#### 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.

##### 5.1. The Cascade’s Basic Geometry Parameters Optimization

The importance of solving the problem of the cascade’s basic characteristics definition can be seen from the following considerations. Let designed axial turbine stage blades at a predetermined height. Under certain parameters before and behind the stage is usually determined the number of blades and profile chords so that with an energy loss minimum satisfy strength and vibration requirements. The simplest solution is to select the “optimal” t/b ratio using known empirical relationships and determining the chords provide reliable operation. Upon closer examination the situation is not so simple: first, the optimum ratio t/b is determined by many factors (the relative thickness of the edge, the Reynolds number and the relative roughness of the surface, relative height and others); secondly, the permissible loss and the vibration characteristics depend on the influence of the previous cascade; third, the stage design can be carried out both from the set of standard profiles or suggest subsequent entirely new cascades profiling. Consideration of these circumstances makes the task of optimization of the basic cascade parameters quite challenging and promising in terms of using hidden in complicated situations reserves to increase efficiency and reduce consumption of materials in the created turbomachine design.

The calculation of the kinetic energy loss on the basis of empirical relationships has repeatedly been considered and, as experience shows, in the form set out in Chapter 2, is a reliable tool to assess the various components of the losses in the cascade. Calculation of the geometric characteristics of the profiles is carried out using a dependency suitable for working and nozzle profiles, including an elongated front portion. The stresses in the diaphragms, nozzle and rotor blades, as well as restrictions on the vibrational reliability calculated by the well-known and, as far as possible, the exact dependence.

When optimizing an isolated cascade the following problem statements species are considered.

I. Profile presentation method

• – I.1. Standard profile. The geometric characteristics are determined by the tabular data and restated for a specific profile stagger in the cascade, which provides the desired output stream angle at a known relative pitch.
• – I.2. “Macromodel”. The form of the profile is not known beforehand, but its defining geometrical characteristics can be estimated by empirical dependence of the type [26].
• – I.3. Profiling. In addition to the previous statement can be built demo profile, designed by a faster way. It is possible geometrical and strength characteristics evaluation on its configuration.

II. Variable parameters.

• – II.1. Optimization of chord when t/b = const.
• – II.2. Optimization of t/b when b = const.
• – II.3. The chord and the relative pitch optimization. In constructing the cascade of the standard profiles the profiles chord selection is in sequential enumeration of profiles of this type, but of different size [20, 33].

III. Boundary conditions.

• – III.1. Geometric, kinematic and gas-dynamic parameters in the first approximation are given from stage thermal calculation.
• – III.2. Cascade optimization process is conducted directly to the stage (multistage flow path) thermal calculation and optimization. In this case, the design of the cascade is embedded in an iterative process instead of the verifying energy losses in cascades, as is usually done.

Optimization is made by LP- search, and where this is not possible, brute force at defined ranges of variable parameters and the number of sampling points. The calculation is carried out in designer’s dialogue with a computer, which significantly reduces the time to find an acceptable solution.

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