Blade design

2.3.1 Blade cascade geometry

After the three-dimensional flow angles are determined, the rotor and stator blade shapes need to be designed so that the designed flow is achieved.

In compressor blade design, a quasi three-dimensional approach is often used. The flow is assumed to only have an axial and tangential velocity component that vary over the radial position. The blades are designed by considering the two-dimensional flow angles at every radius. Next, the blade sections from every radius are ’stacked’ to create the total blade shape. The figure below shows a part of a cascade with the important geometry parameters indicated:

Figure 2.6: Three blade sections part of a typical cascade with the important angles and dimensions. Figure adapted from [4].

Here α⁰₁ and α⁰₂ are the blade in- and outlet angle, α1 and α2 are the air in- and outlet angle, V₁ and V₂ are the in- and outlet air velocity, θ = α⁰₁ − α₂⁰ is the camber angle and ζ is the angle between the chord and the axial direction called the stagger angle or blade setting. The chord is an imaginary line connecting the leading and trailing edge of the blade section and its length is given by c. Furthermore, s is the pitch or spacing between the blades,

i = α₁− α⁰₁ is the incidence angle, AoA = α₁− ζ is the angle of attack and δ is the deviation angle.

Note that when discussing velocity triangles, α denotes the air angles with respect to the fixed world and V denotes the velocity with respect to the rotor blade. From now on this use of the variables will be called the velocity triangle notation. When discussing blade section design α and V always denote the air angle and air velocity with respect to the blade, both in rotor and stator design. This will be called the cascade notation.

From the large number of parameters shown in Figure 2.6 it may appear that many parameters need to be determined to get to a section design. However, note that many of the parameters above are equivalent. For example, both the incidence angle as the angle of attack describe the position of the blade relative to the air. Given that α1 and α2 are known and that the section has a symmetrical mean-line, only 4 parameters need to be determined to get a fully constrained blade design: c, s, θ and ζ. In case of an asymmetrical mean-line the exact shape of the mean-line must be known to calculate α⁰₁ and α⁰₂. All other parameters mentioned above can be calculated from these four parameters and the air angles using simple equations that follow directly from the geometry.

2.3.2 Blade section shape

As already mentioned in section 2.2.3 different blade section shapes can be used depending on the blade Mach number. Examples of blade sections are the NACA 65-series and double-circular-arc(DCA) sections. The former is recommended for blade Mach numbers up to 0.78, the latter for blade Mach number between 0.70 and 1.20 [5]. For higher blade Mach numbers other shapes have been developed. However, these Mach numbers will not be encountered within this project. Many systematic cascade tests have been done on NACA 65-series blade sections. Using this data the performance of a blade design can be estimated.

In general blade sections shapes are defined using a mean-line and a thickness distribution with respect to this mean-line. The construction of a NACA 65-series section is visualized in the figure below:

Figure 2.7: Construction of a NACA section. The upper and lower surfaces are defined using a thickness distribution with respect to a specific mean line. Figure from [6].

For a NACA 65-series section the mean-line is defined with respect to the chord line. The shape of the mean-line depends on the camber, which for NACA sections is defined by the design lift coefficient of the isolated air foil C_l0. The mean-line for a NACA 65-series section with C_l0= 1.0 is described by the stations x along the chord line and the mean-line ordinates ym given in table A.1 [7]. To obtain a mean-line for a different design lift coefficient the

ordinates can be multiplied by the desired C_l0 [6][7]. As is common in NACA wing section literature, all stations and ordinates are given in percentage of the chord length. The mean-line slope dy_m/dx is tabulated in table A.1. The mean-line slope at the leading and trailing edge are not given because for those stations the slope is theoretically infinite. The mean-line angle φ can be calculated from the mean line slope using simple goniometry.

Next, the upper and lower surfaces of the section are defined by a thickness distribution perpendicular to the mean-line. Table A.1 gives the thickness ordinates in percentages of the chord for a NACA 65-series section with a maximum thickness of 0.1c. To calculate the thickness profile of a section with a different maximum thickness the ordinates can simply be scaled. The upper and lower surface coordinates can now be calculated using:

x_U = x − ytsin φ (2.35)

y_U = y_m+ y_tcos φ (2.36)

xL= x + ytsin φ (2.37)

y_L= y_m− y_tcos φ (2.38)

Here x_U and y_U are the coordinates for the upper surface and idem for the lower surface.

Normally the blade in and outlet angle are defined by the mean-line slope at the leading and trailing edge. However, since the mean-line slope is theoretically infinite for this does not make sense for NACA 65-series profiles. Instead an equivalent camber angle is defined based on an equivalent circular-arc mean-line. For cambers 0.0 < C_l0 < 2.4 the equivalent camber angle in degrees is given by[5]

θ = 24.7C_l0 (2.39)

Since a circular arc mean-line is symmetrical, the blade in and outlet angles can be cal-culated by:

α⁰₁ = ζ +1

2θ (2.40)

α⁰₂ = ζ −1

2θ (2.41)

For blade Mach numbers above 0.78 the NACA 65-series sections are not suitable. Instead DCA sections can be used. Double-circular-arc blade(DCA) section, both the suction and pressure surface of the blade consist of a circular section. Because of this, the mean-line is also a circular arc. One downside of the use of DCA section is the limited amount of available test data. An example of DCA blade section is shown below:

Figure 2.8: Example of a double circular arc blade section. Recommended for blade Mach numbers between 0.70 and 1.20.

2.3.3 Blade design using empirical data

Blade design is often done using empirical data from two-dimensional experiments. In such experiments a row of blades is positioned in a two-dimensional flow wind tunnel. Next, for a fixed blade shape, the solidity c/s, the incidence angle and the air inlet angle are varied. The achieved deflection , losses and forces acting on the blades are measured. To use this data to design blade shapes, the data has been transformed into diagrams or empirical formulas that can easily be used by the engineer.

To come to a complete blade section design one should determine the chord length c;

the blade spacing s; the camber; the blade setting or stagger angle ζ; or other parameters equivalent to those. Note that the camber can be expressed in camber angle θ or as the design lift coefficient C_l0. The latter is common in NACA 65-series design.

The first step to a complete blade design is to choose the chord length. Chord length is often determined by considering the aspect ratio l/c of the blades. Here l = r_t− r_r is the length of the blade which is known because the root and the tip radius already have been determined using the theory in section 2.2.1 and 2.2.2. The aspect ratio is important when concerning blade losses. A good first estimate for the aspect ratio is l/c = 3.

Next the blade spacing s needs to be determined. In American literature the blade spacing is often considered in terms of the solidity σ = c/s. Note that the blade spacing directly depends on the number of blades and that the spacing varies over the blade length. In practice the choice in number of blades is limited by the available test data and the fact that the number of blades is preferably a prime number. By choosing a prime number, one avoids common multiples in the rotor and stator. This reduces the chance of introducing resonant forcing frequencies.

With the chord length and blade spacing known, only the camber and stagger angle need to be determined. Two possible methods are discussed in the sections below.

Method 1: determine θ and ζ using an empirical formula expressing δ To finish the blade design, a camber angle θ and blade setting ζ must be chosen so that the desired deflection  is achieved. One method is to choose a value for the incidence i, and to then determine the camber angle using an empirical equation for δ. Here δ is the deviation. Ideally the air would leave the blade cascade exactly in the direction the trailing edge points. In real life however, a deviation δ is found. Often i = 0° is chosen. The camber angle can be calculated by:

θ = α₁− i − α₂+ δ (2.42)

The deviation can be calculated using empirical equations:

δ = mθp

1/σ (2.43)

Here m is a parameter that can be calculated using equations 2.44 [4] or 2.45 [5]:

m = 0.23 2a Note that all angles are in degrees! Here a is the point of maximum camber measured along the chord line from the trailing edge. In case of a symmetrical mean-line 2a/c = 1.

Furthermore, α2 is the air outlet angle and σ is the solidity of the blade cascade.

Now, given that the mean-line is symmetrical, ζ can be calculated by:

ζ = α₁− i − 1

2θ (2.46)

Method 2: determine θ and ζ using figures from [8] A second method to determine the camber θ and stagger ζ angle uses carpet plot from [8]. In [7] two-dimensional cascade test were done on NACA 65-series blades, systematically varying the geometry. In [8] this data was summarized and visualized in carpet plots that can easily be used by engineers to determine the blade geometry once the blade inlet air angle α1, the deflection  and the solidity σ are known. The figures cover a range of α1 = 30 − 70°,  = 2−22° and σ = 0.5−1.5.

First, the camber C_l0must be determined using Figure 1 from [8], which shows the camber Cl0 as a function of air inlet angle α1, the desired deflection  = α1 − α₂ and the solidity σ = c/s. Next, the design angle of attack AoA can be determined using Figure 2 from [8], which shows the design angle of attack as a function of C_l0 and σ. Appendix C explains how to use the carpet plots. Now, the camber angle can be calculated using equation 2.39 and the blade setting ζ can be calculated by:

ζ = α₁− AoA (2.47)

2.3.4 Compressor blade efficiency

In equation 2.14 the isentropic efficiency is needed to determine the compressor pressure ratio.

Below, the calculation of ηs is shown.

The blade row efficiency in terms of pressures is given by:

η_b = p₂− p₁

Here p⁰₂ is the static pressure at station 2 in case of isentropic compression and p₂ is the actual static pressure. Furthermore, ¯w is the total pressure loss and ∆p_th is the theoretical pressure rise over the blade row. The denominator is called the pressure rise coefficient and can be calculated using:

∆pth 1

2ρV₁² = 1 −cos² α1

cos² α2

(2.49) The enumerator of equation 2.48 is called the loss coefficient and can be calculated using:

¯ Here α_m is the mean flow angle is the blade row and C_D is the overall drag coefficient given by:

α_m= 1/2 (tan α₁+ tan α₂) (2.51)

CD = CDp+ CDA+ CDS (2.52)

Here CDp is the profile drag coefficient over the blade section, CDA is the annulus drag coefficient and C_DS is the secondary losses drag coefficient. The profile drag coefficient of a NACA 65-series section can be determined from Figures 7 to 84 from [7]. C_DA and C_DS are given by:

Here l is the blade length and CL is the section lift coefficient that can be from the same figures as the profile drag coefficient[7] or with the equation below:

CL= 2 1 σ



(tan α1− tan α₂) cos αm− C_Dp tan αm (2.55) Now the blade efficiencies for the rotor and stator can be determined using the equations above. Although the η_b and the isentropic efficiency η_sare different things, their actual values are very close and without much error η_b = η_s can be used. The isentropic efficiency can be calculated using:

η_s = Λ η_b,rotor+ (1 − Λ) η_b,stator (2.56)