*3.2. Results of the Geometric Model*

This paragraph shows the comparison between the geometric values determined by the model previously described and the actual geometry of the machines measured in the laboratory (Table 6). Furthermore, the difference between the two corresponding values is highlighted to be able to observe the reliability of the procedure used. Since this is a study based on statistical graphs, it is foreseeable that this difference will be minimal for some values, while it will be substantial for others, depending on the actual geometry of the machine, which adapts to the model itself. It is the designer's task to establish, and then evaluate, whether to accept this obtained gap, obviously depending on the sensitivity that the parameter itself has towards the result. This procedure was the basis for the verification of the model and its refinement. The curves that have been obtained from the entire module, which includes the fluid dynamic and geometric model obtained for the machine models under examination, have reported tolerable results for the purpose that had been set.



As previously specified, we notice significant differences for some values. These differences, however, are relative and calculated as a function of the specific measured value. This analysis was carried out on a sufficiently large number of models, in such a way as to allow their improvement. Obviously, this research, being still ongoing, has the potential to provide even better results.

## **4. Sensitivity Analysis**

The proposed model provides more accurate results when using the measured geometry of the machine. The influence of the single geometric parameters on the model results was analyzed to carry out a sensitivity analysis. The geometric parameters to which the model is most sensitive have been identified and are as follows:


A sensitivity analysis was carried out for each of these parameters in correspondence to percentage variations ±Δ/2% and ±Δ%. For both pump and turbine operation, the following parameters are determined:


The percentage error is then evaluated as follows:

$$\frac{(Q\_{\prime}H)\_{P/T}meas - (Q\_{\prime}H)\_{P/T}calc}{(Q\_{\prime}H)\_{P/T}meas} \tag{49}$$

Table 7 summarizes the influence of the geometric parameters considered on the values of flow rate and head at the BEP, in direct (*QP*, *HP*) and inverse (*QT*, *HT*) operation. The stars (\*) indicate a greater or lesser degree of sensitivity in proportion to their number.


**Table 7.** Sensitivity to geometric parameters.

For example, (\*) means that the model is not very sensitive to this parameter, for (\*\*\*\*) the model is very sensitive to this parameter.

As can be seen from Table 7, in determining the head at BEP in turbine operation (*HT*), the most critical parameters for what is necessary to have a value as accurate as possible are the diameter of the hub (*d1m*), the height of the inlet blades (*b*1), the suction diameter (*d*0), and the dimensions of the volute (*hv*, *b*). Regarding, instead, the determination of the flow rate at the BEP for the turbine operation (*QT*), the influence of the height of the inlet blades *b*<sup>1</sup> is always negligible. As an example, Tables 8–10 show the results relating to parameters *b* and *hv*, to which the model is most sensitive, for three representative pump models: P40-335, P40-250, and P80-220. During the analysis, the range of variation of each parameter is between ±20%, except for three parameters which showed a higher sensitivity (diameter of hub and suction band diameter of ±10%, width of the volute band of ±5%).


**Table 8.** Sensitivity analysis for pump 40-335.

**Table 9.** Sensitivity analysis for pump 40-250.


**Table 10.** Sensitivity analysis for pump 80-220.

