## Abstract

This paper presents a fast simulation model for predicting the dynamic response of a motor-pump system to startup event. The purpose is to analyze the effect of the impeller acceleration time, the final flow rate and the impeller geometry on the pump transient flow during starting operations. The motor speed and torque variations were predicted by simulating the transient law of the three-phase induction motor by adopting the d-q axes theory. The pump model was built by solving the unsteady flow governing equations with the method of characteristics (MOC). The whole model was validated with available tests from literature. Accordingly, the computation of impeller acceleration, the motor torque, the unsteady pressure and flow rate was made for various starting conditions. The results have revealed that during its starting time, the pump hydraulic transients are well influenced by the motor speed acceleration, the flow inertia and the impeller geometry. Through the analysis of the simulation results, the conclusion was that the accuracy of the present method is reasonable, and it can be used for assisting pumping system design.

## HIGHLIGHTS

Radial pump startup transient was investigated using the method of characteristics (MOC).

Pump fast startup can cause a significant head impact and high torque oscillations.

Startup hydraulic transients are well influenced by the motor acceleration time and the impeller geometry.

## INTRODUCTION

Water distribution networks (WDNs) represent one of the largest infrastructure assets of an industrial society (Darweesh 2018). Their security and efficiency are of paramount importance to human health and economic development (Duan *et al.* 2020). The leading causes behind hydraulic network anomalies may be essentially associated with leakage, corrosion, deformation, and hydraulic transients. Centrifugal pumps are the most commonly used devices for pressure generation and fluid transmission in water systems. The variation of their operating conditions such as startups, shutdown or rapid changes in control valve settings will lead to transient flow disturbances which are closely related to the safety and reliability of the WDN_{S} (Yang *et al.* 2015; Triki 2018).

During the past few decades, there have been numerous studies on centrifugal pump startup characteristics. Saito (1982) experimentally and theoretically explored the transient characteristics of a diffuser pump during starting time. His results have clearly shown the significant differences between pump startup transients and the steady state performances. In addition, his conclusion was that the response of a centrifugal pump to startup events is influenced by three important factors which are the mass of water in the pipes, the downstream valve openings and the starting time. On a similar subject, a theoretical and experimental study was made by Tsukamoto & Ohashi (1982). As a result of this study, they demonstrated that the impulsive pressure and the lag in circulation formation around the impeller vanes are the main reasons for the difference between the dynamic and quasi-steady characteristics of the turbo pump during the starting period. For the purpose of predicting the dynamic behavior of a centrifugal pump during operating transients, Thanapandi & Prasad (1994, 1995) developed a numerical model using the method of characteristics (MOC). The numerical results were validated with experiments and the conclusion was that the change of the pump behavior during normal startup and stopping is quasi-steady. They also deduced that the quasi-steady model can be used to predict the pump transient characteristics where the impeller acceleration is small. Farhadi *et al.* (2007) theoretically studied transient pump startup in the primary cooling system of a nuclear reactor. They investigated the effect of the moment of inertia on the pump flow rate during starting time. It has been found that for a pumping system with a high moment of inertia, the fluid flow is established gradually and it requires more time to reach its final value. However, for a pump having a small moment of inertia the steady-state operating flow will be approached much more rapidly. To investigate the mechanism of pump startup transients, Wu *et al.* (2009) established a three-dimensional model for simulating the unsteady flow in a centrifugal pump during the starting period. The pump transient performances were analyzed when the impeller started from stationary to maximum speed, and the effects of different pipeline resistances were also evaluated.

In recent years, Duplaa *et al.* (2010) experimentally tested a centrifugal pump fast startup in cavitating and non-cavitating conditions. They found that at high flow rate, large scale low-frequency oscillations are obtained on the pressure signal at the delivery, whereas at a lower flow rate, only a drop at the end of startup is detected. They also reported, according to the numerical study presented in Dazin *et al.* (2007), that centrifugal pump fast startup results in severe transient effects that are mainly governed by the speed acceleration and the flow rate increase.

Transient flows in a centrifugal pump during its starting period were also experimentally and numerically investigated by Li *et al.* (2011). They used the two-dimensional particle image velocimetry technique (PIV) to capture the transient flow evolutions in the pump's diffuser. They also established a three-dimensional model with a dynamic slip region (DSR) method to simulate the transient flow evolutions during startup operations. The conclusion was that in the early stage of the startup, the transient vortex evolution between blades is the main reason leading to the transient head coefficient being lower than the steady state value*.* Based on the experimental data of performance curves, Wan & Huang (2011) established an inversion model to obtain the performance curves of a centrifugal pump. The proposed method is adopted for hydraulic transient simulation of centrifugal pump failure and startup. Meanwhile, the MOC numerical model was applied to compute the gradually unsteady process with different hydraulic pressure. It was found that the inversion method can give more reasonable results to describe the pump's transient process and it can be used as an effective tool to achieve the complete characteristic curves of pumps.

Lucius & Brenner (2011) presented a numerical method to investigate the effect of the rotating stall on the internal flow field of a centrifugal pump. The paper describes how the fluctuations of fluid properties during rotating stall can be investigated in order to determine their importance for the excitation of structural vibrations. As a conclusion, they found that the data analysis in relative frame can deliver major excitation frequencies not visible in stationary frame analysis. The numerical method was also successfully applied in pump startup simulations by Elaoud & Hadj-Taïeb (2011), Chalghoum *et al.* (2016) and others. They have proposed various models by solving the unsteady flow governing equations with the 1D method.

More recently, Page *et al.* (2019) proposed a pump speed controller to solve the pressure deficiency in water system distribution (WDS). As a function of the average pressure deviation, the controller adjusts the pump speed to supply the adequate pressure. Dutta *et al.* (2020) adopted a machine learning (ML) algorithm as a predictive control method for the identification of water hammer problems in a pumping system. It is an experiment-based work intended to discover the hydraulic anomalies through the analysis of the pump motor current signal. Duan *et al.* (2020) proposed an important comprehensive literature review of the hydraulic models for predicting and describing the transient flows that are commonly used for the effective design and management of urban water supply systems (UWSS). First, the review illustrates the importance of transient flow phenomena and its impact on the safety, reliability and availability of pipe systems. Thereafter, the authors discussed the advances and limitations of current transient models and methods, including the derivations of governing equations in 1D and 2D forms, unsteady friction, turbulence formulas, viscoelastic models and numerical simulation methods. On the other hand, the utilizations of transient flows for pipeline diagnosis and fault detection are reviewed. Four types of common pipe defects, namely leakage, discrete partial blockage, extended partial blockage, and unknown branch, are elaborated and evaluated. To control the water hummer processes in the water supply systems, Chen *et al.* (2021) proposed a solution for determining the size of the air tank in the actual operation process. The operating parameters of the air tank were optimized by introducing the influence of friction and the hydraulic impedance of the connecting pipes.

Despite various contributions, the unsteady flow phenomena in turbo pumps remains unclear, complex and difficult to investigate. In addition, the characteristic method (MOC) is not fully exploited in this field. The impeller speed, on the other hand, is often assumed to follow linear or exponential law. These approximations hide the natural fluctuations of the impeller velocity that occur at the very beginning of starting time.

At the outset, this paper presents a simple and fast numerical method applicable to radial pump startup analysis. The unsteady flow equations of continuity and motion are numerically solved with the 1D method of characteristics. To get the impeller speed variation, the induction motor is modeled with the d-q axis transformation theory. The proposed model is validated with experimental data presented in Duplaa *et al.* (2010). Thereafter, the numerical solution is used to investigate the influence of several parameters (e.g. the impeller acceleration time, the final flow rate, the impeller width, the impeller blade exit angle and the number of impeller vanes) on the unsteady flow during pump starting period. Through the interpretation of the instantaneous motor-pump performance; it is found that the impeller geometry, final flow rate and moment of inertia are the key parameters for predicting the hydrodynamic pump characteristics. Accordingly, this work proposes a new coherent model, including the centrifugal pump and its drive motor which can provide useful information regarding the electro-hydraulic transient regimes in a new pumping system.

## METHODS AND MATERIALS

### Modeling of 3-phase induction motor

*et al.*1984). It is introduced herein to facilitate the pump motor analysis and reduce computation time (Orille

*et al.*1999). This method consists of transforming the motor equations in the stationary reference frame, from the real three phase axes to the rotating reference frame described by direct quadrature (d-q) transformation (Figure 1). In general, for any arbitrary value of , the transformation of the stator and the rotor variables from the three phases stationary coordinates to the (d-q) rotating coordinates is carried out through Park's transformation described as follows (Krause

*et al.*2013): where:The three phase voltages supplied to the induction motor in a stationary reference frame are given as:in which

*t*is time, is the stator voltage pulsation, is the RMS (root mean square) value of voltage and is the frequency.

*et al.*2004):where , are the stator direct-axis and quadrature axis currents (A); , are the rotor direct-axis and quadrature axis currents (A); are the resistances of the stator and rotor (

*Ω*); are the cyclic inductances of the stator and rotor (H);

*M*is the mutual cyclic stator-rotor inductance (H); is the magnetic leakage factor (–);

*g*is the sliding coefficient (–);

*p*is the motor pole number (–);

*I*is the inertia of the rotating parts (kgm

^{2}); are the rotor and stator pulsation (rad/s); is the motor electromagnetic torque (Nm); is the load torque (Nm).

The numerical integration of the motor equations with the pump resistant torque expression that will be seen later allows the motor speed and the torque evolution to be obtained. More information can be found in Okoro (2003) and Mimouni *et al.* (2004).

### Equations of motion for coupled motor pump

*Q*(Elaoud & Hadj-Taïeb 2011):

The hydrodynamic torque coefficients *k*_{1}, *k*_{2} and *k*_{3 }are obtained from the pump steady state characteristic curves.

### Model exit angle effect

*r*

_{2}the outer impeller radius and is impeller blade exit angle.

*is*the tangential component of the flow absolute velocity

*V*

_{2}.

*Z*is the number of impeller vanes.

### Hydraulic loss model

*et al.*2012):where is the hydraulic diameter of the impeller passage section (), and is the friction coefficient that depends on the Reynolds number and the surface roughness of the impeller passage.

*et al.*2019):

In this equation denotes the impeller nominal velocity. The shock loss coefficient *k*_{s} is obtained from the pump steady state performance curves.

### Pump modeling

#### Equations for unsteady flow through the impeller

*et al.*(2016). The basic equations of continuity and motion, in a reference frame rotating with the impeller, after neglecting unimportant terms, can be expressed as:where

*x*is the space distance along the rotating passage, is the impeller passage section, is the fluid discharge, is the pump head,

*a*is the pressure wave speed, is the hydraulic diameter of the section

*A*,

*b*is the blade height,

*l*is the blade length and is the impeller angular velocity. The term in Equation (20) represents the centrifugal force developed by the rotating impeller.

#### Method of characteristics (MOC)

*C*

^{+}and (Figure 4). The ordinary equations obtained after transformation are identified as:in which

*J*is calculated from Equations (16) and (17) as:

*H*and

*Q*are known initial values at each sections of the grid along

*t*= 0. The integration of Equations (21) and (22) along the

*i*+ 1 characteristic lines can be written in the following forms:

*P*. Hence, in order to get the solution until an elevated time level, it is necessary to introduce the appropriate boundary conditions (BC). The pumping system boundary conditions are presented in Figure 5, including the junctions between the pump, the suction and discharge pipe with their appropriate equations. It is also important to note that in the junction between the impeller and the volute, we have to introduce the number of the impeller vanes in the expression of the flow rate. In addition, the boundary conditions at the discharge valve are not presented herein, they can be found in Wang

_{i}*et al.*(2019) and Chalghoum

*et al.*(2016). Flows in the casing volute and pipes are analyzed with the same procedure without consideration of the centrifugal force term. Thus, the following equations are used:in which

*J*is obtained from Equation (17) as:

## RESULTS AND DISCUSSION

We consider the experimental tests carried out by Duplaa *et al.* (2010) to verify the simulation results and validate the present numerical model. Experimental tests were carried out in the fluid Mechanics Laboratory of Lille (LMFL) with the pump fast startup test facility. The test loop is a closed circuit with the suction and delivery pipe connected to a single tank (Figure 6). In this study the close loop test facility is modeled with the 1D approach including the centrifugal pump, the induction motor, the pipes and the discharge valve, whereas experimental test conditions listed in Tables 1 and 2 are respected.

Impeller speed (rpm) | 2,900 | Inlet vane angle (deg) | 32.2 |

Rated flow (Q) _{n} | 23 m^{3}/h | Outlet vane angle (deg) | 23 |

Differential head | 50 m | Number of vanes Z | 5 |

Inlet diameter d1 (mm) | 38.5 | Outer width b2 (mm) | 7 |

Outlet diameter D2 (mm) | 202.5 | Impeller inertia (Kg.m^{2}) | 0.0085 |

Impeller speed (rpm) | 2,900 | Inlet vane angle (deg) | 32.2 |

Rated flow (Q) _{n} | 23 m^{3}/h | Outlet vane angle (deg) | 23 |

Differential head | 50 m | Number of vanes Z | 5 |

Inlet diameter d1 (mm) | 38.5 | Outer width b2 (mm) | 7 |

Outlet diameter D2 (mm) | 202.5 | Impeller inertia (Kg.m^{2}) | 0.0085 |

Output power (W) | 7,500 | Mutual inductance M (Ohm) | 0.0831 |

Speed (rpm) | 2,875 | Stator inductance L (Ohm) _{s} | 0.0875 |

Stator resistance R (Ohm) _{s} | 0.5 | Rotor inductance L (Ohm) _{r} | 0.0875 |

Rotor resistance R (Ohm) _{r} | 0.85 | Moment of inertia (Kg.m^{2}) | 0.03 |

Output power (W) | 7,500 | Mutual inductance M (Ohm) | 0.0831 |

Speed (rpm) | 2,875 | Stator inductance L (Ohm) _{s} | 0.0875 |

Stator resistance R (Ohm) _{s} | 0.5 | Rotor inductance L (Ohm) _{r} | 0.0875 |

Rotor resistance R (Ohm) _{r} | 0.85 | Moment of inertia (Kg.m^{2}) | 0.03 |

At the beginning of this simulation, the pump is started to reach its design operating point, all experimental conditions are respected and the motor pump specifications are correctly introduced. Thereafter, a fast startup simulation is carried out to analyze the motor pump transient behavior under the critical condition of quick startup. Subsequently, a series of simulations are reserved to study the effect of certain impeller geometric parameters on the pump transient head and the flow rate evolution.

### Experimental validation

For a given rotational speed *N* = 3,000 rpm and a total inertia *,* the pump performance curves are obtained by considering different opening positions of the discharge valve. With the change of the final flow rate, the operating point will change along the pump characteristic curves. A comparison between calculated and measured results is presented in Figure 7(a). All performance curves are in good agreement except in Figure 7(c), the shift between experiment and simulation curves is probably due to the approximation of the impeller geometry and the volute shape. To compare the predicted startup characteristics with available experimental data, the impeller speed is imposed to follow the actual speed evolution and the discharge valve is adjusted to reach the nominal flow rate (*Q* = 23 m^{3}/h). Figure 7(b) shows a comparison between theoretical and actual impeller speed variations. Only a small shift at the beginning and the earlier step of starting time are noted. These differences are mainly due to the load torque approximation. The computed flow rate and the pump head variations are compared with experimental data and are respectively presented in Figure 7(c) and 7(d). It is also clear from Figure 7(d) that the pump head variation is in a good agreement with the experimental curve.

### Fast startup simulation

To follow the actual operating conditions, a rotational speed of 2,900 rpm () is considered. The motor and pump are directly connected to the same rotating shaft. The response of the coupled system is simulated under fast startup condition. This typical simulation () is considered as a reference case. Figure 8(a) shows that the motor starting time is close to 0.4 s and this allows the impeller speed to rise from zero to 303 rad/s. As can be seen in Figure 8(b), the motor torque starts with high oscillations which gradually disappear after 0.2 s. The peak instantaneous torque is more than six times rated torque. Figure 8(c) shows that the peak instantaneous current starts at 106 A while the nominal value is only 15 A. The motor output power presented in Figure 8(d) is accompanied by significant fluctuations followed by a peak exceeding twice its rated value. Therefore, the motor torque oscillation which corresponds to the large startup current causes stator voltage drop and affects the pump dynamic responses.

The inlet, outlet and differential head variations are plotted in Figure 9(a). The outlet head rises approximately as fast as the impeller speed and it peaks before slowly stabilizing, while the inlet head goes through a negative peak exceeding cavitation limit. As presented in Figure 9(b), the pump flow rate rises gradually with a lag due to the inertial effect of the mass of water in the pump and pipes. It requires more than 0.6 s to stabilize at its nominal value. Figure 9(c) depicts the startup efficiency curve. At time *t* = 0.5 s the efficiency reaches the maximum value of 52%, while at the first stage of starting time (*t* < 0.2 s) the pump operates at almost zero efficiency.

According to Dazin *et al.* (2007), the pump transient head is mainly governed by the impeller acceleration () and the flow inertia (). To focus upon this idea, the impeller speed and flow rate accelerations are plotted in Figure 9(d). The two parameters are normalized by division with their corresponding maximum value at best operating conditions and time is normalized by which corresponds to 63% of the impeller final speed. It is similar to the pump nominal acceleration time () used in Tsukamoto & Ohashi (1982). This coefficient is related to the motor acceleration and it is often used to characterize the pump starting time. As can be seen from Figure 9(d), during the first stage of starting time (), the term is dominant and it shows an oscillating profile explaining the harmful vibration induced by pump startup. Because of the flow inertia effect, the term is very low. Subsequently, the speed acceleration outweighs the inertial influence. Thereafter, during the second period (), the transient flow is governed by the sum of the flow rate acceleration and impeller acceleration. Before the end of this stage, the flow inertial term becomes dominant. In the last period of starting time (), the unsteady flow is only governed by the flow inertial effect.

### Parameters affecting the pump startup

A series of numerical simulations have been performed to investigate the effect of several parameters on the pump startup characteristics. Because of their importance in hydraulic transients, only the responses of pump head and flow rate acceleration are analyzed here.

#### Effect of the acceleration time and the final flow rate

The impeller acceleration is mainly affected by the mechanical inertia of the coupled rotating parts. However, it is not practically affected by the final flow rate. In Figure 10(a), the difference between the impeller acceleration curves is noticeable. For a large starting time the impeller accelerates slowly and the size of fluctuations at the first stage decrease remarkably. The same effect is seen in Figure 10(b)–10(d), low acceleration time leads to fast startup and induces a large head impact, whereas large acceleration time causes slow startup and reduces the head impact.

The effect of final flow rate on the flow acceleration and pump head variation are plotted in Figure 11. The flow rate acceleration traces a symmetric path whose delay is proportional to the final discharge (Figure 11(a)). The pump head evolution shows a significant peak before stabilizing. This impact is also proportional to the quantity of water delivered by the pump. Furthermore, as can be seen from Figure 11(b), when the pump operates under small flow rate the pressure head increases gradually without exceeding its final value because of the reduction of the hump in flow rate acceleration.

#### Effect of the impeller geometry

Four important parameters are considered to analyze the effect of the impeller geometry on the pump head evolution. These parameters are the inlet diameter, blade height, blade exit angle and blade number. All simulations are carried out with the same final speed (2,900 rpm). The impeller outlet diameter (D_{2}) is not changed and the discharge valve is kept in the same position (full open).

The influence of the inlet diameter on the transient head evolution is evaluated in Figure 12(a). Smaller diameter (d_{1} = 40 mm) results in the highest head impact (40.9 m), but there is no significant effect on the required starting time. Figure 12(b) presents the effect of the impeller blade height on the pump head evolution. The large head impact (49.2 m) is obtained with the highest impeller blade (b = 12 mm), such a pump impeller requires more time to reach its stationary regime. It is evident from Figure 12(c) that a higher blade angle (60°) leads to a larger head impact (43.15 m). Indeed, there is a decrease tendency on the required starting time with decreasing the blade exit angle. This is because of the impeller slip factor effect and the transient axial thrust of the moving water. The influence of the impeller blade number (Z) on the pump transient head is plotted in Figure 12(d). The large head peak (42.8 m) is obtained with the highest blade number (12 blades), whereas the starting time tends to decrease with the impeller blade number.

Figure 13 shows the effect of four parameters of the impeller geometry on the impact head, which is defined as the maximal difference between the suction and the delivery pressure. It is clear from Figure 13(a) that the peak head increases with decreasing the impeller inner diameter. Figure 13(b) shows that, for a given pumping system, increasing the blade height will increase the transient head impact. This is because of the large change in the mass of the moving water. It is also evident from Figure 13(c) that the peak of the transient head is proportional to the impeller exit angle. It is finally demonstrated in Figure 13(d) that the number of the impeller blade has no important consequence for the startup head impact.

## CONCLUSIONS

In this paper, the radial pump startup transients are analyzed by numerical method. The dynamic response of the coupled motor pump is investigated for different startup conditions. The predicted results have a similar trend with experimental data. Furthermore, the influence of the impeller acceleration time, final flow rate and impeller geometry on the pump transient head is studied. Through the analysis of numerical results the following conclusions are found:

The impeller acceleration () and the flow rate acceleration () are two important parameters that can determine the pump transient head evolution.

Fast startup causes high torque oscillations and large head impact, however, large acceleration time leads to slow startup and reduces the pressure head fluctuation.

The transient head fluctuation shows a tendency to increase with the increase of pump final flow rate. However, when the pump operates under small flow rate, the pump head increases gradually without exceeding its final value.

The unsteady flow in a centrifugal pump depends on its impeller geometry. A large change in the transient head curve is observed with different blade heights. The impeller blade exit angle has also a remarkable effect on the startup head evolution. Nevertheless, only a small change in the head evolution is noticed by varying the impeller blade number. In addition, the transient head impact is well influenced by the impeller inner diameter.

With acceptable accuracy and simple implementation, the characteristic method proves its effectiveness for the approximation of flow governing equations and investigation of the transient behavior of pumping systems.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## REFERENCES

*3rd edn*