# The Estimation of Centrifugal Pump Flow Rate Based on the Power–Speed Curve Interpolation Method

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}/h, 0.2047 m

^{3}/h, 0.2197 m

^{3}/h, and 0.1979 m

^{3}/h. Additionally, the average relative errors are 2.0816%, 3.2875%, 3.6981%, and 2.9419%. Hence, this method fully meets the needs of centrifugal pump monitoring and control.

## 1. Introduction

_{2}emissions, which accounts for 70% of the total electricity consumption of industry [1,2]. As per the European Commission report [3], pumping systems occupy nearly 22% of the energy consumed by electric motors worldwide and consume approximately 8.7% of global electricity [4]. Hence, a feasible way to alleviate the energy crisis and reduce carbon emissions is to improve the energy efficiency of pumps [5]. According to statistics, even a 1% increase in pump efficiency could reduce CO

_{2}emissions by at least 572 tons per day in the European region [6]. For this reason, centrifugal pumps, as a vital part of pump systems, consume 72% of the pump system’s energy and have great potential for energy and carbon savings [7]. Meanwhile, this also helps to extend the life of equipment by improving the pumps’ energy efficiency [8,9]. A practical method to reduce the energy consumption of centrifugal pumps is to optimize the control strategy using frequency converters [10], and the optimization process typically requires real-time monitoring of the centrifugal pump’s operational state [11]. However, monitoring by external sensors not only requires more installation space but also brings more risk of centrifugal pump failure [12]. The estimation of motor operating parameters by modern frequency converters makes it possible to monitor a centrifugal pump’s operational status without sensors [13].

## 2. Experimental Setup

_{nom}= 15 m

^{3}/h, H

_{nom}= 70.5 m, and n

_{nom}= 2900 rpm. The frequency converter (Siemens, Berlin, Brandenburg, Germany, 2020) adjusts the speed of the pump, and a motorized valve controls the flow rates of the pump. The controller is a Siemens S7-200smart ST30 (Siemens, Berlin, Brandenburg, Germany, 2020), and the analog signals are collected through the expansion module Siemens EM AM06 (Siemens, Berlin, Brandenburg, Germany, 2020). The frequency converter communicates with the PLC through the Modbus protocol. The software was written in LabVIEW (Version 18.0, National Instruments, Austin, Texas, United States, 2021), and the PLC communicates with the computer through Ethernet.

## 3. Estimation Methods

#### 3.1. QP Method

_{nom}is the rated speed, rpm; Q is the actual flow rate, m

^{3}/h; Q

_{nom}is the rated flow rate, m

^{3}/h; H is the actual head, m; H

_{nom}is the rated head, m; P is the actual power, kW; P

_{nom}is the rated power, kW.

_{est}can be calculated using the motor shaft power P

_{act}, and then the head H

_{est}can be computed by the above flow rate Q

_{est}. In the QP method, the transformation of the QP curve based on the motor speed and the calculation of the flow rate using the shaft power are two of the critical steps. Hence, the speed and shaft power of the centrifugal pump are vital variables in the flow rate prediction process.

#### 3.2. PN Curves Interpolation Method

_{0}and Q

_{n}PN curves divided the graph into three parts. Q

_{0}is the PN curve with the lowest flow rate, generally taken as 0 m

^{3}/h. Therefore, when the centrifugal pump works below the Q

_{0}curve, namely the S

_{1}region, the estimation flow rate is 0 m

^{3}/h.

_{2}region contains a lot of PN curves, the flow rate from Q

_{0}, Q

_{1},… Q

_{j}, Q

_{j+1},… Q

_{k−1}, Q

_{k}covers a wide range of centrifugal pumps with small flow rates, rated conditions, and high flow rates. If t lies between the Q

_{j}and Q

_{j+1}flow curves in the S

_{2}region, the flow rate at point t (N

_{t}, P

_{t}) can be obtained by the internal interpolation method, which is shown in Equation (4).

_{j}is the power–speed function at the flow rate Q

_{j}, and f

_{j+1}is the power–speed function at the flow rate Q

_{j+1}.

_{3}region represents the centrifugal pump working above the Q

_{n}curve. Typically, pumps operate in the S

_{2}and S

_{1}regions. When the valve opening of the system is so large that the pump is in extreme operating condition, the flow rates are in the S

_{3}region. This paper provides an external interpolation method to calculate the flow rates in the S

_{3}region, as shown in Equation (5).

_{k−1}is the power–speed function at the flow rate Q

_{k−1}, and f

_{k}is the power–speed function at the flow rate Q

_{k}.

#### 3.3. Estimation Model Based on PN Curves

^{3}/h flow rate at 50 Hz in the QP curve. As the flow rate increases, dP/dQ gradually decreases, causing ΔP to be smaller. The QP curves for the other input frequencies have similar power variations to 50 Hz. This means that to make the power difference between PN curves approximately equivalent, the flow rate difference between adjacent PN curves should be increased appropriately while in the high flow rate region.

_{0}= 0 m

^{3}/h, Q

_{1}= 3 m

^{3}/h, Q

_{2}= 6 m

^{3}/h, Q

_{3}= 9 m

^{3}/h, Q

_{4}= 15 m

^{3}/h, and Q

_{5}= 21 m

^{3}/h. The fitted equations are shown in Equations (6)–(11).

^{2}of all six fitted curves is close to 1, and the residual sum of squares (RSS) is close to 0, which indicates that the six curves have high accuracy and the results are reliable.

_{0}, P

_{1}, P

_{2}, P

_{3}, P

_{4}, and P

_{5}of curves Q

_{0}, Q

_{1}, Q

_{2}, Q

_{3}, Q

_{4}, and Q

_{5}at different speeds. The flow rate of the centrifugal pump is obtained from the collected power and the calculated power. For instance, the predicted flow rate Q

_{est}= 0 m

^{3}/h when P ≤ P

_{0}; when P

_{j}≤ P ≤ P

_{j+1}, the flow rate can be calculated by the internal interpolation method; when P ≥ P

_{5}, the flow rate can be calculated by external interpolation method.

## 4. Influence Factors on the Estimation Accuracy

#### 4.1. PN Curves

^{3}/h in area A of the PN curve because the curve in this area is relatively dense, which means that the power variation of the adjacent PN curve is slight. When the pump operates in area B, the error in its flow prediction is between 3 and 6 m

^{3}/h. When the pump works in area C, the error in its flow prediction is smaller than 3 m

^{3}/h, where the curve is sparse, and the power of the adjacent PN curve varies widely. It indicates that centrifugal pumps operating in the low-speed region are more sensitive to power variations. Therefore, the location of the centrifugal pump’s operating point significantly influences the prediction results.

#### 4.2. Impact of Shaft Power

^{3}/h, 9 m

^{3}/h, and 18 m

^{3}/h, respectively. As shown in Figure 11, the fluctuation of motor shaft power is more significant at high input frequencies than at low input frequencies. However, the prediction results are less affected by power fluctuations when the input frequency is high since the PN curves of the centrifugal pump are sparse in this region (see Figure 8). Meanwhile, motor shaft power fluctuates slightly at the lower speed operating area where the PN curve is dense. Therefore, the impact of power fluctuations is also insignificant at low input frequencies.

## 5. Results and Discussion

^{3}/h, which is much lower than the other two prediction methods. Meanwhile, the mean relative error of the interpolation method is 2.6919%, while the mean relative error of BPNN and QP prediction methods is 16.6261% and 23.3509%, both of which are much higher than the relative error of the interpolation method. This indicates that the PN curve interpolation prediction method is more accurate and stable.

## 6. Application of the PN Curves Interpolation Method

## 7. Conclusions

- (1)
- The PN curves containing speed, power, and flow information were presented, and an interpolated flow prediction model based on these curves was established. The factors affecting the accuracy of the estimation method were analyzed, which mainly include the flow rate and accuracy of the PN curve, the location of the centrifugal pump operational point, and the fluctuation of the shaft power. Particularly in this aspect of the location of the centrifugal pump operational point, centrifugal pumps operating at a low speeds were more sensitive to power variations. Furthermore, the fluctuation of the motor shaft power can cause variations in the estimation flow rate.
- (2)
- Three different estimation models were compared. The average absolute and relative errors of the interpolated flow prediction method are 0.1756 m
^{3}/h and 2.6919%, compared to 1.3851 m^{3}/h and 6.6261% for BPNN and 1.8315 m^{3}/h and 23.35089% for the QP method. This means that the PN curve interpolation method has high accuracy in estimating the flow rates. - (3)
- The PN curve interpolation method was tested in an industrial application, and its average absolute errors at 47.5 Hz, 42.5 Hz, 37.5 Hz, and 32.5 Hz are 0.1442 m
^{3}/h, 0.2047 m^{3}/h, 0.2197 m^{3}/h, and 0.1979 m^{3}/h, and its average relative errors are 2.0816%, 3.2875%, 3.6981%, and 2.9419%. This highly accurate prediction capability fully meets the monitoring and control requirements and improves the accuracy of centrifugal pump flow prediction at small flow rates.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Experimental facilities: (

**a**) closed-loop test bed, (

**b**) vertical centrifugal pump, and (

**c**) control system.

**Figure 6.**PN curves and PN curves interpolation methods: (

**a**) PN curves for different flow rates and (

**b**) PN curve-based flow estimation method.

Sensor Type | Measurement Range | Precision | Manufacturers |
---|---|---|---|

Electromagnetic flowmeter | 0–150 m^{3}/h | 0.2% | Endress + Hauser, Switzerland |

Power meter | Current, voltage, power | 0.5% | Qingzhi instrument, China |

Tachometer | 0–20,000 rpm | ±1 rpm | Onosokki, Japan |

Items | Q_{0} | Q_{1} | Q_{2} | Q_{3} | Q_{4} | Q_{5} |
---|---|---|---|---|---|---|

R^{2} | 0.9999 | 0.9999 | 0.99999 | 0.99996 | 1 | 0.99994 |

RSS | 1.465 × 10^{−4} | 1.051 × 10^{−4} | 4.146 × 10^{−5} | 1.716 × 10^{−4} | 2.455 × 10^{−5} | 3.312 × 10^{−4} |

Items | Errors | |||
---|---|---|---|---|

Maximum | Average | Minimum | ||

PN curves interpolation | δ (m^{3}/h) | 0.5409 | 0.1756 | 0.0021 |

ε (%) | 16.9814 | 2.6919 | 0.0181 | |

BPNN | δ (m^{3}/h) | 2.7565 | 1.3851 | 0.0717 |

ε (%) | 61.9739 | 16.6261 | 2.0429 | |

QP | δ (m^{3}/h) | 3.4191 | 1.8315 | 0.0496 |

ε (%) | 137.6936 | 23.35089 | 1.3291 |

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**MDPI and ACS Style**

Wu, Y.; Wu, D.; Fei, M.; Xiao, G.; Gu, Y.; Mou, J.
The Estimation of Centrifugal Pump Flow Rate Based on the Power–Speed Curve Interpolation Method. *Processes* **2022**, *10*, 2163.
https://doi.org/10.3390/pr10112163

**AMA Style**

Wu Y, Wu D, Fei M, Xiao G, Gu Y, Mou J.
The Estimation of Centrifugal Pump Flow Rate Based on the Power–Speed Curve Interpolation Method. *Processes*. 2022; 10(11):2163.
https://doi.org/10.3390/pr10112163

**Chicago/Turabian Style**

Wu, Yuezhong, Denghao Wu, Minghao Fei, Gang Xiao, Yunqing Gu, and Jiegang Mou.
2022. "The Estimation of Centrifugal Pump Flow Rate Based on the Power–Speed Curve Interpolation Method" *Processes* 10, no. 11: 2163.
https://doi.org/10.3390/pr10112163