SHAPES OF BLADES AND FLOW PARTS OF TURBOMACHINES

Types of profile cascades

The size of the inlet and outlet flow area of the blade cascade is crucial for the resulting working fluid velocities. The size of the inlet and outlet flow area of the blade cascade is crucial for the resulting working fluid velocities. From this, we distinguish three types of profile cascades depending on whether the blade passages create confuser, diffuser (convergent or divergent) passages, or passages with the same flow area between the inlet and outlet.

Changing type of profile cascade by changing stagger angle of profile

The shape, respectively the inlet and outlet flow area of the profile cascade, is determined by the stagger angle (inclination in the cascade). Any basic type of profile cascades with the same pitch can be constructed from the same profile by simply turning the profile or changing the stagger angle of the profile in the cascade, see Figure 1.

1:

(a) confuser cascade - also called overpressure, turbine cascades; (b) pressureless cascade - with same flow area pressure and velocity between inlet and outlet do not change; (c) diffuser cascade - also called overpressure, compressor or turbine cascades in case A₁ is critical flow area. A [m²] flow area. γ [°] stagger angle of profile in cascade; s [m] pitch. The profiles are simplistically drawn as if they were made of sheet.

Blade profile

The profile of the blades is usually shaped like a cambered drop. The camber of the blade is based on the requirements for the camber of the fluid velocity inside the blade passages. However, the shape and size of the profile also depend on the strength and other requirements related to the operation of the turbomachine.

Profile drawing

The blade profile can be recorded using blade profile geometry rules. The choice of the method of recording the blade profile shape depends on the medium of this recording, or the most suitable method in relation to the required form of production documentation. Currently, graphical output (using vector graphics) is sufficient, e.g. in CAD systems, as machine tools and computing software are able to work directly with such output, but other forms of blade profile notation exist. For example, they are written in tabular form in the x; y coordinates and using the shape of the camber line of the profile, see Figure 2.

3:

β_B1, β_B2 [°] inlet and outlet angle of blade profile; i [°] angle of attack; δ [°] angle of deviation; Δβ [°] camber of flow; b [m] width of profile cascade; W₁, W₂ [m·s^-1] attack and outflow velocity; W_m [m·s^-1] mean aerodynamic velocity in cascade; i_m [°] angle of attack of mean aerodynamic velocity.

Flow adhesion to profile as a basic aerodynamic requirement

The profile shape is based on the requirements that the blade must meet in the machine, usually the profile resembles a curved drop. The droplet shape allows the ideal adhesion of the fluid to the blade surface and therefore follows the camber of the blade. The property of a fluid to follow a surface is called the Coanda effect, after the Romanian engineer Henri Coandă (1886-1972), who studied the wrapping of surfaces and bodies. The Coanda effect is clearly visible when water is drawn from a cup that does not have a rim around the neck. Up to a certain angle of inclination of the cup, the water runs down the outer surface of the cup instead of flowing directly to the ground. The pressure of the water stream on the surface of the cup is caused by the viscosity, buoyancy, and lower pressure in the flowing liquid than the ambient air pressure. A similar phenomenon occurs when water flows around a cross pipe, for example in a condenser, etc. The adhesion of the fluid to the wrapping surface has limits, beyond which the flow separation from the wrapping surface, these limits are dealt with in the aerodynamics of profile cascades.

Profile catalogues

A suitable profile can be selected from profile catalogues based on aerodynamic requirements. The shapes and aerodynamic data of thin and low camber profiles can be obtained from extensive airfoil catalogues used in aeronautics, for example [Abbott and Doenhoff, 1959]. If a suitable blade profile is missing from the catalog, it must be developed and experimentally verified.

Range of values of some geometric and aerodynamic variables of blade cascades

The values of geometric and aerodynamic parameters of the blade cascade are usually based on the requirement for maximum internal efficiency of the stage. This requirement may not always be prioritized; blade cascade parameters can be chosen based on the needs performance of machine and cost.

Number of blades based on blade cascade density

The required camber of flow is a function of the camber, of pitch, of chord respectively the number of blades, where the ratio between the length of the chord and the pitch of the profile cascade is referred to as a density of profile cascade, see Formula 6.

6:

σ [1] density of profile cascade.

Estimation of optimal density of profile cascade

With a higher density of the profile cascade, a greater camber of flow can be expected and vice versa. On the other hand, the number of blades and frictional losses in the cascade increase with the density of the profile cascade. Thus, there is an optimum value for the density of the profile cascade. In the case of diffuser profile cascades, the optimum density should be sought to be one at which the c/a_m ratio is around 2,5, where a_m is the mean width of blade passage [Pfleiderer and Petermann, 2005, p. 408], see Formula 7. In the case of confuser cascades, this ratio is usually less than 2,5. These ratios are valid for profile cascades composed of thin, low camber profiles.

7:

a_m [m] mean width of blade passage. The derivation of the equation is shown in Appendix 5.

Optimal density of heat turbine blade cascades

The density of profile cascade with highly camber heat turbine profiles can be approximated by the Zweifel coefficient. The Zweifel coefficient C_L,θ is the ratio of the tangential component of the force on the blade from the fluid flow F_θ to the product of the blade area and the dynamic pressure of the relative velocity at the outlet of the blade cascade, see Formula 8. The value of this coefficient for the designed profile cascade should be in the range of 0,75...0,85 for modern profiles with high blade material strength up to 1 [Japikse, 1997, p. 6-17].

Maximum camber of flow in diffuser profile cascades

Diffuser profile cascades are most commonly used for the working machine stages. Diffuser profile cascades are more sensitive to flow separation from the blade as the camber increases and therefore a small camber of flow Δβ is required, ranging from 15° to 30°. The goal is to achieve the smallest possible V_1θ values for the maximum absolute values of the Euler work.

Shapes of blades

The shape of the blade determines the final shape of the blade channel. The simplest blade channels have the same shape along the entire length of the blade. Such blade channels can be created using straight blades – blades with the same shape and profile size along their entire length. In the case of axial stages, the radius and pitch of the profile cascade change. Variable blade channel shapes can be created using twisted blades, which have a variable profile shape along their length. In addition to these two basic shapes, there are also radial blade shapes for radial stages, whereby straight blades are also used for types without axial parts, see Figure 9.

Straight blades are based on 1D calculation of stage

Straight blades of axial stages are usually used where a small ratio between blade length and mean blade radius can be achieved, so that the spatial character of the flow is not so apparentand 1D calculation is sufficient for stage design. The advantage of straight blades is the simplicity of design, manufacture and cost. They are often manufactured by drawing as circular wires, see Figure 10.

10:

left-static steam turbine blade made of drawn profile with machined groove for attachment at the base of the blade; right-rotor steam turbine blade made of drawn profile with machined tip and foot made of forging.

12:

(a), (b) examples of radial straight blades; (c) example of radial baldes with axial part

The index _t indicates the tip of the blade, _h the root of the blade or the radius of the shaft. b [m] width; r_m [m] mean radius of blade.

Shapes of meridional surfaces of radial stages

The curves ψ_h and ψ_t ( meridional surfaces) at the root and tip of the blades (Figure 12c) are either circles or other smooth, easily describable and manufacturable curves.

Radial blade leading and trailing edge shapes

The axial part of radial blades is used in turbine stages and working machines. In turbines, it has the function of uniformly transferring flow from radial to axial direction with a reduction of tangential velocity component V_2θ. The axial part is called the inducer in the case of working machine stages. In both cases, this part of the radial stage is calculated as twisted (see 2D-calculation of inducer), because there is usually a large difference between the blade speed at the root and tip of the blades. In hydrodynamic pumps and Francis turbine, the axial part is usually not pronounced (Figure 13) to avoid large differences in relative velocity angles compared to the optimal angle preventing cavitation. In this case, the blade edges are bounded by the C_edge curve. The C_edge curve is a smooth curve that follows the radius at the tip r_t approximately perpendicularly (λ_t≈90°), the angle λ_h is usually less than 90° [Pfleiderer and Petermann, 2005, p. 157]. Only for rotors with small relative blade width b is the C_edge curve replaced by a straight line (Figure 13b). The drawing of individual streamlines, respectively the boundaries between the individual elementary stages, is based on the simplified rule that for a drawn arbitrary continuous curve C_x (Figure 13a) the Equation 13c holds. The manufacture of this part of the blade by machining is difficult and therefore this type of rotor consists of several parts - see Figure 14.

13:

C_edge-curve of leading edge (hydrodynamic pumps) or trailing edge (Francis turbines). The derivation of Equation 13c is shown in Appendix 7.

Calculation and shapes of spiral casings

The basic design of the spiral casing is based on the theory of potential flow, in which the streamlines have the shape of logarithmic spiral. The basic spiral casings shapes are shown in Figure 15, with some shapes not satisfying the conditions of the potential flow equations, resulting in vortices. However, they have other advantages - in particular, they reduce the calculated diameter of the spiral casing, which results in a much larger diameter than the rotor diameter for a constant casing width and potential flow. Spiral casings can also be reduced by ending them at less than 360°, see Problem 4 - shortened casings can be used where size is a more important parameter than efficiency.

15:

(a) rectangular (constant casing width - mainly used in fans); (b) trapezoidal (gradual extension leads to lower losses than step extension and its shape is very close to the condition for potential flow); (c) circular; (d) tangential outlet casing. b [m] casing width.

Basic branches shapes for axial stages

Figure 16 shows some designs of axial and side branches. In the case of Figure 16b, the jet engine axial beveled branch that allows for more optimal pressure distribution of the first stage of compressor. The maximum engine power (air consumption) is during takeoff, and therefore the angle α approximately corresponds to the pitch angle during takeoff. More details on this issue, including the calculation of the optimal deflection angle α, are given in [Mattingly et al., 2002, p. 424].

16:

(a) axial inlet (e.g. combustion turbine inlet); (b) jet engine inlet; (c) axial outlet (e.g. axial fan outlet); (d) side branches (e.g. axial compressor side branches). α [°] deflection angle.

(a) rotor; (b) stagger angle detail; (c) manufacturing drawing of blade. ω [°] auxiliary angle.

References