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.

The objective function (4.1) can be calculated with the known distribution of the kinematic parameters of the flow in the gaps, which are determined by solving the direct problem of the stage calculation using the models set out in Chapter 2.

Mathematically, the assigned stage optimization problem has been reduced to a problem of the theory of optimal control of distributed parameter systems, including integrated criterion of quality (4.1) and system of constraints, which comprises:

  • – a system of equations in section 1 for the nozzle (2.39);
  • – a system of equations in section 2 of the rotor (2.40);
  • – isoperimetric condition (4.2), ensuring operation at a given stage heat drop;
  • – restrictions on the control variables (4.3).

The velocities c1 w2 and the radii r1 r2 are the phase variables; the independent variable stream function ψ plays the role of time.

From physical considerations it is obvious that the control functions α1(ψ) and β2 (ψ) must be sufficiently smooth, at least continuously differentiable, i.e.
does not have discontinuities of the first kind, and kinks. For this purpose it is convenient to use parametric angles as dependencies


which allow to investigate the influence of coefficients m1 and m2 on the stage efficiency.

The parameters m1 and m2 characterize twist angles gradients. For m>0 are obtained the angles increasing to the periphery (direct twist) and for m<0 – decreasing (reverse twist). Twist law cur = const corresponds to the values m1 =1, and m2 = -1, that under the simplified equation of radial equilibrium, provides a minimum of the output velocity losses for the stage with cylindrical contours.

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