Scaling Laws in Turbomachinery Design and Operational Optimization

Turbomachines are undoubtedly complex. While designing them from scratch has the best potential to maximize performance, it is not always the best route.

With the help of similarity concepts and the associated nondimensional parameters, the preliminary design of a new machine can be based on features of an existing machine, even one which may have been designed for a different fluid, other flow conditions, or a different rotational speed.

Let’s say we have a turbomachine, in this case, a one-stage Centrifugal Compressor. It was designed for a specific mass flow rate and rotational speed value to achieve a certain pressure ratio at the best efficiency possible.

Table 1. Specification for Baseline Compressor
Table 1. Specification for Baseline Compressor
Figure 1. Baseline Centrifugal Compressor
Figure 1. Baseline Centrifugal Compressor

One would be able to get the same performance at any value of mass flow rate or rotational speed required just by scaling the machine (scaling is the process of changing a geometry while preserving similarity between the prototype and the model). And not just this specific point, but the whole performance map could be moved either to lower or higher mass flow rates. This is possible thanks to the concept of “Similarity”.

Figure 2. Performance map of Baseline Compressor (Pressure Ratio vs Mass Flow Rate) at different Rotational Speeds
Figure 2. Performance map of Baseline Compressor (Pressure Ratio vs Mass Flow Rate) at different Rotational Speeds

To have similitude between different turbomachines 3 conditions must exist between them:

 

  • Geometric similarity
    • Model and prototype must have the same shape.
    • All linear dimensions of the model are related to the corresponding dimensions of the prototype by a constant scale factor.
  • Kinematic similarity
    • Velocities at corresponding points in the two flows are in the same direction and are related by a constant scale factor in magnitude.
    • Flow regimes must be the same.
  • Dynamic similarity
    • All forces (forces of inertia, friction, and pressure) in the model flow scale by a constant factor to corresponding forces in the prototype flow.

 

It is important to note that in practice when scaling a machine, some features tend to remain the same size in absolute dimensions, such as the surface roughness or the fillet radii, or small clearance gaps and leakage flow paths, and so these parameters may in relative terms become progressively larger when a compressor is scaled down in size. Mechanical or manufacturing limitations may also cause the relative thickness of blades to increase in very small machines.

There exist two common dimensionless parameters used in turbomachinery design, these are the work and flow coefficients respectively.

Formula 1

Where Δh is the enthalpy rise, ρ the fluid density, is the mass flow rate, N is the rotational speed, and D is a characteristic length, in this case, the outlet diameter of the impeller.

By using the Work and Flow Coefficients of the baseline machine, considering that we would need the same values for the scaled machine, it is possible to obtain a scale factor (SF) that would multiply all linear dimensions of the machine (geometric similarity) and that would modify the mass flow rate and rotational speed accordingly:

Formula 2

Considering the subscripts 1 and 2 for the Baseline and Scaled Compressors. This procedure achieves geometric and kinematic similarity but disregards dynamic similarity, so it is expected that performance would not be exactly the same. Nevertheless, for pressure rise, since the baseline and scaled compressors have similar velocity triangles, the Euler turbomachinery equation states that the total enthalpy rise is proportional to the square of the tip speed, this means that if the tip-speed Mach number of the machine is the same, the pressure rise would be maintained.

To illustrate this procedure two examples are shown; one with half the MFR which moves the performance map to the left, and the other with doubled MFR which moves the performance map to the right.

Table 2. Specification for Scaled Compressors
Table 2. Specification for Scaled Compressors
Figure 3. Performance Map of Baseline and Scaled Compressors
Figure 3. Performance Map of Baseline and Scaled Compressors

After scaling the machine and performing a meanline analysis in only a few “clicks” using the AxSTREAM software, we can see the differences in performance.

Figure 4 shows three sets of performance maps. The first (blue) is a theoretical map developed by the direct application of the scaling equations 3, 4, and 5, which based on the requirements of mass flow, speed, or dimension a scale factor is obtained. The second map (brown) shows the actual predicted dynamic effects using a 1D flow path analysis, using loss models, these models consider drawbacks like real incidence, calculates actual leakages, secondary flows, etc. This design is with the scaled clearances directly obtained by the scaled factor. The third map (pink), takes the impeller tip clearance back to the baseline value, leaving all the other dimensions scaled to account for the practical mechanical aspects of scaling.

Figure 4. PR vs MFR of Baseline and Scaled Compressors
Figure 4. PR vs MFR of Baseline and Scaled Compressors

In figure 5, the comparison of efficiency curves at rotational design speed is made. In these curves, the impact of maintaining the practical clearances required for manufacturing and efficiency can be seen. In this specific case at design point this impact is nearly 2%, but this could be a limitation in different cases where efficiency is a high priority.

Figure 5 Efficiency vs MFR of Baseline and Scaled Compressors
Figure 5 Efficiency vs MFR of Baseline and Scaled Compressors

These differences are mainly because the previous methodology disregards dynamic similarity (Reynolds Number, Mach Number, etc). It is also important to note that since we are scaling all linear dimensions of the machine by a scale factor, this also includes all kinds of clearances, so the scaled clearance could be so low that it loses its manufacturability (especially when scaling to smaller machines). In this case, even considering manufacturable clearances (0.3mm) which are higher than the scaled one (0.15mm), the differences are still negligibly far from the choke line.

The present discussion has examined the possibility of designing centrifugal compressors by the application of a geometrical scale factor to an existing baseline compressor. This is the inverse problem compared to the classic challenge of scaling test results to a different operating point. The method proposed relies on the application of scaling laws derived by the application of the similarity principles. The scaling methodology presented could be a practical and faster way if there is already an existing and validated machine to estimate the performance in different conditions. Table 3 shows, in summary, the advantages and disadvantages of the scaling methodology presented.

Table 3. Summary of advantages and disadvantages
Table 3. Summary of advantages and disadvantages

Are you leveraging scaling in your development strategy? Interested in learning more about the tools used in this article? Reach out to us at Info@softinway.com to arrange a time to chat!

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