# Model-Based Assessment of Energy Efficiency in Industrial Pump Systems: A Case Study Approach
## 4 Energy-Efficiency Model of the Centrifugal Pump
### Analytical Modelling of the Pump Efficiency Characteristics for Variable-Speed Operation
#### 4.1.2 Analytical Enhancement of Pump Characteristics  
While interpolation within the range specified by the manufacturer is straightforward, extrapolation beyond this range can lead to significant deviations from the true values. To ensure that all OPs from the dataset are covered, we have extended the diagram of the pump properties specified by the manufacturer by manual extrapolation. In addition to the factory-specified characteristics shown in Figure 9a,b, the analysis contains manually created curves that cover lower efficiencies and extend the operating range for lower speeds. The isospeed curve for 560 rpm is shown below and to the left of  
Figure 9: \(Q\)-\(H\) diagrams of the pump characteristic curves (237 mm impeller diameter) with system operating points drawn in: (**a**) only the pump characteristic curves specified at the factory and (**b**) additionally modelled ones. the factory isoefficiency curves, together with the generated curves for 20%, 30%, 40%, and 50% efficiency. In order to model the 560 rpm isospeed curve, discrete points were first generated at specific flow rates of 0 m\({}^{3}\)/h, 25 m\({}^{3}\)/h, 50 m\({}^{3}\)/h, \(\ldots\) 150 m\({}^{3}\)/using the third-order polynomial approximation  
\[H(n)=a_{3}\cdot n^{3}+a_{2}\cdot n^{2}+a_{1}\cdot n\text{ for }Q=\text{ const.}, \tag{9}\]  
taking into account that the function \(H(n)\) intersects with the coordinate origin (0, 0). The points for determining the coefficients \(a_{n}\) were taken from isospeed curves specified by the manufacturer at a given flow rate. To illustrate this process, Figure 10a shows graphically the method for generating the polynomial function \(H(n)\) (9) for a pump with an impeller diameter of 237 mm at a flow rate of 0 m\({}^{3}\)/h. The markers (dots) with the corresponding head values represent the data extracted from the individual isospeed curves. The generated dashed approximation curve connects these four points and the coordinate origin. The equation of the curve with the values of the coefficients \(a_{n}\) is shown in the graph. The independent variable x stands for the rotational speed of the pump and the dependent variable y for the delivery head. This equation was then used to calculate the value of the delivery head at 560 rpm, which is marked as a square in Figure 10a. The isospeed curve for 560 rpm was then defined from the generated discrete points using a polynomial function  
\[H^{560}(Q)=b_{3}\cdot Q^{3}+b_{2}\cdot Q^{2}+b_{1}\cdot Q+b_{0}. \tag{10}\]  
In order to create additional isoefficiency curves in the low efficiency range, polynomial approximations of the efficiency  
\[\eta(Q)=c_{4}\cdot Q^{4}+c_{3}\cdot Q^{3}+c_{2}\cdot Q^{2}+c_{1}\cdot Q \tag{11}\]  
are generated from the factory-set discrete points on an individual isospeed curve. In Figure 10b, the discrete efficiency points for the aforementioned pump are represented by round markers. These points are connected to the coordinate origin by a dashed curve representing the functional relationship between efficiency and flow \(\eta(Q)\) (11), where the coefficients \(c_{n}\) were determined at 1450 rpm. This time, the independent variable x represents the flow rate, and the dependent variable y represents the efficiency. The flow rates at predefined efficiencies below 60% are then determined by specific evaluation of the polynomial. The calculated points are shown as squared markers in Figure 10b. Finally, the  
Figure 10: (**a**) Determination of the functional dependence of the head on the speed and (**b**) determination of the functional dependence of the efficiency on the flow rate (x—independent variable, y—dependent variable, round markers—factory provided data, square markers—calculated data). head values were calculated using the approximation polynomial for each factory-specified isospeed curve.