# Model-Based Assessment of Energy Efficiency in Industrial Pump Systems: A Case Study Approach
## 2 Case Study on Water-Cooling System: Operation, Data Collection, and Efficiency Modelling
### System Efficiency Model of Existing VSDP  
Based on the measured OPs at three preset valve openings (30%, 60%, and 100%), the dependence of the pressure difference \(\Delta p\) on the flow rate \(Q\) is modelled by a second-degree polynomial for each pre-set speed (1500 rpm, 1200 rpm, 900 rpm, and 600 rpm):  
\[\Delta p(Q)=p_{2}\cdot Q^{2}+p_{1}\cdot Q+p_{0}\text{ for }n=\text{const}. \tag{1}\]  
This results in four isospeed curves, which are shown in Figure 3a. Hydraulic power is calculated from the measured \(\Delta p\) and \(Q\) (Q in 1/s)  
\[P_{hyd}=Q\cdot\Delta p \tag{2}\]  
and the efficiency of the VSDP system \(\eta_{sys}\) is calculated as the ratio between the hydraulic power \(P_{hyd}\) and the directly measured electrical input power \(P_{el}\)  
\[\eta_{sys}=\frac{P_{hyd}}{P_{el}}. \tag{3}\]  
A third-degree polynomial is then determined for the VSDP efficiency as a function of flow [42], using three efficiency values at a constant, predetermined frequency  
\[\eta(Q)=e_{3}\cdot Q^{3}+e_{2}\cdot Q^{2}+e_{1}\cdot Q\text{ for }n=\text{const}. \tag{4}\]  
Furthermore, this fit must intersect the origin, as the VSDP efficiency without flow is zero regardless of the speed.  
The polynomial efficiency fit (4) was evaluated for a vector of flow values in steps of 1 m\({}^{3}\)/h, e.g., from 0 m\({}^{3}\)/h to a suitable upper limit depending on the respective isospeed. This additional set of points is shown as red dots in Figure 4. Then, a fourth-degree surface polynomial model for the flow and third degree for the head (\(H=\Delta p/\rho g\)) was fitted to the entire dataset (see the contour plot in Figure 4). The details of the fitting process are presented later in Section 3.4, as all efficiency maps in this paper have the same polynomial model structure.  
Figure 3: (**a**) Measured pressure differences and flows at predefined OPs (the hydraulic power \(P_{hyd}\) in kW is indicated in the labels), cloud of annual OPs, and modelled curves of pressure dependence on flow rate and (**b**) measured electrical power supplied at predefined OPs (the system efficiency is indicated in the labels) at four speeds and three valve openings.  
In this way, we have obtained a detailed model that can predict the efficiency of the system for all OPs in the annual dataset (grey dots in Figure 4). A closer look shows that the average and maximum electrical power is only 3.50 kW and 17.62 kW, respectively. Consequently, the maximum efficiency of the individual OPs in the dataset does not exceed 55.74%, with an average efficiency of only 34.22%. A comparison of these energy-efficiency indicators with the nominal VSDP data (Table 1) suggests that VSDP1 and VSDP2 are considerably oversized, resulting in an overall efficiency performance that is unacceptably low by current standards (Figure 4).  
However, the low efficiency is also a consequence of the use of outdated technology, as the existing motors only correspond to the efficiency class IE1. Furthermore, the almost-w twenty-year-old pumps are inferior to the modern, CFD-optimised pumps available today and are likely to suffer performance losses due to impeller wear and fouling [43,44]. Therefore, it is important to accurately estimate the potential energy savings before embarking on VSDP revitalisation to ensure that the replacement is economically justified. It is not sufficient to simply apply general estimates such as those of [PERSON] and [PERSON] [7], which assume savings of between 20% and 30%. The reason for this is that only the VSDPs can be replaced, while optimising the chilled water supply is not feasible due to the load dynamics dictated by the consumers in the industrial process.