Significant impact on the stage efficiency have leakage of the working fluid through the seal gaps and discharge openings. The dependence of the leakage (and associated losses) of the stage bounding surfaces parameters can dramatically affect the distribution of the optimal parameters along the radii and, hence, the spatial structure of the flow therein. The latter, in turn, is determined by the shape and twist law of guide vane and impeller.
Development of algorithms for the axial turbine stages crowns twist laws optimization demanded the establishment of appropriate in the terms of computer time methods for calculating the quantities of leaks and losses on them, allowing the joint implementation of the procedure for calculating the spatial parameters of the flow in the stage.
The leakage calculation is necessary to conduct together with a spatial calculation step, as the results of which the parameters in the calculation sections are determined, including the meridian boundaries of the flow path. The flow capacity depends on the clearance (or leakages) values, in connection with which main stream flow calculation is made with the mass flow amplification at fixed the initial parameters and counter-pressure on the mean radius, or clarifying counter-pressure at fixed initial parameters and mass flow. The need for multiple stage spatial parameters calculation (in the optimization problem the number of direct spatial calculations increases many times) demanded a less time-consuming, but well reflecting the true picture of the flow,
methods of spatial stage calculation in the gaps described above (Fig. 2.3).
When calculating stage in view of leakage the continuity equation is convenient to take as :
where μ – the mass transfer coefficient, which allows to take into account changes in the amount of fluid passing through the crowns, and at the same time to solve a system of ordinary differential equations in sections in front of and behind the impeller like with a constant flow rate.
The leakage mass transfer coefficients  is defined as follows:
In the case of wet steam flow with loss of moisture, crown overall mass transfer coefficient is given by
where ψm,i flow coefficient, is usually determined in function of the degree of humidity and pressure ratio .
As shown in Chapter 2, the calculation of stage spatial flow with the known in some approximation the shape of the streamlines is reduced to the solution in the sections z1 = const and z2 const (Fig. 2.3) a system of ordinary differential equations (2.39) and (2.40), wherein as the independent.
As discussed in Chapter 2, the solution of (2.54) (2.55) for a given flow rate is reduced to finding the roots of the two independent transcendental equations in the hub velocities c1h, w2h.
For a given backpressure to the number of defined values the parameter ψ* is added and the problem reduces to solving a system of three equations. As a third equation the heat drop constraint (2.45) is added which can be symbolically written as
Systems of equations are solved using the methods of nonlinear programming.
The calculation of stages with reverse twist using the proposed method allows to obtain a valid reaction degree gradient and circumferential velocity component of the stage, while the calculation provided cylindricality flow gives results that differ significantly from the experimental data. The technique allows to take into account also the effect of the rotor twist law on the parameters distribution in the gap after stator. This is evidenced by comparison of stages with the same nozzle assembly.
Effect of Dm/ι ratio
With the methods described above, you can put the task of the stage parameters optimization along the height, taking into account the spatial flow and leakages. To explain the physical meaning of certain optimum blade units twist laws depending on different characteristics of stages, such as D/ι ratio, radial gap, level of reaction degree at the mean radius, the presence of suction, and other factors, there is convenient to use the angle dependencies in the form (4.4).
The stages characteristics changing may be presented as constant level lines (isolines or topograms) in the plane of the variables m1 and m2. Computational studies were subjected to axial turbine stages with Dm/ι =19…3.2, which have been tested on an experimental air turbine . Some of them are at the same axial dimension and Dm/ι have different levels of reaction at the mean radius that allows us to estimate the impact of the last factor on the crowns optimal twist laws. Separately the impact of radial clearance and suction at the root on optimal stage parameters was studied.
Influence of leakage through the radial gap
An important part of the kinetic energy loss in the axial turbine stage is a loss from radial clearance leakage, which is defined on the one hand, design and dimensions of the peripheral seal of the rotor blade and on the other – the pressure difference in the axial clearance in the outer radius. The analysis shows that in the stages of steam turbines for the amount of leakage significantly affects the Dm/ι ratio: in the stages with relatively short blades, where δr/ιb large, leakage losses greater of stages with a small Dm/ι ratio, despite the higher degree of reaction at the outer radius of the latter. Higher losses from leaks have stage without rim seals.
Large turbine units operating experience shows, that the radial clearances may increase from 1.5 mm to 5 mm, which results in reduced efficiency due to leaks more than 2 %. In some turbines due to increased near-the-shroud gaps efficiency drops 2…3 % or even 5 %. In order to analyze the impact of the radial clearance leakage losses to optimal axial turbine stage crowns twist laws calculation were conducted for the above stages with all sorts of combinations of parameters m1 and m2. and the radial clearance values. As a result of numerical experiments parameters level lines built that characterize the efficiency in the plane of m1, m2.
As an example, the results of the calculations are shown in Fig. 4.1, 4.2. Each point of the topogram was produced by the method of the spatial calculation in gaps with the streamlines refinement. The calculations were performed in different statements: with a given flow rate with a predetermined heat drop, with a predetermined flow rate and heat drop adjusting when you change the angle α 1m.
Effect of parameters m1, m2 on the reaction gradient degree varies with different diameter to blade length ratios. Thus, for stages with relatively long blades(Dm/ι <5) substantial reaction gradient alignment has been observed when m1 = -1, whereas for short blades(Dm/ι >10), requires values m1 reduction up to -8…-10. Significant impact on the reaction gradient degree with the reverse twist of the guide blades have also stage axial dimensions, especially nozzle relative width (or chord). At lower values the effect of the reaction gradient degree leveling is stronger. Reaction gradient degree alignment not only affects the magnitude of flow leakage and loss of these, but also manifested in the increase of uneven circumferential velocity component along the radius for the rotor, which leads to increased losses from the exit velocity the more (depending on m1 and m2) the smaller the Dm/ι ratio. The value of the leakage losses in the radial gap determined by the relative leakage flow rate, which depends, inter alia, on the radial clearance size and its design.
The results of numerous calculations indicate that the crowns optimum twist laws (parameters m1 and m2) for the stages of the various Dm/ι ratio at different values of radial clearance is mainly determined by the ratio between the amount of output velocity losses, and losses from leaks in the radial clearance. Influence of hydraulic losses in the guide vane and the rotor has a significant impact on the level of the degree of reaction for very small leak quantities into over-shroud space.
At zero clearance maximum of peripheral efficiency of the cylindrical stage for small Dm/ι located in a neighborhood of the point with a minimum exit loss (Fig. 4.1, 4.2): the guide vanes twist close to the cur= const law, and impeller should be twisted a little more intense (m2 = -1…-2).
With increasing Dm/ι ratio the maximum peripheral efficiency shifts toward twist laws with m1>1 due to the hydraulic losses influence. The amount of displacement depends on the method of calculation (with a given flow rate or heat drop) and the reaction degree at the mean radius: offset is stronger when calculating with a predetermined flow rate due to changes in the angle α 1m at the middle (on flow rate) radius as a result of the streamlines lifting, as well as an increase in the average reaction degree.
The increase of the relative radial clearance leads to a shift of the maximum internal stage efficiency point in the direction of m1 downward and m2 increase, which is a sharper, then Dm/ι relation is greater. This entails the reaction degree gradient alignment due to the streamlines inclination to the hub after the stator and some thrown at the periphery of the impeller. For example, in sages with Dm/ι =8.3 in the absence of leakage in a radial gap some flow preload to the periphery in the gap between the vanes is expedient. As the radial clearance increases it is appropriate to decrease the level of the reaction degree at the mean radius and its gradient. When δr = 1.5 mm is advantageous to almost completely eliminated the reaction gradient (m1 = -4…-5, m2 = 0…1) The calculated results are in good agreement with the experimental study .
In the stages with even shorter blades (Dm/ι=19) for large radial clearances m1 optimum value drops to -9…-11, and m2 increases to 4…5. For large Dm/ι values the stages with reverse twist for all real values of clearance have higher efficiency than the stages of traditional design. Winning increases with decreasing gap and the degree of reaction at the mean radius. Experimental research of stages Dm/ι = 19  fully confirms the conclusions.
The character of the stage relative loss change qualitatively is the same for all types of stages, with both small and large Dm/ι ratios (Fig. 4.2). Isolines of the exit velocity loss form closed curves, surrounding the minimum point with m1 = 1, m2 = -1, corresponding to the constant circulation twist law. Isolines of the relative hydraulic loss regardless of the size of radial clearance are in the nature of saddle points: the relative losses in the guide vanes have in the saddle point m1 maximum and m2 minimum, while the relative losses in the rotor blades on the contrary, have m1 minimum and m2 maximum. With the Dm/ι ratio increase hydraulic losses in the crowns becoming less dependent on the laws of another crown twist, acquiring the form of lines extended along the respective axes. This applies in particular to losses in the guide vane.
The leakage losses in the radial clearance in the plane of the variables m1, m2 achieved the highest value in the upper left corner of the topogram, where the peripheral degree of reaction is maximal, and the least – in the lower right corner, where the reaction degree gradient is minimal.
Influence of suction in the near-the-hub gap
To investigate the suction effect on the crowns best twist laws at fixed parameters on the mean radius were selected three experimental air turbine stages, with different blades elongation and reaction degree at the mean radius Dm/ι = 3.6, 8.3, 14.1 and Rm= 0.2, 0.02, 0.01, respectively). Calculations were made for various values of the radial gap and flow suction introduced by changing the reduced gap of the diaphragm seal, to provide thereby suction value in stages with Dm/ι =3.6 – 0.5%, with Dm/ι = 8.3 – 1 % and with Dm/ι = 14.1 – 2 %.
The influence of the flow suction does not change the conclusions regarding the optimal crown twist laws made above. It should be borne in mind that in actual turbine stages discharge holes presence results in a large impact on the suction flow of the pressure difference at the inner radius of the impeller. When properly selected reaction degree in the root and the appropriate size of the discharge holes relative flow rate of the jet can be almost reduced to zero.