# Software for Monitoring the In-Service Efficiency of Hydraulic Pumps
## 2 Algorithm for Assessing Pump Efficiency
The value of the volumetric flow rate \(Q\) [m\({}^{3}\)/s] is obtained from the equality of the relationships (3) and (4) by solving the 2 nd-degree equation below. \[Q^{2}\{h_{2}-\frac{8}{g\pi^{2}}[\frac{1}{D_{d}^{4}}(\lambda_{d}\frac{l_{d}}{D_{ d}}+\zeta_{d}+1)+\frac{1}{D_{s}^{4}}(\lambda_{s}\frac{l_{s}}{D_{s}}+\zeta_{s}-1)]\}+h _{1}Q+[h_{0}-(z_{d}-z_{s}-\frac{(p_{d}-p_{s})}{g\rho})]=0 \tag{5}\]  
The 2 nd-degree equation for the volumetric flow rate in Equation (5) is written in the following manner:  
\[t_{2}Q^{2}+t_{1}Q+t_{0}=0 \tag{6}\]  
for which the following terms have been used:  
\[t_{0}=h_{0}-\Delta z-\frac{(p_{d}-p_{s})}{g\rho} \tag{7}\] \[t_{1}=h_{1}\] \[t_{2}=h_{2}-\frac{8}{g\pi^{2}}[\frac{1}{D_{d}^{4}}(\lambda_{d} \frac{l_{d}}{D_{d}}+\zeta_{d}+1)+\frac{1}{D_{s}^{4}}(\lambda_{s}\frac{l_{s}}{ D_{s}}+\zeta_{s}-1)]\]  
where \(\Delta z\) is defined by Equation (8):  
\[\Delta z=z_{d}-z_{s} \tag{8}\]  
The solutions of the 2 nd-degree polynomial equation presented in Equation (6) for flow rate are:  
\[H=\frac{(p_{d}-p_{s})}{g\rho}+\frac{8Q^{2}}{g\pi^{2}}[\frac{1}{D_{d}^{4}}- \frac{1}{D_{s}^{4}}]+(z_{d}-z_{s}) \tag{2}\]0  
Of the two solutions of Equation (6), the one with the positive value shall be retained (\(Q_{1}>0\)). Once the \(Q_{1}\) value is known, the coefficients \(h_{2}\), \(h_{1}\) and \(h_{0}\) for the \(H(Q)\) curve in Equation (4), the \(p_{3}\), \(p_{2}\), \(p_{1}\) and \(p_{0}\) coefficients for the \(P_{m}(Q)\) curve in Equation (10), respectively, \(e_{2}\), \(e_{1}\) and \(e_{0}\) coefficients for \(\eta_{P}(Q)\) curve in Equation (11) are used for calculating the pumping head value \(H(Q_{1})\), the mechanical power value \(P_{m}(Q_{1})\) and the pump efficiency \(\eta_{P}(Q_{1})\). These coefficients are determined by a fitting procedure, applied to the available data. \[H=\frac{(p_{d}-p_{s})}{g\rho}+\frac{8Q^{2}}{g\pi^{2}}[\frac{1}{D_{d}^{4}}- \frac{1}{D_{s}^{4}}]+(z_{d}-z_{s}) \tag{2}\]1  
\[H=\frac{(p_{d}-p_{s})}{g\rho}+\frac{8Q^{2}}{g\pi^{2}}[\frac{1}{D_{d}^{4}}- \frac{1}{D_{s}^{4}}]+(z_{d}-z_{s}) \tag{2}\]2  
The value of the volumetric flow rate corresponding to the BEP is obtained from Equation (12):  
\[H=\frac{(p_{d}-p_{s})}{g\rho}+\frac{8Q^{2}}{g\pi^{2}}[\frac{1}{D_{d}^{4}}- \frac{1}{D_{s}^{4}}]+(z_{d}-z_{s}) \tag{2}\]3  
The value is positive for all the cases \(Q(\eta_{P_{BEP}})>0\) because the coefficient \(e_{2}<0\) is associated with the efficiency curve of \(\eta_{P}\) pump, which is a vertex-up parabola. The \(\eta_{P_{BEP}}\) value is obtained by introducing the value \(Q(\eta_{P_{BEP}})\) in Equation (11). This value of the maximum efficiency \(\eta_{P_{BEP}}\) corresponding to each constant speed pump is used as a reference value to define the pump's operating regime. We notice that the maximum efficiency value \(\eta_{P_{BEP}}\) of the BEP is determined from the input values and is different for each pump. The pump's operating regime is determined by comparing the efficiency value \(\eta_{P}\) of the pump's duty point with the reference value corresponding to the pump's maximum efficiency \(\eta_{P_{BEP}}\). Three operating regimes are defined for a hydraulic pump using color vision deficiency codes, containing both text and color flag information as follows:  
* Normal operation (NO) from an efficiency point of view (\(\langle\)GREEN\(\rangle\)): \[H=\frac{(p_{d}-p_{s})}{g\rho}+\frac{8Q^{2}}{g\pi^{2}}[\frac{1}{D_{d}^{4}}- \frac{1}{D_{s}^{4}}]+(z_{d}-z_{s}) \tag{2}\]4 Normal operation (NO) means the pump is operating at the proper efficiency. That is, the costs of energy consumption for pumping are justified.