including the turbine cascades) the transverse gradient of pressure arises, under influence of which whirlwind is forming at the ends of the channel.
Interaction of the boundary layer, accumulated on the end wall in front of the blade with the leading edge of the blade. For this reason, a horseshoe-shaped whirlwind is formed, which is then divided into two parts on both sides of the blade.
Vortex wedge. In almost every corner areas, which are generated between vortical structures and walls of turbine channel cascade, the forming or dissipating of corner vortexes may take place. Some from them are there constantly, some are dissipating depending rom the flow parameters and the type of the wedge.
The created profiling algorithms have allowed to design a series of profiles of turbine cascades.
As a starting (1O) was taken the standard profile P2 with a high aerodynamic quality. Wherein were accepted such flow conditions that ensure the smallest possible profile P2 (1O) losses: t– = t/b=0.722, βb= 76°26′,β1 = 29°30′.
Retaining the basic, necessary for the machine profiling raw data:
with the help of the developed algorithms were obtained new profiles: 1MMC (for the geometric quality criteria – the minimum of maximum curvature) and 1MPL (the minimum of profile loss).
From technological considerations subsequently profile 1MMC contour was approximated by the radii (Fig. 5.4, 5.5, Table 5.2). Fig. 5.6–5.8 shows the distribution of the velocity and the parameter B (the Buri boundary layer separation criterion) along the contours of the original and newly created profiles.
The calculated profile loss ζpr values correspondingly are 3.35, 3.16 and 3.00%. Attention is drawn to the different law of the parameter B variation along the profiles contours. Apparently, the possibility of the boundary layer separation, or the intensity of its thickening (which leads to increased losses) must be judged not only by the maximum value of the parameter B, which (usually) achieved at cascade’s oblique cut, but also the character of its change within the channel prior bevel, particularly on the convex side of the profile.
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 .
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 :
The constants B and m can be taken equal to: B = 0.013…0.020, m = 6.
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 . 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