Key Symbols

Indexes and Other Signs

Abbreviations

##### 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” [38] 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.

##### 7.1 Multi-Criterion Optimization of HPC of Powerful Steam Turbines at Nominal Operational Mode

##### 7.1.1 A Preliminary Study of Influence of Quality Criteria Weights Coefficients on the Optimization Results

Practice of the optimal design of axial turbines cylinders has showed that when optimizing steam turbine cylinder with extraction of working fluid for regeneration and heat supplying at least two criteria – the efficiency of the cylinder flow path and its capacity must be taking into account [38, 40-42].

Using the convolution of quality criteria in accordance with (1.37) allows efficiently solve the multi-criterion optimization problems corresponding the Pareto front.

As an example of the effectiveness of the use of convolution (1.37) the results of the optimization of HPC FP of a powerful steam turbine by two criteria – power and cylinder efficiency for different values of the weight coefficients μ_{i} are presented in Table 7.1 and Fig. 7.1.

Numbers on the curve corresponds to the numbers of optimization problem in the Table 7.1.

##### 7.1.2 Optimization of HPC Parameters of the 220 MW Capacity Turbine for Nuclear Power Plant

The number of optimization parameters – 33:

- – level 1 (cylinder) – optimized for 19 parameters:
- – Root diameter and height of the nozzle blades of the first stage of the cylinder.
- – Meridional disclosing of the channels of the nozzle and working cascades.
- – Effective exit angles of the nozzle and working cascades of all turbine stages.

- 2-nd level (stage) – optimized for 14 parameters:
- – The number of the blades in the nozzle cascades for all turbine stages.
- – The number of the blades in the working cascades for all turbine stages.

Quality criteria applied when optimizing – the criterion vector that includes the normalized values of internal relative efficiency of the cylinder η_{oi} and its power (N) with equal weight coefficients.

The results of the optimization of the HPC FP of the 220 Mw capacity turbine [39] are listed in Table 7.2 and in Fig. 7.2, where η_{d} – Moliere diagram efficiency of FP η’ – the ratio of efficiency of the stages to Moliere diagram efficiency of the initial variant of the cylinder η_{oi} – internal efficiency of FP; Δη_{d} – gain of the internal efficiency of the optimal FP; *N* – power; Δ*N* – the power gain of the optimal variant of the HPC FP.

Improvement of the quality indicators of the optimized FP obtained through:

- – rational distribution of the cylinder heat drop, having in its disposal, between the stages;
- – some decreasing of the axial speed components and ensuring closer to axial outlet working fluid from the stages, resulting in reducing the exit velocity losses;
- – reducing the incidence angles, that provides the improving efficiency of the nozzle and working cascades;
- – increasing the mean diameter of the stages, that led to obtaining the optimal values of the ratio of the velocities
*u*/C_{0} - – reducing the specific weight of the losses near the hub and the shroud boundaries by increasing the height of the blades;
- – the optimal value of the nozzle and working cascades relative pitch, which also led to an increase of their effectiveness.

The final variant is obtained by optimization taking into account the technological restrictions on the production of the flow path parts. This explains the slight decreasing of efficiency and cylinder capacity compared to the best option without restrictions.

The optimal variant of HPC FP of the 220 MW capacity turbine for nuclear power plant is obtained, which characterized by high perfection levels of aerodynamic indices, providing a boost of power on 5.4 MW, of internal efficiency on 2.71% and Moliere diagram efficiency on 2.27% as compared to the initial version of FP.

##### 7.1.3 Optimization of High-Pressure Cylinder Parameters of the 330 MW Capacity Turbine

The number of optimization parameters – 55:

- – level 1 (cylinder)-optimized for 44 parameters:
- – Root diameter and height of the nozzle blades of the first stage of the cylinder.
- – Meridional disclosing of the channels of the nozzle and working cascades.
- – Effective exit angles of the nozzle and working cascades of all turbine stages.

- – 2-nd level (stage)-optimized for 11 parameters:
- – The number of the blades in the working cascades for all turbine stages.

Quality criteria applied when optimizing – the criterion vector that includes the normalized values of Moliere diagram efficiency of the cylinder (η_{d}) and its power (*N*) with equal weight coefficients.

The results of the optimization of the HPC FP of the turbine 330 MW capacity turbine are listed in Table 7.3 and in Fig. 7.3, where η_{d} – Moliere diagram efficiency of FP; η’ – the ratio of efficiency of the stages to Moliere diagram efficiency of the initial variant of the cylinder; η_{oi} – internal efficiency of FP;Δη_{oi} – gain of the internal efficiency of the optimal FP; *N* – power; Δ*N* – the power gain of the optimal variant of the HPC FP.

Improvement of the quality indicators of the optimized FP obtained through:

- – more rational distribution of the cylinder heat drop, having in its disposal, between the stages;
- – application of the optimal configuration of meridional shape of FP with a slightly reduced heights blades;
- – increasing value of the effective nozzle exit angles, providing the reduction of the incidence angles on the working cascades;
- – improving the efficiency of working cascades through the optimal choice of stagger angles and numbers of the blades, resulting in a significant reduction of losses from the incidence angle;
- – reducing the degree of reaction level of the stages and, as a consequence, reducing the losses from root and radial leakages.

Practical application of the developed optimization theory provided the solution of the task: the optimum variant HPC PF of the 330 MW capacity turbine was obtained, which characterized by high perfection levels of aerodynamic indices, providing a boost of power on 6.2 MW, of the relative internal efficiency on 5.76% and Moliere diagram efficiency on 3.94% in comparison with the initial version of FP.

##### 7.1.4 Optimization of the HPC Flow Path Parameters of the 540 MW Capacity Turbine

Features of the initial variant of the HPC FP:

- – FP of the 9 stages HPC has high enough quality integral indicators, which have been achieved thanks to the very high level of aerodynamic perfection of the flow path of the cylinder:
- – numbers of the nozzle and working cascades blades are close to the optimal values;
- – the inlet flow incidence angles at the nozzle and work cascades are close enough to the possible minimum values given used profiles and blades production technology;
- – the root degrees of reaction provide fairly low levels of hub leakages;
- – the use of highly effective radial seals has significantly reduced radial leakages.

However, in the construction of FP reserves of possible efficiency gains were identified associated with not quite rationally distribution of disposable heat drop between the cylinder stages and somewhat inflated level of root leakages in first stage.

The number of optimization parameters of HPC FP of the turbine 540 MW capacity – 55:

- – level 1 (cylinder) – optimized for 37 parameters:
- – Root diameter and height of the nozzle blades of the first stage of the cylinder.
- – Meridional disclosing of the channels of the nozzle and working cascades.
- – Effective exit angles of the nozzle and working cascades of all turbine stages.

Due to the fact that in the initial variant of the HPC FP number of the nozzle and working cascades blades near by the optimal values, the second level of optimization (stage) was not used in this task.

Quality criteria applied when optimizing – the criterion vector that includes the normalized values of Moliere diagram efficiency of the cylinder (η_{d}) and its power (*N*) with equal weight coefficients. The results of the optimization of the HPC FP of the turbine 540 MW capacity are listed in Fig. 7.4, where (*N*) – power and η’ – the ratio of efficiency of the stages to Moliere diagram efficiency of the initial variant of the cylinder.

Improvement of the quality indicators of the optimized FP obtained through:

- – a more rational distribution of the disposal cylinder heat drop, between the stages, thereby improving the integral indicators of the cylinder quality;
- – some decrease of axial velocity component and ensuring closer to axial outlet of working fluid from the stages, that reduced the exit velocity losses, improving inlet conditions for nozzles cascades (which led to an improvement in their effectiveness);
- – close to optimal values of velocities ratio (u/C
_{0}), obtained by increasing the mean diameter of the stages; - – reduction in the share of the losses near the hub and the shroud boundaries associated with increasing the heights of the blades;
- – using in 6–9 stages of the blades a highly effective 1MMC profile (Chapter 5), which provided a good matching flow inlet angles and the geometric inlet angles of the working cascades, that resulted in increasing their efficiency;
- – obtaining the optimal twist laws of the β
_{2e}angles at the outlet of the working wheels of 6–9 stages that contributed to the rational distribution of the gas-dynamic parameters along the radius of these stages.

So, the practical application of the developed optimization theory secured the solution of the requested task: the optimum variant of HPC FP of the 540 MW capacity turbine was obtained, which characterized by high perfection levels of aerodynamic indices, providing a boost of power on 1.4 MW, of inner efficiency on 1.52% and Moliere diagram efficiency on 1.63% in comparison with the initial version of FP.