Physical Modelling of the Set of Performance Curves for Radial Centrifugal Pumps to Determine the Flow Rate

Abstract

To depict the pump power characteristics of radial centrifugal pumps, a physical model was developed. The model relies on established empirical equations. To parameterize the model for specific pumps, physically interpretable tuning factors were integrated. The tuning factors are identified by using the Levenberg–Marquardt method, which was applied to the characteristic curve at a constant speed. A cross-validation of the physical model highlighted the advantage of representing the set of performance curves with less deviation compared to approximation functions. Calculating the entire set of performance curves requires only one pump characteristic curve at a constant speed. Therefore, only a single measurement is necessary. Furthermore, the physical model can be used to calculate the changes in the set of performance curves due to prewhirl. This increases the accuracy of flow rate calculations when prewhirl occurs.

1. Introduction

The flow rate of centrifugal pumps plays an important role as a command variable for controlling pumping systems. Pumps are used to generate a flow rate, which needs to be adapted to the requirements of the specific application using one of various possible control methods. In particular, the flow rate is considered the main control variable in many applications (e.g., in the process industry) [1,2]. The flow rate ($Q$) can be determined either by direct measurements or by using numerical algorithms. Various algorithms to determine the flow rate are known in the literature. Those algorithms require the characteristic curve of the pump, which is typically published by the manufactures at a constant pump speed. The application of speed-controlled pumps has increased significantly. Therefore, the set of performance curves is essential for determining the flow rate. Depending on the kind of algorithm, the measured values, and their approximation functions, either of the delivery head $H\left(Q\right)$ or the power of the pump $P_K\left(Q\right)$ are required. A frequently employed algorithm involves measuring either the pump head ($H$) or power ($P_K$) then determining the flow rate by applying the inverse of the $H\left(Q\right)$ or $P_K\left(Q\right)$ approximation functions. J. Tamminen et al., utilized this algorithm for both pumps and fans [3]. They amalgamated flow rate calculations from both approximation functions to estimate the overall flow rate. S. Hammo & J. Viholainen have also investigated this algorithm [4]. S. Hammo & J. Viholainen, along with J. Tamminen et al., computed flow rates across various speeds. Therefore, they apply the following affinity laws: $Q\propto n$, $H\propto n^2$, and $P_K\propto n^3$. The relative deviations between the measured and calculated flow rates in [4] range from 0% to 4%. S. Leonow (2015) provided an overview of the most common algorithms to determine the flow rate [2]. These algorithms can be divided into static and dynamic methods. S. Leonow & M. Mönnigmann (2013) introduced the boundary curve method, wherein the stator current of the drive motor is measured as a function of the flow rate [5]. Two boundary curves are obtained by measuring the stator current characteristic diagram at various speeds for $Q=0$ and $Q=\mathrm{m}\mathrm{a}\mathrm{x}\left(Q\right)$. This algorithm closely resembles the approach in D. Kernan et al.’s patent, where they also utilized a boundary curve measured across different speeds [6]. Both algorithms require the manufacturer’s published performance curve as the initial data. Even more recent publications, such as that of M. Rakibuzzaman et al., or A. Shankar et al., require the manufacturer’s performance curve [7,8]. A. Shankar et al., adjusted the pump characteristics at variable speeds by using the affinity laws [8]. They achieved an average error of about 2% in estimating the flow rate. S. Pöyhönen et al.’s algorithm employs data about the flow rate and head at the best efficiency point (BEP) of the pump ($Q^{\ast }, H^{\ast }$) during nominal speed ($n_N$) to compute the pump characteristics [9]. Hence, the specific speed ($n_q$), as determined by Equation (1), is required.

$$n_q=n_N\frac{\sqrt{Q_{opt}}}{{H_{opt}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}$$

(1)

Through using the generic pump characteristic curves described by J. Gülich and A. Stepanoff [10,11], one can calculate the pump curves for specific speeds. Subsequently, the flow rate estimation is achieved through interpolation and polynomial approximation utilizing the $H\left(Q\right)$ or $P_K\left(Q\right)$ characteristic curve. The flow rate can be calculated with an error of 7–15% through using this approach. Variable speeds also involve the utilization of the affinity laws. R. Susan-Resiga et al., showed that substantial deviations arise when applying the affinity laws for calculations, particularly with a significant reduction in rotational speed [12]. They enhanced the approximation functions, leading to a more accurate reproduction of the data. Y. Wu et al., confirmed that the affinity laws lead to an increasing error in the characteristic curve when the speed is greatly reduced [13]. They measured the pump characteristic curves at different speeds and subsequently created plots that relate power to discrete flow rates as a function of the pump speed. They interpolated between the discrete curves to estimate the flow rate. They contrasted the results of their flow rate estimation method using interpolation with those obtained from a back-propagation neural network (BPNN) and the application of affinity laws to the $Q\left(P_K\right)$-curve. They achieved an average deviation of 2%. In addition to the previously mentioned polynomial flow rate calculation approaches, M. Chai et al., presented a method that uses Gaussian Process Regression (GPR) [14]. This method stores the flow rate within kernels as a function of both the pump speed and the position of the discharge control valve. Linked to a mechanical model, other parameters such as pump efficiency and pump head can be determined. This approach achieves an average deviation in flow rate estimation of less than 2%, but again, additional measurements are required.

As shown, the deviations of the affinity laws can be improved by modifying approximation functions, the boundary curve method, or the GPR approach, which requires additional measurements. Additionally, the pump performance curves might change when centrifugal pumps are used within pumping systems. Deliberate or unintended design modifications to the suction pipe, such as deviations from the standard, can profoundly impact the pump curve. M. Roth shows that, depending on the installation situation of flange flap valves and elbows, losses in pumping head up to 2.8% and at shaft power values up to 1.7% can be expected at the best efficiency point (BEP) of the pump [15]. M. Roth named prewhirl as the main reason for such changes [15]. She showed that prewhirl has different impacts depending on its direction. The effect of prewhirl on the performance curves of centrifugal pumps with medium to high specific speeds has already been investigated in several studies. These studies mainly focus on prewhirl control, which is a common control method for these types of pumps [10]. The review of M. Liu et al., cites various studies dealing with prewhirl control with respect to centrifugal pumps, in which the influence of the prewhirl on pump curves was studied experimentally as well as numerically [16]. Here, it is shown that the characteristic curves of pumps can also be influenced by a prewhirl guide valve at medium specific speeds. L. Tan et al., focused on the performance optimization of a pump with a specific speed of $n_q=38 {\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$ at partial load operation [17]. These authors suggested an appropriate design of the prewhirl guide valve for adjusting the flow at the impeller inlet. They even showed that the prewhirl may have a positive effect on the flow when the pump is working under partial load operating conditions. W. Wu et al., numerically studied the influence of prewhirl on the inlet flow at an impeller of an axial pump operating at different flow rates [18]. H. Hou et al., in their numerical investigations, demonstrated that the BEP of the pump shifts depend on the prewhirl direction [19]. X. Zibin analysed the behaviour of the characteristic curve under the influence of prewhirl [20]. He showed that the prewhirl direction affects the specific work. Several publications have shown that the impact of prewhirl on the specific work increases with the specific speed of the pump [10,21,22]. I. Schröder also examined a pump with a lower specific speed in terms of prewhirl impact on the pump curves and was able to show that a significant influence can also be expected for such pumps [21].

There are two aspects of the published studies presented here that require further investigation. The accuracy of flow rate determination for variable speed pumps can only be improved by additional measurements. This requires extra effort. In addition, studies on the influence of prewhirl on the pump characteristic curves and the prewhirl guide valves clearly show that even pumps with low and medium specific speeds lead to changes in the characteristic curves. Changes in the pump curves due to installation conditions are not considered in all empirical models for calculating the flow rate. In the next section of this paper, the development of a physical model is described for the performance curves of radial centrifugal pumps. This can be customized for specific pumps by employing the manufacturer’s performance curves. This eliminates the need for additional measurements. The physical interpretability also enables the consideration of curve alterations induced by the pump system.

2. Derivation of the Physical Model of the Set of Performance Curves

2.1. Derivation of the Physical Model Equations

The functional relationships of the pump performance curve ($P_K$) are represented by a physical model expressed as $P_K:{\mathbb{R}}^2\to \mathbb{R}$. The physical model is adapted to a specific pump with the help of a tuning parameter vector ($t$). The derivation of the physical model starts with the calculation of the specific impeller work ($\tilde{Y}$) using Equation (2), known as the Euler Equation for turbomachinery. The left term represents the whirl at the impeller blade outlet, consisting of the circumferential speed ($u_2$) and the circumferential component of the absolute speed ($c_{u2}$). Similarly, the right term describes the whirl at the inlet to the impeller blade with the circumferential speed ($u_1$) and the circumferential component of the absolute speed ($c_{u1}$).

$$\tilde{Y}≔c_{u2}{u}_2-c_{u1}{u}_1$$

(2)

The circumferential speed at position $i$ of the pump impeller ($u_i$) is generally determined from Equation (3) if the pump speed $n$ and the diameter $d_i$ are known.

$$u_i≔\pi n d_i$$

(3)

In Figure 1a, a meridian view of an impeller for a radial centrifugal pump is presented. Index 1 at the parameters denotes the inlet to the impeller blade. Index 2 indicates the exit of the impeller blade. Figure 1b illustrates the velocity triangle, showing the velocity components and angles at each location within the pump impeller. The absolute velocity ($c$) splits into its circumferential velocity ($u$) and relative velocity ($w$) components. The absolute velocity can be further decomposed into a circumferential component ($c_u$) and a meridional component ($c_m$). The meridian portion refers to the portion of the absolute velocity that follows the flow along the meridian section. Similarly, the circumferential component refers to the portion of the absolute velocity magnitude that is parallel to the circumferential velocity direction. The relative velocity describes the flow velocity relative to the rotating pump impeller. The angle $\alpha$ is the angle between $u$ and $c$ and characterizes the orientation of the absolute flow to the pump impeller. Angle $\beta$ quantifies the blade angle.

After several transformations, Equation (2) can be expressed as follows, assuming an incident flow without prewhirl ($c_{u1}=0$):

$$\tilde{Y}=\left(\gamma -\frac{\tilde{Q}}{\pi {\left(b d u\right)}_2}\cot \left({\beta }_{2B}\right)\right){u_2}^2.$$

(4)

The inclusion of the power factor ($\gamma$) considers the variation between the actual flow and the flow congruent with the blades. When considering a flow congruent with the blades, it is assumed that the pump impeller has an infinite number of blades (denoted by the index $\infty$) [10]. The flow then precisely follows the shape of the blade contour. The power factor is defined by Equation (5) [10].

$$\gamma ≔1-\frac{c_{u2\infty }-c_{u2}}{u_2}$$

(5)

Considering the power factor, the blade angle at the outlet (${\beta }_{2B}$) can be used in Equation (4). To estimate the power factor $\gamma \in \left[0;1\right],$ empirical equations such as in [10,11] can be used. In Equation (4), $\tilde{Q}$ is the flow rate delivered by the impeller. This exceeds the pump flow ($Q$) due to the presence of the gap flow ($Q_{Sp}$):

$$\tilde{Q}≔Q+Q_{Sp}.$$

(6)

This is due to the backflow from the blade outlet to the blade inlet via the side spaces between the pump impeller and pump casing. If we assume that $c_{u1}\ne 0$, Equation (4) can be expanded to give the following:

$$\tilde{Y}=\left(\gamma -\frac{\tilde{Q}}{\pi {\left(b d u\right)}_2} \left(\cot \left({\beta }_{2B}\right)+\frac{4 b_2{d}_{1m}}{d_{1a}^2-d_{1i}^2} \mathrm{t}\mathrm{a}\mathrm{n}(∆{\alpha }_1)\right)\right){u_2}^2.$$

(7)

Equation (7) outlines the inflow conditions when the prewhirl is engaged, with the prewhirl angle (${∆\alpha }_1$) serving as the defining parameter, as shown in Figure 1b. The mean diameter at the blade inlet ($d_{1m}$) is calculated from Equation (8).

$$d_{1m}≔\sqrt{\frac{1}{2}\left(d_{1a}^2+d_{1i}^2\right)}$$

(8)

$\tilde{Y}$ does not include any losses. The flow losses due to hydraulic efficiency (${\eta }_h$) can be considered using Equation (9). The parameter $Y$ signifies the pump delivery work, which can be transformed into the pump head in accordance with Equation (10), where $g$ denotes the acceleration due to gravity ($g=9.81\frac{\mathrm{m}}{\mathrm{s}²}$).

$$Y≔\widetilde{Y }{\eta }_h$$

(9)

$$H≔\frac{Y}{g}$$

(10)

Apart from hydraulic losses, the pump also incurs mechanical losses. The mechanical power ($P_m$) includes the mechanical losses that arise from components like bearings or the mechanical seal. Moreover, recirculation could potentially manifest within the pump. The recirculation efficiency (${\eta }_{Rez}$) factors in any rise in pump power that could arise from the presence of recirculation zones within the pump impeller. The efficiency ${\eta }_{Rez}$ is calculated using Equation (11). In addition, the previously discussed gap loss can be accounted for by the volumetric efficiency (${\eta }_V$), as shown in Equation (12).

$${\eta }_{Rez}≔1-\frac{P_{Rez}}{P_K}$$

(11)

$${\eta }_V≔\frac{Q}{\tilde{Q}}$$

(12)

In addition to the losses mentioned earlier, the pump also requires power for the impeller friction ($P_{RR}$). This is caused by friction between the impeller discs and the pump housing.

The pump efficiencies, power factor, impeller friction power, mechanical power, and geometric parameters are indispensable for the $P_K(Q,n)$ calculation. The power factor and geometric parameters can both be extracted from design data. Due to the difficulty of determining efficiencies, tuning parameters ($t$) are used to adjust efficiencies for a specific pump. To achieve this, knowledge of the pump’s discrete measurement points ($Q_{exp,i}, H_{exp,i}, P_{K,exp,i}, n_{exp,i}$) is essential. These points can often be extracted from the pump manufacturer’s data booklets. Recognizing that the power factor calculation is an approximation, it is necessary to introduce the tuning parameter $t_0$. This estimates the actual reduced power factor by summing it with the calculated reduced power factor. For the gap current ($Q_{Sp}$), an empirical equation for calculation is given in [10]. In this equation, the relation $Q_{Sp}\propto \sqrt{H}$ is specified. When calculating the actual gap current for the pump under certain measurements, it is assumed that

$$Q_{Sp}=t_2 \sqrt{H}.$$

(13)

Since the tuning factors can be identified using the manufacturer’s measurements, the prewhirl term with the specification of ${∆\alpha }_1$ in Equation (7) is omitted. Therefore, the specific impeller work can be parameterized with the tuning parameters $t_0$ and $t_2$ to the specific pump using Equation (14).

$$\tilde{Y}=\left(\gamma +t_0-\frac{Q+t_2\sqrt{H}}{\pi {\left(b d u\right)}_2}\cot \left({\beta }_{2B}\right)\right)u_2^2$$

(14)

The wheel friction power is replaced by the parameter $t_1$ ($t_1\stackrel{\wedge}{=}{P}_{RR}$). The power of the recirculation ($P_{Rez}$) diminishes linearly from $Q=0$ to $Q_{Rez,min}$ and becomes negligible at a flow rate ($Q_{Rez,ic}$) [10]. S. Leonow (2015) formulated the model for $Q_{Rez}$ using an PT1-Controller [2]. This approach can effectively emulate the behaviour, but it is a discontinuous approximation. The sigmoid function $s\left(x\right)$ (Equation (15)) was inserted into the physical model to model the recirculation efficiency.

$$s\left(x\right)≔\frac{1}{1+e^x}$$

(15)

The sigmoid function is defined as $s\left(x\right)\in [0;1]$, which makes it appropriate for modelling purposes. The sigmoid function is modified by the tuning parameters $t_3$ and $t_4$. For the variable $x$, $x=Q/Q^{\ast }$ applies. Equation (16) represents the functional approach for the recirculation efficiency.

$${\eta }_{Rez}=\frac{1+t_4}{1+e^{-t_3\frac{Q}{Q^{\ast }}}}-t_{4.}$$

(16)

The physical model, incorporating the tuning factors $t_i\in t, 0\le i\le 4$, is expressed through the following system of equations:

$$\begin{gathered} u_2=\pi n d_2 \\ \tilde{Y}=\left(\gamma +t_0-\frac{Q+t_2\sqrt{H}}{\pi {\left(b d u\right)}_2}\cot \left({\beta }_{2B}\right)\right)u_2^2 \\ {P}_K=\tilde{Y} \rho (Q+t_2 \sqrt{H})+t_1+P_m+P_{K,exp}\left(1-\left(\frac{1+t_4}{1+e^{-t_3\frac{Q}{Q^{\ast }}}}-t_4\right)\right). \end{gathered}$$

(17)

For the variable speed calculation, it should also be noted that $t_1\propto n^3$, $t_2\propto n$, and $Q^{\ast }\propto n$. It is assumed that the mechanical power loss $P_m$ is known.

2.2. Identification of the Tuning Parameters

The tuning parameters can be determined utilizing the Levenberg–Marquardt method. This method is applied to address multidimensional, non-linear, and unconstrained optimization problems. The solution requires a function $\Phi \left(t\right)$ that is twice continuously differentiable [23]:

$$\Phi \left(t\right)≔\frac{1}{2}‖{z\left(t\right)‖}^2.$$

(18)

The objective of the optimization problem is to find an optimal vector $t_{opt}$ that minimizes $\Phi \left(t\right)$, thereby representing the optimal tuning parameters for the specific pump [23]:

$$t_{opt}≔\arg \underset{t\in \mathbb{R}}{\min }\Phi \left(t\right).$$

(19)

The vector $z\left(t\right)$ defines the difference between the pump power calculated using the physical model (Equation (20)) and the measurements taken at the specific pump’s data points. In the following, $z\left(t\right)$ is designated as the objective function:

$$z\left(t\right)≔{\left(P_{K,1}-P_{K,exp,1},P_{K,2}-P_{K,exp,2},\cdots ,P_{K,n}-P_{K,exp,n}\right)}^T$$

(20)

The Levenberg–Marquardt method is iterative in nature, meaning a new vector $t^{k+1}$ is calculated for each iteration step $k$ by Equation (21) [24].

$$t^{k+1}≔t^k-{\left({\left(J^k\right)}^T{J}^k+\mu I\right)}^{-1}{\left(J^k\right)}^T{z\left(t\right)}^k$$

(21)

$J^k$ denotes the Jacobian matrix at time step $k$ and $I$ the identity matrix. The Jacobian matrix is a $n\times m$ matrix that represents the total differential of $z\left(t\right)$ at $n$ measurement points and $m$ tuning parameters (Equation (22)).

$$J≔Dz\left(t\right)$$

(22)

In the Levenberg–Marquardt method, the objective is not only $\Phi \left(t\right)\to min!$ but also ${‖t^{k+1}-t^k‖}^2\to min!$. The weighting of the two objectives is determined by the parameter $\mu \in [0,\mathrm{\infty }]$. The matrix ${\left({\left(J^k\right)}^T{J}^k+\mu I\right)}^{-1}$ is always positive definite and thus not singular [24]. Physical interpretability must be taken into consideration when identifying the tuning parameters. To achieve this, it is necessary to consider inequality constraints. However, the Levenberg–Marquardt method operates without any constraints. For this reason, $P_K\left(t\right)$ estimated by the physical model in $z\left(t\right)$ was augmented with a logarithmic barrier term to $b\left(t\right)$ [25]. The definition of $b\left(t\right)$ is as follows:

$$b\left(t\right)≔P_K\left(t\right)-\lambda \sum LN\left(t_i-r_{i,j}\right).$$

(23)

Inequality restrictions are taken into account via $r_{i,j}\in R$ in the right-hand term of $b\left(t\right)$. The factor $\lambda$ controls how close the tuning factor may be taken to the interval limits. During the identification process, a value of $\lambda =1e^{-6}$ was consistently assumed. Algorithm 1 outlines the procedure for identifying the tuning parameters using the Levenberg–Marquardt method with the involvement of barrier terms. The method is based on that described in [24].

Algorithm 1: Identification of the tuning parameters

$N_{max}≔1000$  $t^1≔\left[\begin{array}{ccc}0& P_{K,exp}\left(Q^{\ast }\right)·0.02& \begin{array}{cc}9e-5& \begin{array}{cc}6& 0\end{array}\end{array}\end{array} \right]$  $\mu ≔0$  ${\beta }_0≔0.5 \wedge {\beta }_1≔0.9$  $\epsilon ≔0$

FOR $k=1:N_{max}$          WHILE $\epsilon <{\beta }_0$                    Calculation of ${b\left(t^k\right)}^k$                    Calculation of $J^k$                    $z\left(t^k\right)≔P_{K,exp}-{b\left(t^k\right)}^k$                    Calculation of $t^{k+1}$                    $\epsilon ≔\frac{{‖z\left(t^k\right)‖}^2-‖z\left(t^{k+1}\right)‖²}{{‖z\left(t^k\right)‖}^2-{‖z\left(t^k\right)-J^k\left(t^{k+1}-t^k\right)‖}^2}$                    IF $\epsilon <{\beta }_0$                             $\mu ≔2\mu$                    END         END         IF $\epsilon >{\beta }_1$                    $\mu ≔\frac{\mu }{2}$         END         IF $‖P_{K,exp}-P_K‖<60$[W]                    BREAK         END END

2.3. Algorithm for Calculating of the Flow Rate by Using the Physical Model

To evaluate the utility of the physical model for determining the flow rate, an algorithm was devised. The calculation of the flow rate ($Q_{calc}$) is treated as a parameter within an optimization problem. To optimize $Q_{calc}$, the one-dimensional Newton method [25] can be used to fit it to the measured pump power. The objective function is formed using the physical model. Two low-pass filters can be used to reduce the fluctuations of the input signals as well as the calculated flow rate. In addition to the parameters for the calculation of the physical model, the speed ($n_{exp}$) and the power consumption ($P_{K,exp}$) are required for the algorithm. As suggested in [5], these quantities can be taken from a frequency converter. In cases where the measured pump head is unavailable, an approximate function ($H_{calc}\left(Q_{calc}\right)$) can be assumed. Algorithm 2 describes the flow rate calculation algorithm. The damping characteristics of the low-pass filters can be modified by adjusting the damping coefficients ($k_1, k_2$). The BEP is taken as the initial operating point. During the calculation of the Newton step ($∆Q$), the derivative of the physical model with respect to the flow rate (${P_K}^{\prime }$) is incorporated into the algorithm. In the final stage, the Newtonian step is combined with the calculated flow rate, yielding the updated conveying flow rate for the present time step.

Algorithm 2: Flow rate determination

$a_1≔\frac{1}{k_1} \wedge a_2≔\frac{1}{k_2}$ $n_{TP}≔n_N$ $H_{TP}≔H^{\ast }$ $P_{K,TP}≔{P_K}^{\ast }$ $Q_{calc}≔Q^{\ast }$

WHILE          $n_{TP}≔n_{exp}{a}_1+n_{TP}(1-a_1)$          $H_{TP}≔H_{exp}{a}_1+H_{TP}(1-a_1)$          $P_{K,TP}≔P_{K,exp}{a}_1+P_{K,TP}(1-a_1)$          Calculation of $P_K$          Calculation of ${P_K}^{\prime }≔\frac{d{P}_K}{dQ}$          $∆Q≔\frac{P_{K,TP}-P_K}{{P_K}^{\prime }}$          $Q_{calc}≔\left(Q_{calc}+∆Q\right)a_2+Q_{calc}(1-a_2)$ LOOP

2.4. Experimental Setup to Measure the Set of Performance Curves by Varying the Prewhirl Angles

To validate the physical model, measurements were conducted on two radial centrifugal pumps. The pumps are denoted as test objects (PO) in the following context. They exhibited varying specific speeds. R. Susan-Resiga et al. [12] validated their approach based on data from J. Delgado et al. [26]. These data can be obtained by measuring the set of characteristic curves of a centrifugal pump subjected to a significant speed reduction. Incorporating inequality constraints into the parameter identification process for tuning parameters enables the development of a physical model, even when assuming pump parameters. This model’s validation will be carried out using the data from [26].

Table 1. Parameters of the two measured pumps (PO001 and PO002) and the assumed parameters of the pump in the work of [26] (PO003) for the initialization of the physical model.

Parameter	PO001	PO002	PO003
specific speed ($n_q$)	$43{ \mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$	$54 {\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$	$24 {\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
outer diameter inlet ($d_{1a}$)	$124 \mathrm{m}\mathrm{m}$	$122 \mathrm{m}\mathrm{m}$	$65 \mathrm{m}\mathrm{m}$
inner diameter inlet ($d_{1i}$)	$46 \mathrm{m}\mathrm{m}$	$47 \mathrm{m}\mathrm{m}$	$23 \mathrm{m}\mathrm{m}$
diameter outlet ($d_2$)	$210 \mathrm{m}\mathrm{m}$	$180 \mathrm{m}\mathrm{m}$	$158 \mathrm{m}\mathrm{m}$
width impeller outlet ($b_2$)	$22 \mathrm{m}\mathrm{m}$	$33 \mathrm{m}\mathrm{m}$	$10 \mathrm{m}\mathrm{m}$
number of blades ($z_L$)	$7$	$6$	$6$
blade angle outlet (${\beta }_{2B}$)	$32°$	$35°$	$30°$
mechanical power at $n_N$ ($P_m$)	$330 \mathrm{W}$	$70 \mathrm{W}$	$240 \mathrm{W}$
flow rate at BEP ($Q^{\ast }$)	$98.6\frac{{\mathrm{m}}^3}{\mathrm{h}}$	$113.2\frac{{\mathrm{m}}^3}{\mathrm{h}}$	$39\frac{{\mathrm{m}}^3}{\mathrm{h}}$
head at BEP ($H^{\ast }$)	$10 \mathrm{m}$	$8.2 \mathrm{m}$	$30 \mathrm{m}$
pump power at BEP (${P_K}^{\ast }$)	$3143 \mathrm{W}$	$3375 \mathrm{W}$	$4586 \mathrm{W}$
nominal speed ($n_N$)	$1475 {\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$	$1475 {\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$	$2920 {\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
power factor ($\gamma$)	$0.798$	$0.767$	$0.782$

shows the parameters of the three PO. PO003 indicates the pump measured at [26].

As explained in the introductory section, centrifugal pump characteristics can be affected by prewhirl. The PO001 and PO002 characteristics with prewhirl were measured by using prewhirl bodies. This approach was proposed by I. Schröder [21]. The experimental setup for the measurements is shown in Figure 2. A torque-measuring shaft is installed between the pumps and their motor. Through this, the pump power can be measured. The pump head can be determined by measuring the differential pressure across the pump (${∆p}_P$) in accordance with Equation (24). The differential pressure is determined using a differential pressure gauge. The velocities of the flow on both the pressure and suction sides are computed using the measured flow rate ($Q_{exp}$) and the cross-sectional areas of the pipes.

A magnetic inductive flow meter (MID) is used to measure $Q_{exp}$. The flow rate can be adjusted by regulating a control valve. Moreover, the difference in geodetic height ($∆h$) between the differential pressure gauge and the pump’s centreline is accounted for. As depicted in Figure 2, the pressure loss across the prewhirl bodies (${∆H}_{pb}$) is factored into the head calculation. Both the pressure losses and the effective prewhirl angles following the existence of the prewhirl bodies were determined through a CFD simulation.

$$H_{exp}≔\frac{{∆p}_P}{g\rho }+\frac{v_p^2-v_s^2}{2g}+∆h+{∆H}_{pb}$$

(24)

The prewhirl angle was altered within a range of ${∆\alpha }_1\in [-34°;34°]$. Additionally, the pump’s speed can be modified. It varies within the interval $n\in [1180 {\mathrm{m}\mathrm{i}\mathrm{n}}^{-1};1475 {\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}]$ by a variable frequency converter (VFD).

3. Analysis and Validation of the Physical Model

3.1. Parameter Identification and Regression Analysis

The tuning parameters of the three test objects were identified using Algorithm 1. The restriction matrix for the parameter identification is defined by the following:

$$R≔\left[\begin{array}{cc}-0.2& 0.2\\ P_{K,exp}\left(Q^{\ast }\right)·0.005& P_{K,exp}\left(Q^{\ast }\right)·0.2\\ 1e-6& 1e-2\\ 5& 15\\ -0.5& 15\end{array}\right].$$

(25)

C²-splines were fitted to the original measurement data. From these, a varying number of equidistantly distributed measurement points were extracted and fed into Algorithm 1 to identify the tuning parameters. Figure 3 shows the ratio of the tuning factors to their average value with the varying number of measuring points ($N$) for the three PO. It is conspicuous that $t_4$ for PO001 and PO003 deviate from the average value as the number of measurement points increase. Figure 4a illustrates the graph of recirculation power ($P_{Rez}$) for the number of measurement points $N\in [9;20]$ of PO001. The parameter affects the position of $Q_{Rez,ic}$ and $Q_{Rez,min}$. However, the changes are negligible.

In addition to the tuning parameters, the power consumption values of the PO001, PO002, and PO003 were approximated by using a third- and fourth-degree polynomial. Figure 4b shows the measurement points (Data), the physical model (Physically), and the fourth-degree approximation polynomial (Polynom 4th) for PO001. The measured points are well presented by the physical model as well as the polynomial. There is a 0.3% deviation between the physical model and the measurement points. The deviation of the polynomial is 0.6%. Section 1 demonstrated the common utilization of approximation functions in the flow rate calculation. For assessing the physical model against the third- and fourth-degree polynomials, the quadratic norm of the objective function ($‖z\left(t\right)‖²$) is taken into account, following the optimization criterion outlined in Equation (19). Both the polynomials and the physical model were parameterized using ten data points. As depicted in Figure 4c, the physical model replicates the measured data with comparable accuracy to the fourth-degree polynomial. In the case of PO002, even greater accuracy is achieved. Furthermore, it is evident that for both PO001 and PO002, the accuracy using the fourth-degree polynomial and the physical model can be notably improved compared to the third-degree polynomial.

In contrast to polynomial approximations, the physical model offers an advantage in terms of its physical interpretability. Consequently, data points located between the measured values can also be mapped with minimal error. Two sets, each comprising five measurement points, were established for PO001, PO002, and PO003. The counting of measuring points occurs at position $n$, with $i$ values ranging from $1$ to $5$. The first set comprises measuring points at positions $n=2i$, while the second set encompasses all measuring points at positions $n=i+(i-1)$. For the two sets with five data points, the polynomials and tuning parameters of the physical model are identified in each case. The cross-validated values for $‖z\left(t\right)‖²$ are shown in Figure 4d for PO001, in Figure 4e for PO002, and in Figure 4f for PO003. Even with a limited number of measurement points, the physical model can significantly improve the mapping of data points between measured values compared to polynomial approximations.

Demonstrably, the physical model can be effectively parameterized even when utilizing a limited quantity of measuring points. The increase in the number of measuring points directly contributes to the improvement of the recirculation performance characteristic. The representation accuracy of the physical model corresponds to the precision of a fourth-degree polynomial.

3.2. Validation with Varying Speeds and Prewhirl Angles

The set of performance curves of PO003 was calculated following the work of the authors of [26]. The plot shown in Figure 5a displays the measured data (Data) and the characteristics calculated by the physical model (Physically), assuming a mechanical power of $P_m=240 \mathrm{W}$. In addition to the solid curves calculated by the physical model, dashed curves representing the calculated curves using the affinity laws (Affinity) are also shown. According to the authors of [12], the mechanical power has a large influence on the calculation of the set of performance curves. For the calculation, it was assumed that $P_m\propto n^{1.3}$ [10]. The physical model effectively captures the set of performance curves when mechanical power is correctly defined. Figure 5b illustrates the measured data points (Data) and the characteristic curve calculations derived from the physical model (Physically) for PO001 (left-hand side) and PO002 (right-hand side). The motor frequency ($f_M$) has been adjusted to three different values $f_M\in [40 \mathrm{H}\mathrm{z}, 45 \mathrm{H}\mathrm{z}, 50 \mathrm{H}\mathrm{z}]$, thus changing the pump speed. The physical model can also accurately replicate the set of performance curves for these two pumps. In contrast to PO003, the speeds of PO001 and PO002 have been slightly reduced. The mechanical power of these two pumps would be determined via measurements.

Figure 5c shows the measurement data points (Data) and the characteristic curves (Physically) for PO001 (left-hand side) and PO002 (right-hand side), calculated by the physical model at different prewhirl angles (${∆\alpha }_1$). The characteristic curves show a high degree of correspondence with the measured values. In the case of PO002, the physical model exhibits a deviation within the range of $Q=0-30 {\mathrm{m}}^3/\mathrm{h}$. According to this, the recirculation power ($P_{Rez}$) is also influenced by the prewhirl bodies. It is notable that the recirculation performance increases for both positive and negative prewhirl angles. The recirculation power also changes for PO001. In comparison to PO002, there are only insignificant changes.

3.3. Evaluation of the Physical Model in the Flow Rate Calculation Algorithm

For both PO001 and PO002, the flow rate was calculated by using the physical model in accordance with Algorithm 2 (outlined in Section 2.3). The flow calculation algorithm was applied to the time signals of the measurements. The average values of the time signal results are shown in the following figures. Figure 6a depicts the error $\left|Q-Q_{calc}\right|$ for PO001, while Figure 6b represents the same for PO002. In the spectrogram on the left-hand side, the flow rate was calculated by using the physical model, whereas the spectrogram on the right-hand side was constructed based on using the polynomial approximation and the affinity laws. Within the range $Q=100-120 {\mathrm{m}}^3/\mathrm{h}$ at $n\approx 1200 {\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$, PO001 exhibits a maximum error of $\left|Q-Q_{calc}\right|=5 {\mathrm{m}}^3/\mathrm{h}$. In the calculation, which was developed using the affinity laws, the maximum deviation also appears at this speed but in the range of the maximum flow rate. In general, the relative error for PO001 using the physical model and the affinity laws amounts to approximately 3%. For PO002, the calculation of the flow rate over the whole range shows a small error of $\left|Q-Q_{calc}\right|<2 {\mathrm{m}}^3/\mathrm{h}$. The average relative error from the calculation with the physical model and the affinity laws is about 1.5% for PO002.

Figure 7 shows the errors $\left|Q-Q_{calc}\right|$ for calculating the flow using the physical model for PO001 (Figure 7a) and PO002 (Figure 7b). The flow rate ($Q$) was plotted on the y-axis, and the prewhirl angle (${∆\alpha }_1$) was plotted on the x-axis—both were varied. For PO001, the average relative error in calculating the flow rate is 1.6%. The average relative error for PO002 is 3.9%. In contrast to PO001, PO002 exhibits increased errors in flow rates at $Q<50 {\mathrm{m}}^3/\mathrm{h}$ for prewhirl angles $∆{\alpha }_1\ne 0°$. As mentioned in Figure 5c, the recirculation power is influenced by prewhirl bodies, which is not represented in the physical model. For PO001, an unstable characteristic curve occurs from a prewhirl angle of ${∆\alpha }_1>15°$, and for PO002, the same occurs from a prewhirl angle of ${∆\alpha }_1>20°$. The calculation of the flow rate becomes ambiguous in this case, as mentioned in [2,5].

4. Conclusions

Currently approximation functions are used in algorithms for calculating the flow rate of centrifugal pumps. Polynomials are often used; however, their parameters are not physically interpretable. When calculating the set of performance curves, the affinity laws are primarily employed by the approximation functions. This approach leads to deviations, particularly when the pump speed is significantly reduced. Procedures to improve the calculation of the set of performance curves often require additional measurements. In addition, the approximation functions cannot calculate changes in the set of performance curves due to prewhirl. Through the utilization of the developed physical model to depict the set of performance curves, enhancements can be achieved in the calculation of these curves across a broad speed spectrum. Moreover, the model’s physical interpretability enables one to adjust the set of performance curves to account for the influence of prewhirl. The findings of this study can be summarized as follows:

• The physical model relies on established empirical equations. The model can be adapted to specific pump performance curves via parameter tuning. A modified Levenberg–Marquardt method is used to fit the tuning vectors.
• The accuracy of mapping the performance curve by the physical model is in the range of fourth-degree polynomial accuracy. However, in comparison to the polynomial approximation, the model can be parameterized with significantly fewer measurement points.
• Due to the separate consideration of the mechanical losses of the pumps, the physical model offers superior accuracy in flow rate determination compared to the affinity laws. Deviations of less than 2% can be achieved. It can be concluded that the need for additional measurements can be avoided using a physical model.
• It is possible to calculate the changes in the set of performance curves resulting from prewhirl. This also resulted in deviations of less than 2% when determining the flow rate. The improved accuracy of the physical model in determining the flow rate is due to its ability to reproduce the set of performance curves with minimal deviation despite the large number of influencing variables.