# Software for Monitoring the In-Service Efficiency of Hydraulic Pumps
## 2 Algorithm for Assessing Pump Efficiency  
A robust algorithm is developed to monitor the efficiency of in-service pumps. The \(\eta_{P}\) pump efficiency equation is:  
\[\eta_{P}=\frac{P_{h}}{P_{m}}=\frac{g\rho QH}{P_{m}} \tag{1}\]  
where \(P_{h}=g\rho QH\) [W] is the hydraulic power, representing the power transferred to the liquid, \(Q\) [m\({}^{3}\)/s] is the pump discharge (volumetric flow rate), \(H\) [m] represents the pumping head, \(\rho\) [kg/m\({}^{3}\)] is the density of the working liquid (water), \(g\) [m/s\({}^{2}\)] is the gravity acceleration, and \(P_{m}\) [W] represents the mechanical power at the pump shaft. The pumping head \(H\) is presented in Equation (2). \[H=\frac{(p_{d}-p_{s})}{g\rho}+\frac{8Q^{2}}{g\pi^{2}}[\frac{1}{D_{d}^{4}}- \frac{1}{D_{s}^{4}}]+(z_{d}-z_{s}) \tag{2}\]  
This equation is valid if the instruments are installed on the pump flanges. In practical terms, the instruments cannot be installed on the pump flanges in most cases. That is why the algorithm will be considered with a generalized form of the equation of the pumping head \(H\) given in Equation (3). This equation includes the distributed and local hydraulic losses due to the positioning of the instruments in relationship to the pump flanges and the static pressures measured on the suction and discharge lines. \[H=\frac{(p_{d}-p_{s})}{g\rho}+\frac{8Q^{2}}{g\pi^{2}}[\frac{1}{D_{d}^{4}}( \lambda_{d}\frac{l_{d}}{D_{d}}+\zeta_{d}+1)+\frac{1}{D_{s}^{4}}(\lambda_{s} \frac{l_{s}}{D_{s}}+\zeta_{s}-1)]+(z_{d}-z_{s}) \tag{3}\]  
where \(p_{s}\) and \(p_{d}\) are the static pressures measured at the pump's suction and discharge, \(l_{s}\), \(D_{s}\), \(\lambda_{s}\), \(\zeta_{s}\) are the length, diameter, coefficient of distributed hydraulic loss, or Darcy's coefficient, and the local hydraulic loss coefficient, in connection with the position of the instrument installed on the suction line compared to the pump flange, \(l_{d}\), \(D_{d}\), \(\lambda_{d}\), \(\zeta_{d}\) are the length, diameter, coefficient of distributed hydraulic loss, or the [PERSON]'s coefficient and the local hydraulic loss coefficient, in connection with the position of the instrument installed on the discharge line compared to the pump flange, and \(z_{d}-z_{s}\) is the quota difference between the position of the instrument installed on the discharge line and the suction line. The generalized equation of the pumping head (3) is reduced to (2) when the instruments are installed on the pump flanges, because the distributed hydraulic loss coefficients \(\lambda_{s}=\lambda_{d}=0\) and the local hydraulic loss coefficients \(\zeta_{s}=\zeta_{d}=0\). It is important to mention that for the present algorithm, we need only the readings of the static suction pressure \(p_{s}\) and discharge pressure \(p_{d}\) to determine the efficiency \(\eta_{P}\) of the pump. The remaining values needed for the calculus are **determined** (e.g., the volumetric flow rate that has passed through the pump \(Q\)--a value difficult to determine in industrial applications), or **assessed** (the mechanical power by the pump \(P_{m}\)). The pumping head is defined as the specific energy difference of the fluid between the inlet section and the outlet section of the pump. The pumping head is given by the \(H(Q)\) curve from the pump supplier's catalog or by the measurements performed in situ. As a result, the \(H(Q)\) curve can be expressed in the form of a 2 nd-degree polynomial for a radial centrifugal pump. The coefficients \(h_{2}\), \(h_{1}\) and \(h_{0}\) are determined by a fitting procedure using the available data. \[H(Q)=h_{2}Q^{2}+h_{1}Q+h_{0} \tag{4}\]  
The algorithm for assessing the efficiency of hydraulic pumps is based on knowing the pump's catalog features (or the pump's curves, determined experimentally) and the determination of geometric data in situ, which will be introduced as parameters of the algorithm.