An alternative method for centrifugal compressor loading factor modelling

The loading factor at design point is calculated by one or other empirical formula in classical design methods. Performance modelling as a whole is out of consideration. Test data of compressor stages demonstrates that loading factor versus flow coefficient at the impeller exit has a linear character independent of compressibility. Known Universal Modelling Method exploits this fact. Two points define the function – loading factor at design point and at zero flow rate. The proper formulae include empirical coefficients. A good modelling result is possible if the choice of coefficients is based on experience and close analogs. Earlier Y. Galerkin and K. Soldatova had proposed to define loading factor performance by the angle of its inclination to the ordinate axis and by the loading factor at zero flow rate. Simple and definite equations with four geometry parameters were proposed for loading factor performance calculated for inviscid flow. The authors of this publication have studied the test performance of thirteen stages of different types. The equations are proposed with universal empirical coefficients. The calculation error lies in the range of plus to minus 1,5%. The alternative model of a loading factor performance modelling is included in new versions of the Universal Modelling Method.

center of pressure

Aim of the work
For flow path design, an instrument is necessary for calculation of centrifugal compressor gas dynamic performance. The authors use the primary design procedure described in [1]. The Universal Modelling Method presented in [2] is applied to calculate performance of a primary design and of possible better candidates. Calculated non-dimensional performance of a stage is presented in Figure 1. (1) Efficiency prediction is possible with new versions of the Universal Modelling PC programs without special skill and experience [2][3][4]. Prediction of the work coefficient performance ( ) i f y = F still requires experience and intuition. The authors' aim is to offer an alternative way of the work coefficient performance calculation that is simple and precise. Presented work is based on ideas and results of Y. Galerkin and K. Soldatova who had studied loading factor performances of impellers with inviscid flows.

Scheme of work coefficient modelling
In accordance with the scheme proposed previously in [5] the head transmitted to gas by an impeller i h consists of three parts. The main part of engine's mechanical head T h is transferred to gas by blades of the impeller. Two additional parts appear due to parasitic losses. The part of the head lc h is lost in labyrinth seals. The friction on outer surfaces of hub and shroud transmits additional mechanical head df h but this head does not increase gas pressure: Non-dimensional presentation of this equation: impeller with small flow coefficient des F » 0,015 this sum is about 0,055-0,065. It is not large part of a head coefficient. Semi-empirical formulae in [1] and CFD-calculations [9] are good instruments for modelling of these coefficients. Therefore the main problem is modelling of a loading factor.
As a rule there is no velocity tangential component at an impeller inlet in industrial compressors. Then impeller blades transfer the head to a gas in accordance with the Euler equation: This head is presented by the non-dimensional loading factor: (6) The modelling of a loading factor performance is facilitated by the fact that it is a linear function - Figure 2.

Figure 2. Linear function of a loading factor for ideal and real impellers
The linear nature of it is evident for an "ideal" impeller with infinite number of infinitely thin blades: The similar equation is valid for real impellers: The existing way of modelling [2][3][4] exploits this fact. Two values of a loading factor are necessary to determine a linear performance In the method described in [1] these two values are a loading factor at a design flow rate T des y and at zero flow rate 0 T y . The design flow rate corresponds to non-incidence inlet of the critical streamline.
This condition is 1 Exit and inlet velocity triangles demonstrates influence of blades' load and blockade on a critical streamline direction - Figure 3. The empirical coefficient 1 K µ > represents influence or real viscid character of flow. Its value for different types of impellers may be between 1,5-2,3. There is no satisfactory correlation with an impeller configuration. The close analog is necessary. The alternative is to model a loading factor performance by values of 0 T y and angle T b that are shown in Figure 2: The aim of this work is to model loading factor performance of impellers in real viscid flow on the basis of several impeller geometry parameters.
Test data for model stages and factory test data of several pipeline compressors were reduced. The information on the objects of modelling is presented in Table 1. All stages and compressors were tested at u M =0,60 or 0,80. Model stage names mean the following. For example, 0,0604-0,527-0,290 means that the design flow rate coefficient of the stage is des F = 0,0604, design loading factor is T des y = 0527, hub ratio is 2 / h D D = 0,29. Names of compressors are given by their manufacturers on their own principles. Data on compressors are taken from [4]. Symbol * means that 2D impeller has an arc blade mean line. Mean lines are designed by the Method presented in [1] in other cases. In columns 3-9 geometry parameters of impellers are presented, they are included in the presented below equations for loading factor performances modelling.  The sample of data on model stages from the "IDENT" program is presented in Figure 5.
Most of 2D impellers designed by the authors have either an arc mean line of blades, or a mean line is optimized by analysis of velocity diagrams. Two mean lines and corresponding velocity diagrams are presented in Figure 7.  Table 1). For impellers of this compressor the empirical coefficient is µ K = 2.62 (eq. 10). It is an unusually large value. The authors have no explanation for this fact. The modelling error in all other cases is within admissible limits. Figure 10 presents the comparison of performance of one of model stages.
Graphics with individual adjustment of the loading factor performance are presented on the left. Right -loading factor is calculated by eq. (14, 16). Efficiency performance is calculated by 6 th version of the Math model in both cases. Despite visually noticeable difference of load performances, an error of calculation T des y is 1,2%. It means that an impeller designed by this method will have +1,2% of head input. It is acceptable for design practice.

Conclusion
The authors plan to apply the presented method of a loading factor performance calculation in parallel with the existing method in their future projects. In case of a positive result the new method will be incorporated into the Math models. This method is simple and does not require users' high experience and corrections with analogs.