`Advances in Mechanical EngineeringVolume 2013 (2013), Article ID 743201, 10 pageshttp://dx.doi.org/10.1155/2013/743201`
Research Article

## Numerical Simulation of the Transient Process of Power Failure in a Mixed Pump

Institute of Process Equipment, Zhejiang University, 38 Zheda Road, Hangzhou 310027, China

Received 29 December 2012; Accepted 16 March 2013

Copyright © 2013 Xudan Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A hydraulic-force coupling method was used to simulate the transient process of power failure condition. Computational fluid dynamics (CFD) was used to study the three-dimensional (3D), unsteady, incompressible viscous flows in a mixed flow pump in power failure accident. The dynamic mesh (DM) method with nonconformal grid boundaries was applied to simulate the variation of rotational speed of the field around the impeller. User-defined function (UDF) was used to obtain the rotational speed by solving the momentum conservation equation. External characteristics, such as rotational speed, head, flow rate, and hydraulic torque, were obtained during the transient process. Numerical speed and flow rate were compared with results calculated by semiempirical equation and they were in good agreement. The differences between transient and quasisteady results were also studied. Transient head and quasisteady head did not differ too much. The reason that caused this deviation was theoretically analyzed. The difference was explained to be caused by the inertia effect of the fluid contained in the pump and the pipeline. Internal flow field was also shown. Relative velocity vectors showed that the stall form and existence time in transient simulation were different from those in the quasisteady simulation. It is suspected to be one reason for head deviation.

#### 1. Introduction

Transient process of pumps existed in various occasions. Power failure was one of the most common accidents in pump operation. The study on idle rotation of a coolant pump in a power failure accident was of great importance. For example, in a nuclear power plant, coolant pump was one of the main equipment in reactor coolant system and was the only rotating equipment. It was considered to be the heart of the nuclear power plant. The reliability and security in special processes were particularly emphasized. Power failure of a coolant pump was a serious accident. During the blackout of a nuclear power plant, the coolant pump started with an idle rotating process. The refrigerant quantity through the reactor core decreased abruptly and brought threat to the safety of reactor core components. In order to ensure the security of the reactor core, the coolant pump required a longer idle rotating period.

In the studies of power failure accidents, there were following limitations that could be improved. Firstly, the rotational speed was assumed to be changing in a hypothetical law. In fact, the rotational speed was unknown in power failure accidents. Once the physical model was set, the speed variation was predetermined. The hypothetical law was not identical with the practical situation. Because the rotational speed played an important role in pump head, it was essential to obtain a more practical rotational speed variation law. Secondly, the quasisteady curve was often used to analyze the performance of a transient process especially in theoretical studies. This was obviously convenient but with great deviations in severe transient processes. The transient and quasisteady results would be discussed in this paper.

Nowadays, with the development of CFD technology, numerical simulation was greatly developed and widely used in the design and optimization of hydraulic machineries. CFD technology had been successfully used in the transient calculation of pumps during starting and stopping periods [69]. With this method, not only the external characteristic but also the internal flow field could be obtained clearly. In this paper, a numerical method was proposed to simulate the evolution of the transient flow of power failure process in a mixed pump. The major objective of this study was to develop a hydraulic-force coupling method to get the realistic external and internal characteristics of a mixed pump in power failure condition by CFD method.

#### 2. Numerical Model

The computational model used in this paper is a circulation pipeline system. It consists of a mixed pump, a reservoir, and a circular pipe. Since it is difficult to identify the specific boundary conditions of the pump in the stopping period, a circular pipe system is used in the study. It is convenient for the self-set of boundary conditions. Water from the reservoir is pumped by the mixed pump and flowed into the reservoir through the outlet pipe. 3D model is shown in Figure 1. Pipe dimensions are indicated in Figure 1(b). The reservoir is 815 mm high, 700 mm in diameter. The volume ratio of the reservoir and the pump is about 6.

Figure 1: Pumping circulation system.

3D model of the mixed pump impeller is shown in Figure 2 and the parameters are listed in Table 1. The 3-D model is exactly the same scale with the experimental testing bench. Some steady characteristics of this pump have been tested by previous studies.

Table 1: Specifications of the pump model.
Figure 2: Geometry of the impeller and the 3-D blade model.

The model is meshed in Gambit. The grid independence and temporal test have been done in former studies [8] and here is a brief introduction. The mesh information is listed in Table 2. The impeller, inlet pipe, and volute are meshed separately. Nonconformal boundary conditions are set between the impeller and inlet pipe, the impeller and the volute. In order to decrease the deviation caused by the nonconformal boundaries, the same size and structure mesh type are applied on the interfaces. The nonconformal boundary conditions are described in Figure 3.

Table 2: Mesh information.
Figure 3: Details of dynamic mesh method.

#### 3. Computational Method

##### 3.1. Mathematical Model

The revolution of rotor followed the momentum conservation law. The total input torque of the pump would overcome the hydraulic torque to sustain the medium flow in the pump, the rotor friction torque, and inertia moment of rotor. The relationship between these parameters could be described by the following equation: where is the total input torque, is the hydraulic torque of the pump, is the rotor friction torque, is the rotary inertia of the pump system, and is the rotor angular speed.

When the pump loses power supply which leads to (2), the input torque disappears and the pump starts with inertial motion:

So (1) becomes.

Equation (3) is the torque equation which determines the rotational speed variation law during power failure accident. In (3), could be obtained by flow field computation of the transient process. Compared with friction torque , hydraulic friction constitutes a high proportion in a pump. Assuming is linear with : is an empirical factor and usually set as 0.01–0.03. In this case hydraulic friction is not considered and it is set as 0. Equation (3) can be converted into a difference schema: can be obtained by Fluent simulation. Time step and inertia of the pump system are known. The initial value of total torque can be obtained by unsteady simulation with in Fluent 6.3. Once the initial condition is given, according to (5), transient characteristic of power failure can be calculated. As time goes on, the rotational speed decreases very slowly which makes the simulation lasts very long. So the transient process is considered to be completed when the final rotational speed is smaller than . The whole procedure can be described as in Figure 4.

Figure 4: Calculating procedure.
##### 3.2. CFD Method

The simulation of transient process is taken by commercial software Fluent 6.3. Fluent is a solver based on finite volume method and contains various models. In this case of simulation, three different models [10] are used to calculate the motion of the impeller in quasisteady condition, unsteady condition, and transient condition.

In quasisteady condition, the pump works in a specific rotational speed and the flow rate is constant. Because of the simplicity of the quasisteady calculation, in previous studies, quasisteady results are usually used to replace the transient results. Since the flow field in a pump is not steady, an unsteady calculation is an improved and more realistic calculation. It can capture the unsteady characteristic in the pump operation such as rotating stall. And also, the rotational speed and flow rate are constant in unsteady calculation. In transient simulation, operating condition such as the rotational speed or the flow rate is changing from time to time. So in each time step, the condition is different.

In the quasisteady simulation, multiple reference frame (MRF) is adopted. MRF model is the simplest approach for multiple zones. When using MRF model, the grid remains fixed for the computation. The flow in moving cell zone is solved using the moving reference frame equations which contain Coriolis acceleration and centrifugal acceleration. The flow around the moving part can be modeled as a steady-state problem with respect to the moving frame.

Before the transient simulation, an unsteady case with is presimulated to get the initial condition of the power failure process. In unsteady simulation, moving mesh method is used to simulate the motion of the impeller. When a time-accurate solution for the rotor-stator interaction (rather than a time-averaged solution) is desired, the sliding mesh model is the most accurate method for simulating flows in multiple moving reference frames. The interface zones of adjacent cell zones are associated with one another to form a “mesh interface.” The two cell zones will move relative to each other along the mesh interface.

In the transient process simulation, dynamic mesh (DM) technology is used to simulate the motion of the impeller. DM method has been successfully used in transient simulation of a 2-dimensional centrifugal pump during starting period [11]. Then Li et al. [7] pushed on a further application of DM method in a 3-D centrifugal pump to simulate the transient characteristic during starting period. Liu et al. [6] used DM method to simulate the stopping transient process. And Wu et al. [9] used it to simulate transient process during the rapid opening period of the discharge valve in the pump system. In their studies, the motion of the impeller was prescribed. In this paper, the impeller motion is determined by solving the momentum equation (5). According to [7], turbulence model is adopted and the SIMPLE algorithm is used. Unsteady time step size is set as 0.0001 s. At each time step, the maximum iteration is set as 300.

#### 4. Results

##### 4.1. External Characteristics in Power Failure Condition

In the unsteady simulation, the case is considered to be in a “steady-going” state when the torque fluctuates regularly and in small amplitude. When the rotational speed is low enough, the transient process is considered to be completed.

The variation of hydraulic torque is shown in Figure 5. In Figure 5, the solid line is the change of hydraulic torque with time. Assuming that the sum of hydraulic torque and the friction torque are proportional to the rotational speed squared [12] which can be described by

Figure 5: Hydraulic torque variation in transient process.

In Figure 5, the theoretical torque calculated by Fluent is shown with dash lines. From the figure it can be found that they are in good agreement.

With (6), (3) can be written as where is constant. So the transient speed can be described by the following equation is the initial angular speed, and is the time when the instantaneous rotational speed is half of the initial speed.

Integrating (6) and substituting the initial condition, (9) can be obtained:

There are two mathematical models, (8) and (9), to simplify the transient process. In this case, the initial rotational speed is 1500 r/min. According to the simulation result, half-speed time is 0.32038 s. This theoretical result is taken as theory model 1. And in physical model calculated by (8), is 0.28224 s. It is taken as theory model 2. These three results are all shown in Figure 6. In Table 3, there are some specifically given values of speed in different times calculated both by theoretical and CFD methods. The deviation between CFD and theoretical model 1 is small while the deviation between CFD and theoretical model 2 is much larger. But in all, CFD result and the theoretical result coincide well. This result indicates that (7) describes rotational speed variation better. The difficulty is how to get a correct half-speed time . The theoretical model of (9) is convenient but the result is rough.

Table 3: Comparison between CFD and theoretical results.
Figure 6: Angular speed variation with time.

Flow rate is another specifically emphasized parameter. According to the pump similarity law, the flow rate is linear to the rotational speed and the head is proportional to the square of rotational speed. It is concluded that the flow rate changes in the same form as speed variation law. So assuming the flow rate variation can be described by (10): where is the initial flow rate. Similar to the equation of rotational speed variation, is the time when the instantaneous flow rate is half of the initial flow rate. In this case, is 0.54224 s, which is almost twice of the half-speed time. It indicates that the flow rate does not decrease as fast as the rotational speed. It can be explained by the large inertia effect of the fluid on the pump and the pipe. The simulation result and theoretical result are shown in Figure 7. At the beginning, the theoretical and the numerical results fit well. But as time goes on (when  –6 s), numerical results decrease much faster than the theoretical one. After , the deviation becomes small again.

Figure 7: Flow rate variation with time.

Figure 8 is the change of axial and radial forces with time. The axial force changes in the same form as head. Because the sealing clearance is not modeled in this case, the axial force here stands for the hydraulic force applied to the impeller shroud and hub. The radial force, which is the composition of and , decreases with rotational speed. The degree of the force fluctuates with time firstly irregularly and then fluctuates around a specific value. Since the calculation time is too short and data is not sufficient, frequency is not analyzed. But it can be deduced that the variation of the radial and axial force may result in vibration, even damage the pump.

Figure 8: Forces variation with time.
##### 4.2. Comparison between Quasisteady and Transient Result

Figure 11: Derivatives of external characteristic.

Pressure coefficient is nondimensional and stands for the pump characteristic. It is defined as follows: where is the total pressure rise and is the peripheral speed of the impeller at the blade outlet. Figure 12 is the transient and quasisteady pressure coefficient change. Although transient head and quasisteady head do not deviate too much, the pressure coefficient variation is totally different. The quasisteady pressure coefficient decreases with time as the flow rate decreases, but quite the other way the transient pressure coefficient increases. As the speed is not reflected in the pressure coefficient, the difference between the transient and quasisteady coefficients is deduced to be caused by the speed deceleration and the inertia effect of the fluid. Comparing the speed deceleration curve with pressure coefficient curve, it can be found that their absolute value changes in the same trend.

Figure 12: Comparison between quasisteady and transient coefficients.
##### 4.3. Comparison of Flow Field

Figure 13: Flow field comparison.

#### 5. Conclusion

In this paper, the transient characteristic during power failure accident in a mixed pump is studied by CFD method and theoretical methods. The rotational speed, transient head, and flow rate change law are obtained and researched in this study. The quasisteady characteristic in the same speed condition is also studied and compared with the transient result. There are the following conclusions.(1)Hydraulic-force coupling method is available for the power failure stopping process. External and internal characteristics of the pump are obtained.(2)In power failure accidents, the rotational speed decreases much faster than the flow rate. Half-capacity time is almost twice the half-speed time. (3)In pump stopping process, quasisteady head is larger than transient head. The deviation is especially distinct at the beginning of the process ( s). It is caused by the inertia effect of the fluid contained in the pump. Differences are also shown in the internal flow field.

#### Nomenclature

 : Outlet width (mm) : Constant : Inlet diameter one (mm) : Inlet diameter two (mm) : Outlet diameter one (mm) : Outlet diameter two (mm) : Force in direction (N) : Force in direction (N) : Axial force (N) : Nominal total head (m) : Quasisteady head(m) : Rotary inertia of the pump system  (kgm2) : Parameter : Parameter : Inlet length (mm) : Outlet length (mm) : Nominal speed (r/min) : Rotational speed (r/min) : Initial rotational speed (r/min) : Pressure coefficient : Total pressure rise (Pa) : Nominal flow rate  (m3/h) : Total input torque  (Nm) : Initial hydraulic torque (Nm) : Hydraulic torque of the pump (Nm) : Rotor friction torque (Nm) : Time step size (s) : Half-speed time  (s) : Half-capacity time  (s) : Peripheral speed of the impeller at the blade outlet  (m/s).
Greek Letters
 : Empirical factor : Front shroud angle (degree) : Back shroud angle (degree) : Fluid density (kg/m3) : Rotor angular speed (rad/s) : Initial angular speed (rad/s).

#### Acknowledgment

This study was carried out as a part of the National Natural Science Foundation of China (the Project no are 51276213 and 51176168). The support is gratefully acknowledged.

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