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

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.

Lots of researches on transient characteristic during stopping process had been done by numerical, theoretical, and experimental methods. Tsukamoto et al. [1] used both the experimental and the theoretical methods to study the characteristic of a radial pump during stopping period. Their study showed that when the deceleration rate of the pressure and flow rate exceeded a certain limit, the flow deviated greatly from the result of steady state. Kittredge and Princeton [2] studied the transient process during power failure condition by an analytical method. They discussed the methods of integrating equations for an assumed rigid fluid column and elastic fluid column. Zhang [3] built up a set of equations of coolant pump and pipe system by moment conservation law. He used these equations to study the head and flow rate change law during idle rotation. In their study, the speed variation was prescribed and the transient characteristic of the pump was obtained by the steady-state head curve. Thanapandi and Prasad [4] carried out a theoretical and experimental study on the transient characteristics of a centrifugal pump during both starting and stopping periods. In his study, the dynamic characteristic of the test pump had been analyzed by a numerical model using the method of characteristics. Their research showed that the transient head characteristics closely followed the steady-state system head curve. This was a rough result because only the centrifugal action was considered in their pump model. And also in their research, the instantaneous speed was provided as known. Liu et al. [5] studied the transient characteristic of coolant pump idle rotation by experiment. Liu et al. [6] studied the internal and external characteristics in a radial pump during stopping period by computational fluid dynamics (CFD) method, and they revealed the flow field under both transient and quasisteady conditions. Their research also showed that the transient characteristic was different from the quasisteady result.

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 [6 –9]. 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.

Geometric specifications			Hydraulic specifications

Outlet width (mm)	b ₂	66	Nominal speed (r/min)	N ₀	1450
Inlet diameter (mm)	D _i1	45	Nominal flow rate (m³/h)	Q ₀	360
Inlet diameter (mm)	D _i2	187	Nominal total head (m)	H ₀	8
Outlet diameter (mm)	D _o1	175
Outlet diameter (mm)	D _o2	238
Inlet length (mm)	L ₁	52
Outlet length (mm)	L ₂	108
Front shroud angle (deg)	ε₁	34
Back shroud angle (deg)	ε₂	32

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.

Mesh type	Tetrahedron		Hexahedron	Total

Regions	Volute	Impeller	Reservoir and pipe
Mesh numbers	444245	470427	909335	1824007
Node numbers	87226	92735	468796	648758

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:

T = T_{h} + T_{f} + I \frac{d ω}{d t},

(1)

where T is the total input torque, T_h is the hydraulic torque of the pump, T_f is the rotor friction torque, I is the rotary inertia of the pump system, and ω 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:

T = 0 .

(2)

So (1) becomes.

T_{h} + T_{f} + I \frac{d ω}{d t} = 0 .

(3)

Equation (3) is the torque equation which determines the rotational speed variation law during power failure accident. In (3), T_hcould be obtained by flow field computation of the transient process. Compared with friction torque T_f, hydraulic friction T_h constitutes a high proportion in a pump. Assuming T_f is linear with T_h:

T_{f} = α T_{h},

(4)

α 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:

ω (t + Δ t) = ω (t) + Δ t \frac{T_{h}}{I} .

(5)

T_h can be obtained by Fluent simulation. Time step Δt and inertia of the pump system are known. The initial value of total torque can be obtained by unsteady simulation with n = n₀ 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 0.02n₀. 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 n = n₀ 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], k-ε 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. Assumingthat the sum of hydraulic torque and the friction torque are proportional to the rotational speed squared [12] which can be described by

T_{h} = {(\frac{n (t)}{n_{0}})}^{2} T_{0} .

(6)

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

c ω {(t)}^{2} + I \frac{d ω}{d t} = 0,

(7)

where c is constant. So the transient speed can be described by the following equation

ω (t) = \frac{ω_{0}}{1 + t / t_{p n}} .

(8)

ω₀ is the initial angular speed, and t_pn 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:

ω (t) = \frac{ω_{0}}{1 + (T_{h} / I ω_{0}) t} .

(9)

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 t_pn is 0.32038 s. This theoretical result is taken as theory model 1. And in physical model calculated by (8), t_pn 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 t_pn. The theoretical model of (9) is convenient but the result is rough.

Table 3:

Comparison between CFD and theoretical results.

	Time (s)
	0	0.1	0.2	0.3	0.4	0.5

CFD result (r/min)	1500	1141.2	923.7	774.9	648.5	578.1
Model 1 (r/min)	1500	1143.2	923.5	774.6	667.1	585.8
Model 2 (r/min)	1500	1107.6	877.9	727.1	620.5	541.2

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):

Q (t) = \frac{Q_{0}}{1 + t / t_{p q}},

(10)

where Q₀ is the initial flow rate. Similar to the equation of rotational speed variation, t_pq is the time when the instantaneous flow rate is half of the initial flow rate. In this case, t_pq 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 t = 1 s –6 s), numerical results decrease much faster than the theoretical one. After t = 6 s, 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 F_z 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 F_x and F_y, 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

Since the quasisteady head is usually used to substitute for the transient head, in this paper a comparison is also done between the quasisteady and transient results. Experiment had been done to get the head curve of the mixed pump [13]. According to the similarity law, the steady head is a function of rotational speed and flow rate. Figure 9 is the head curve in different rotational speed. Quasisteady head can be obtained by

H_{s} (t) = - 41.96 Q^{2} - 57.44 Q \frac{n (t)}{1500} + 12.68 {(\frac{n (t)}{1500})}^{2} .

(11)

Figure 9:

Steady characteristic.

Figure 10 is the comparison between quasisteady head and transient head variation with flow rate and time, respectively. In Figure 10, it is clearly showed that the quasisteady head is larger than the transient head especially in large capacity (Q = 0.2–0.4 m³/s). So at the beginning of power failure accidents (about t < 0.3 s), quasisteady head and transient head differ greatly. As time goes on, the deviation is getting smaller. The different results between transient and quasisteady results could be explained by Dazin's research [14]. Dazin built a mathematical model to calculate the transient head of a radial pump and applied it to the starting process. According to his research, the transient head can be calculated by the following equation:

H_{t} (t) = H_{s} (t) + K_{ω} \frac{\partial ω}{\partial t} - K_{Q} \frac{\partial Q}{\partial t} .

(12)

Figure 10:

Comparison between quasisteady, and transient head.

In (12), K_ω and K_Q are constant parameters determined by the pump dimensions. The equation shows that compared with the steady head, transient head has extra components of speed acceleration term and inertia term which is related to the flow rate acceleration. The speed acceleration and fluid inertia contented in the impeller and volute will affect the transient total head. In power failure process, speed deceleration will decrease the transient total head, while the flow rate deceleration will increase the transient total head. Obviously, these two factors are not considered in quasisteady results. This is the reason for the deviation between the quasisteady and transient heads. The speed acceleration and flow rate acceleration with time are shown in Figure 11. From Figure 11, it can be found that in the transient process, the speed decreases abruptly and makes very large speed deceleration, while flow rate does not change so fast. This could also be explained by the similarity law. So the decrease by speed deceleration is larger than the increment by flow rate deceleration. This might explain why transient total head is smaller than the quasisteady head. In addition, because the transient head is related to the flow rate and the inertia effect of the fluid, it can be concluded that as the size of the quasisteady pump increases, the transient head and the quasisteady head curve would deviate much larger. And also because of the inertia effect, with a longer pipeline, the transient behavior of the pump would be affected and cause other problems like water hammer vibration.

Figure 11:

Derivatives of external characteristic.

Pressure coefficient is nondimensional and stands for the pump characteristic. It is defined as follows:

P_{coeff} = \frac{2 P}{[ρ u_{2}^{2}]},

(13)

where P is the total pressure rise and u₂ 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

The quasisteady conditions listed in Table 3 are also simulated by CFD method with MRF model to get the internal flow field. Figure 13 is the comparison of the relative velocity vector in the impeller at different times. The reference frame is rotating at the same rotational speed with the blade. Although the quasisteady head and the transient head do not differ too much, distinctive differences are shown in the flow field. In the transient calculation, there is a stall at the beginning of the power failure condition. As time goes on, the stall declines and disappears fast (when t = 0.2 s). But the relative speed is still small at the outlet suction side. After t = 0.3 s, the flow is uniform and there is no stall. In quasisteady flow field, the stall exists in each of the impeller channels by the suction side all the time. On the other hand, the stall form in the transient simulation is also different from that appeared in the quasisteady simulation. It indicates that the reverse pressure gradients appeared in the suction side that are different. In transient case (Figure 13 (a1)), the stall appeases like flow separation. Maybe the fluid is going in a most-energy-saving way. In quasisteady simulation (Figure 13 (a2)), the stall is shown as backflow. Besides, at the same rotational speed, flow rate in transient case is larger than that in quasisteady case. This is caused by inertia effect. Difference of the internal flow field is suspected to be one of the reasons for the deviation of transient head and quasisteady head.

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.

Hydraulic-force coupling method is available for the power failure stopping process. External and internal characteristics of the pump are obtained.

In power failure accidents, the rotational speed decreases much faster than the flow rate. Half-capacity time is almost twice the half-speed time.

In pump stopping process, quasisteady head is larger than transient head. The deviation is especially distinct at the beginning of the process (t < 0.3 s). It is caused by the inertia effect of the fluid contained in the pump. Differences are also shown in the internal flow field.

Footnotes

Nomenclature

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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