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.


To create FMM it is advisable to use the mathematical apparatus of the design of experiment theory to significantly reduce the number of computing experiments with OMM, i.e. obtain sufficient information with a minimum dimension of the vector of observations Y’ . We use two types of FMM – (1.2) and (1.12). To get them methods of the design of experiment theory applied (three-level Boxing-Benkin plans and saturated Rehtshafner’s plans) and cubic spline interpolation.

In a particular implementation of a formal macro modelling methodology need to perform the following steps:

1) the choice of the IMM of the flow path element;
2) the appointment of its performance criteria;
3) choice of OMM parameters, whose influence on performance criteria of the flow path element is necessary to study in detail and the formation on their basis vector of varied parameters Q ⃗;
4) macro modelling area appointment (ranges of components of the vector q ⃗ );
5) DOE matrix formation;
6) active numerical experiments conducting and evaluation of the components of observations vector
7) the processing of the experimental results and the determination of the FMM coefficients.

Steps 1–4 are not amenable to formalization and their implementation should take into account the specific features of macro modelling objects and existing experience of designing elements of axial turbines flow path.

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