USING SIMILARITIES OF TURBOMACHINES IN TURBOMACHINE DESIGN

Basic concepts

The design of the turbomachine is based on entered parameters. The number of entered parameters is usually insufficient for direct calculation of the dimensions, therefore the designer must obtain additional input parameters using similarity theory. Similarity theory uses so-called models for predicting the dimensions, performance characteristics of machines. The basic quantities in similarity theory are similarity coefficients.

Similarity theory as tool for experienced and educated observers

Similarity theory is a tool used by rational observers to predict the development of observed or future processes based on already known similar processes. However, observers must first be able to determine that the process is similar, which requires experience and education in the relevant field of natural or human sciences, as well as the ability to find connections. Similarity theory is used not only in technology, but also in medicine and other natural sciences.

Model as predictable section of reality with required properties

A model is a section of reality that we understand perfectly, is often well described by a theory, and in the time required we are able to determine its properties in the situations we are investigating, while at the same time we think of this section as having the same key properties in these situations as another section of reality we want to make real. Thus, the model can be a similar (model) machine or even a computational model, such as the calculation of a turbomachine for the case of potential flow (flow without losses), calculation under some other simplifying assumption, etc.

In engineering practice, similarity can be quantified using similarity coefficients such as number π

The model must have the same observer-defined similarity coefficients values with the target object or process. The similarity coefficients are for example the number π, which is the ratio of the circumference of a circle to its diameter (3,14159...). Thus, when calculating the circumference of a circle, we assume that circles are similar and that the above ratio holds for all its sizes. Another well-known similarity coefficient, is the Reynolds number used in evaluating flow properties. A typical feature of the similarity parameters is dimensionlessness, since it usually expresses the ratios between the selected quantities.

Values of geometric similarities

The values of geometric similarity parameters for bladed machines are provided in company design documentation and are also widely published in specialized literature; see the references at the end of the article and examples of values for a radial fan with forward curved blades in Table 2.

Flow coefficient

The shape of the velocity triangle is determined by the geometry of the rotor, which was selected based on the geometrical similarity described above. The flow coefficient indicates the usual ratios of the individual sides of the rotor outlet velocity triangle.

Definition of flow coefficient

Flow coefficient is the ratio between the meridional velocity of the working fluid and the blade speed at the rotor outlet measured at the tip of the blades or the circumference of the radial rotor, Formula 3. However, for stages of radial centripetal turbines, the definition of the flow coefficient can be related to the rotor inlet (V_1r, U₁).

– 3: –

(a) definition of flow coefficient; (b) application of flow coefficient to radial stage of working machine. ϕ [1] flow coefficient; V_2m [m·s^-1] meridional velocity of working fluid; V_2r [m·s^-1] radial component of velocity V₂.

Alternative definition of flow coefficient

The given speed ratio is determined by the mass flow of the working fluid through the stage and the flow area, so the formula for ϕ can be converted to a more general form that gives the average value of ϕ for the entire stage, see Formula 4a. Instead of the actual flow area, a reference area equal to the area of a circle with the outer diameter of the rotor is also used, for example in fans. This is possible because the actual flow area A₂ of fans is a function of the diameter d₂, see Formula 1. In this case, however, ϕ no longer directly expresses the ratio of the selected sides of the output velocity triangle, but a multiple of this ratio, see Formula 4b. Due to this ambiguous definition of ϕ, it is necessary to include the formula when specifying the flow coefficient values so that it is clear which value is being referred to.

Specific speed

The specific speed is the speed of the described stage at which the stage would have a power of 1 W, while being loaded with an energy differential of 1 J·kg^-1 and the rotor would be decreased/increased to a diameter of 1 m (it is possible to find other model stage parameters in other units in the literature).

Definition formula of specific speed

Specific speed is defined by Formula 8. In the case of single stage machines, the energy differential is based on the whole machine.

– 8: –

N_S [min^-1] specific speed; N [min^-1] actual machine rotational speed; w_i [J·kg^-1] internal work of stage (machine); d [m] reference diameter (most often rotor diameter); P_i [W] internal power of stage/machine. The derivation of the formula is shown in Appendix 870.

Special variants of specific speeds according to industry standards

Internal work in hydraulic machines is recalculated, for example, to the available water head (for water turbines) based on the Bernoulli equation (foe example Formula 9 for water turbines), or the stagnation pressure change (for pumps and fans). The formula for the specific speed of the fan stage is modified so that the volume flow rate appears instead of the power P_i. Sometimes, the formula for specific speed is even modified so that it no longer has the unit min^-1, but for clarity, the term speed with the unit min^-1 is still used. This is also the case with the Formula 9 for the specific speed of water turbines, where gravitational acceleration, which is ‘the same’ across the entire planet and has no effect on the comparison of two water turbines, is omitted. By removing gravitational acceleration, the specific speed can no longer have the unit min^-1.

– 9: –

g [m·s^-1] gravitational acceleration; Δz [m] difference in water levels. Since water turbines are single-stage machines, the expected power at the coupling is usually compared.

Relationship between specific speed and internal efficiency

In general, it can be said that axial stages have higher internal efficiency at higher specific speeds than radial stages, and vice versa. This is mainly due to the fact that the Euler work of a radial stage will be greater than that of an axial stage at the same relative velocity in the blade passages due to the change in blade speeds, which has an impact on stage losses, see Figure 11. The relationship between specific speeds and the internal efficiency of the machine is demonstrated, for example, in [Ingram, 2009]. The values of optimal specific speeds for different types of blade machines are given, for example, in [Japikse, 1997], [Pfleiderer and Petermann, 2005].

– 11: –

Internal efficiency of compressor stages as function of specific speed: a-radial stage with axial inlet; b-axial stage. η_is [1] internal efficiency of compression stage. Data source in [Japikse, 1997, p. 1-23].

Operational similarities of turbomachine stages

The similarity of turbomachines can be used to predict the values of operating variables , internal losses and efficiency of turbomachines and to optimise their operating states.

Operating variables as a function of flow and head coefficients

The performance characteristics of turbomachines depend on changes in two or more monitored operational variables of the machine, such as power, internal work, compression ratio, increase in total pressure at mass flow, etc. In some cases, based on the theory of similarity, it is possible to derive equations of performance characteristics as a function of head or flow coefficient at constant rotational speed, see Formulas 12.

– 12: –

ε_s [1] compression ratio; c_P [J·kg^-1·K^-1] heat capacity at constant pressure; T_i,s [K] stagnation absolute temperature of working gas at stage inlet; n [-] polytropic exponent. The formula for w_i is derived considering only profile losses; the derivation is shown in Appendix 668.

Internal losses as difference between head coefficient of ideal and real machine

The ideal characteristic does not take into account losses that occur during flow. The difference between Δh_s,is and w_i (in Formulas 6) are internal losses of the stage, so these losses can also be expressed from the difference in head coefficient between ideal flow and real flow, see Equation 15.

– 15: –

(a) for turbine stages; (b) for working machine stages. L_w [J·kg^-1] internal losses of stage.

Actual dimensionless characteristics of turbomachine stages

Losses vary with mass flow, but if we have two similar stages (machines), this dependence will be similar, so if we measure the performance characteristics of one reference stage and create a dimensionless characteristic from it (Figure 16, Figure 17), then we can quite accurately predict the performance characteristics of a completely new stage (Problem 721) and its internal efficiency using Equations 12. Using Equations 15, we can also numerically express the expected losses for the new designed stage.

– 16: –

Left-example of axial turbine stage characteristics; right-example of radial working machine stage characteristics (β_B2>90°). SS-stage stall area – due to incorrect combination of angle of attack i and mass flow, flow separation from blades occurs; i-loss curve due to incorrect angle of attack. η_i [1] internal efficiency of stage (formula valid for turbine and working stages). The index _opt denotes optimal. The characteristics are for N=const.

Determination of position of optimal and nominal operating parameters in ψ-ϕ chart

The ratio ψ/ψ_id must approximately correspond to the internal efficiency of the stage η_i, because the difference between the actual and ideal head coefficient is equivalent to losses. From this, it can be expected that machines with the ideal head coefficient equal to 2 must correspond to the nominal (maximum) and optimal performance. For stages with a positive slope of the ideal head coefficient, optimal performance can be expected at lower mass flows than the nominal ones. For stages with a negative slope of the ideal head coefficient, optimal power can be expected at higher mass flows than the nominal ones. This behavior is mainly caused by a change in the camber of flow when the angle of attack changes – in the first case, the camber of flow increases with flow rate until flow separation occurs, in the second case, the camber of flow increases as the mass flow rate decreases, etc. From the values ψ_opt, ϕ_opt, it is possible to calculate the dimensions of the new machine at which it will most likely achieve optimal parameters (Problem 3).

 

Prediction of power sensitivity to changes in mass flow

The expected shape of the real dimensionless characteristic also depends on the value of the ideal power coefficient slope, as shown in Figure 16. The greater the slope value, the more steep the actual dimensionless characteristic will be. For example, in the case of an axial stage with a reaction of 0, the power will be more sensitive to changes in mass flow than a stage with a reaction of 0,5, etc., see the conclusions in Figure 14.