Evolution Characteristics of Energy Change Field in a Centrifugal Pump during Rapid Starting Period

Abstract

Although the energy change field in the centrifugal pump has been investigated under quasi-steady conditions (QSC), equivalent information is not yet known during the rapid starting period (RSP). A centrifugal pump loop system is constructed to investigate the energy change field in the centrifugal pump during RSP. The RSP is selected as a linear rotational speed from 0 rev/min to 2900 rev/min (design rotational speed) and a constant valve opening of 0.569. Results show that the flow rate lags behind the pump head value with the linear increase in rotational speed. The large values of partial derivations of mechanical energy in normal and tangential components are mainly concentrated at the impeller outlet, whose position is insensitive to rotational speed. The region of dominant energy loss is negatively correlated with rotational speed, and an opposite phenomenon is observed in the region of dominant energy increase. With the rotational speed increasing, the mean energy gradient function in the pump impeller and pump volute gradually increases, and the slope of the former is less than that of the latter. After reaching the design rotational speed, the energy change field gradually approaches that under quasi-steady conditions.

1. Introduction

Centrifugal pumps are general machinery with large quantities and wide ranges, and they play a vital role in aerospace, nuclear, petrochemical, biomedical, marine engineering, and other fields. Centrifugal pumps work under quasi-steady conditions (QSC) most of the time, and their flow rates and rotational speeds do not change with time. However, centrifugal pump often operates in transient conditions, including pump starting. A transient condition means that the working parameters change considerably with time, which increases the intricacy of the internal flow, including the energy change field. The energy field distribution is one of the important characteristics of the flow field inside the centrifugal pump [1,2]. During the rapid starting period (RSP) of the centrifugal pump, its working parameters change considerably with time. To investigate the energy change field of the centrifugal pump during RSP, a centrifugal pump loop system was established to construct a numerical simulation of the fast start of the centrifugal pump was conducted. Therefore, studying the energy change field in the centrifugal pumps during a rapid starting period (RSP) is essential.

Researchers have examined the behaviors of the internal flow in the centrifugal pump during the starting period. Sumio [3] investigated the transient characteristics under different conditions of tube length, valve opening, and starting time. Dazin et al. [4,5] studied the strength and duration of starting transient characteristics considering several factors, such as acceleration, rotational speed, flow rate, and pipeline length. Thanapand et al. [6,7] investigated the start-stop process of the centrifugal pumps under different valve openings. Tsukamoto et al. [8] investigated the instant operating behaviors of the rated speed through numerical calculation and experimental measurement. Lefebvre et al. [9] studied the start-stop process under three impeller starting and acceleration schemes through experiments. Chalghoum et al. [10] studied the starting time and impeller geometrical parameters effects. Elaoud et al. [11] studied the flow and pressure changes in upstream and downstream pipelines under four different starting times. Meanwhile, Grover et al. [12] investigated the effects of rotor inertia and in-pipe fluid inertia on the starting process through a torque balance equation. Based on the Tehran reactor pump, the transient behavior has been studied by Farhadi et al. [13]. To achieve the self-coupling simulation of the centrifugal pump, a closed-loop system is established by Zhang et al. [14]. Moreover, considering the pump start and stop period, they investigated the influences of viscosity on the internal flow [15]. Dai et al. [16] proposed three novel transcritical CO₂ high-temperature heat pump systems to recover waste heat. Besides, in recent years, research on the starting of a multistage centrifugal pump [17] and a centrifugal pump with assisted valve [18] has also made some progress.

These studies did not conduct their analyses from the perspective of the energy field. Some reports in this field have been based on QSC. Several methods can be adopted to study energy losses [19,20,21,22], and among them, entropy production theory is a crucial method to study fluid machinery [23,24,25]. Dou [26] introduced an energy gradient method to express the ratio between energy increase and energy loss, and it has been utilized in several flows [27,28,29,30,31,32]. Recently, Chen et al. [33] investigated the energy change field in the centrifugal pump impeller and proposed the concepts of dominant energy increase (DEI) and dominant energy loss (DEL), whose phenomenon is obviously different between a quarter and design loads. However, no research on the energy change field in the centrifugal pump during RSP has been reported in the public literature. To reveal this mechanism, the present study takes a centrifugal pump loop system as the research object first. Then, we perform a numerical simulation of the research object to investigate the energy change field, whose rotational speed is a linear function of time and whose valve opening is constant. Six numerical simulations under QSC are also carried out to study the RSP effects.

The paper is structured as follows. The evaluation method of the energy change field is reviewed in Section 2. Details regarding numerical considerations are provided in Section 3. In Section 4, the numerical simulation results are presented and the energy change fields of the centrifugal pump during RSP and under QSC are compared. Finally, we sum up our dissertation concludes in Section 5.

2. Evaluation Method of the Energy Change Field

An energy gradient function, a ratio of energy increase and energy loss, is introduced by Dou [26] to analyze the instability of flow and elucidate the mechanism of shifting laminar and turbulent flow. Recently, Chen et al. [33] proposed to characterize the proportion of energy increase and energy loss based on the energy gradient function. The calculation method is as follows: An energy gradient function, a ratio of energy increase and energy loss, is introduced by Dou [26] to analyze the instability of flow and elucidate the mechanism of shifting laminar and turbulent flow. Recently, Chen et al. [33] proposed to characterize the proportion of energy increase and energy loss based on the energy gradient function. The calculation method is as follows:

(1)$K=\frac{\sqrt{u_i^2}\sqrt{{\frac{\partial E}{\partial n}|}_i^2}}{\sqrt{u_i^2}\sqrt{{\frac{\partial E}{\partial s}|}_i^2}+\varphi }$

where u_i is the velocity vector. ∂/∂n and ∂/∂s are partial derivations in the normal and tangential components, respectively. ϕ is the rate of energy dissipation, and E is the total mechanical energy, which can be described as follows:

(2)$\varphi =\frac{1}{2}\mu {\left(\frac{\partial u_j}{\partial x_i}+\frac{\partial u_i}{\partial x_j}\right)}^2, E=p+\frac{1}{2}\rho u^2$

where μ is the viscosity. ρ is the density. p is the static pressure.

Note that, when its threshold equals 1.0, the energy increase and energy loss are mutually balanced. When the threshold value is greater than 1.0, the energy increase is greater than the energy loss, and when the threshold value is less than 1.0, the energy increase is smaller than the energy loss. Chen et al. [33] proposed to use the logarithm of the energy gradient function (LEGF, ${\log }_{10}K$) to represent the energy change results. They also found, ${\log }_{10}K\ge 0.3$ ($K\ge 2.0$) means that when the energy increase is at least twice the energy loss, the local energy increase dominates, namely dominant energy increase (DEI). Similarly, ${\log }_{10}K\le -0.3$ ($K\le 0.5$), indicates that the local energy loss dominates, namely dominant energy loss (DEL).

3. Geometric Model and Numerical Considerations

3.1. Geometric Model and Mesh Generation

Some cylindrical pipes are used to connect the centrifugal pump, valve, and cistern. The diameter of the upstream cylindrical pipes of the centrifugal pump is 0.008 m². The diameter of the downstream cylindrical pipes of the centrifugal pump is 0.005 m². The geometric models of whole loop system, valve, and centrifugal pump are shown in Figure 1, and the main parameters of the centrifugal pump are shown in Table 1.

The pump flow rate is the design value when the valve opening is 56.9% under the quasi-steady condition. To isolate the influence of rotational speed, the valve opening is maintained at 56.9%, and the rotational speed is set as a linear function. The relationship between rotational speed and time is shown in Figure 2, and its mathematical expression is as follows.

(3)$n=\{\begin{array}{l}\frac{n_d}{12t_d}t,\left(t\le 12t_d\right)\\ n_d,(t>12t_d)\end{array}$

where t denotes time.

To investigate the RSP effects, six numerical simulations of the centrifugal pump loop system under QSC are also performed. Based on the constant valve opening, the rotational speeds in QSC are also constant, including $n_d/6$, $2n_d/6$, $3n_d/6$, $4n_d/6$, $5n_d/6$, and $n_d$.

The computational domain of the centrifugal pump loop system is divided by the idea of a regional hexahedron structured mesh used by ANSYS ICEM-CFD 18.0. During the structural mesh generation, the number of interfaces in the computational domain should be reduced, and the number of mesh nodes on both sides of the interface should be consistent or similar. The fluid domain mesh near the wall area should be encrypted to ensure that the accurate flow structure can be captured. Figure 3 shows the meshes of the loop system, including the centrifugal pump, valve, cistern, and cylindrical pipes. The mesh independence of the centrifugal pump has been verified by Zheng et al. [34], and the adopted number is 6.59 million. In addition, the valve has 1.2 million meshes, the cistern has 2.98 million, and the pipeline has 4.35 million. Thus, the loop system has 14.43 million meshes, as presented in Table 2.

3.2. Numerical Approach and CFD Validation

ANSYS Fluent 18.0 is used to simulate the flow field in the centrifugal pump loop system. The fluid domain is solved by the finite volume method. The turbulent model is shear stress transport k–ω with a standard wall function. The coupling of velocity and pressure is achieved by the SIMPLE algorithm. The convective term is discretized by a 2nd upwind scheme. The viscous terms are approximated with a 2nd central difference scheme.

The no-slip boundary condition is adopted for the wall, and the pressure condition is adopted for the upper cistern. The wall related to the impeller is the rotating wall, and the others are static walls. The initial field is static fluid. The time step equals the value when the impeller rotates by 1.5° under the design rotational speed, i.e., Δt = 8.62 × 10⁻⁵ s.

Since several numerical simulations and experimental measurements have been studied under QSC, the data validation based on these conditions is used to verify the result reliability. As shown in Figure 4, the numerical results of the performance curve, including pump head and efficiency, show agree well with the experimental data, whose relative errors are less than 5.4% and 2.9% under the design load, respectively. The distribution of pressure shown in Figure 5 also demonstrates the numerical results are also reliable. The pressure coefficient [35] is defined as follows:

(4)$c_p=\frac{p-p_{ref}}{0.5\rho u^2}$

where p_ref is the reference pressure, and u is the circumferential velocity of the impeller.

4. Results and Discussions

4.1. Pressure and Velocity Field

Figure 6 shows the evolution of the flow rate and pump head value. As can be seen on the left of the figure, the evolution of the pump head and flow rate value is nonlinear when the rotational speed linearly increases. The flow rate lags behind the pump head value. After the impeller rotational speed reaches its design value (i.e., instantaneous time equals 12t_d), the flow rate and pump head value gradually approach 45.4 m³/h (i.e., instantaneous time is approximately equal to 14t_d) and 29.7 m (i.e., instantaneous time is approximately equal to 20t_d), respectively, which are approximately equal to the results under QSC, as shown in Figure 6 (right).

Figure 7 shows the distributions of static pressure on the center plane of the centrifugal pump during RSP. Static pressure increases as the axial position increases, and the change of static pressure is relatively flat at the instantaneous rotational speed of n_d/6. With the rotational speed increasing, the magnitude of static pressure increases dramatically. A similar trend can be observed from the results under QSC shown in Figure 8.

The distributions of streamline are also shown in Figure 7 and Figure 8. Note that the streamline during RSP is more chaotic than that under QSC, especially in the low rotational speed.

4.2. Partial Derivation of Mechanical Energy

According to the description in Section 2, the partial derivation of mechanical energy in the normal (∂E/∂n) and tangential (∂E/∂s) components are two important physical quantities to determine the energy change field. This section mainly discusses the evolution characteristics of the two physical quantities during RSP. They are displayed in the logarithmic form to be intuitive and clear.

Figure 9 shows the distributions of log₁₀|∂E/∂n| on the center plane of the centrifugal pump during RSP. The magnitude of log₁₀|∂E/∂n| gradually increases with the rotational speed increasing. The larger value is mainly concentrated near the impeller outlet, whose position is basically independent of rotational speed. At the same time, the value near the pump volute is smaller than that in the impeller passage.

Although a broadly similar distribution trend can also be observed from log₁₀|∂E/∂n|, some different distributions can still be observed, as shown in Figure 10. For example, compared with the distribution of log₁₀|∂E/∂n|, the distribution of log₁₀|∂E/∂s| is more detailed. The growth rate of log₁₀|∂E/∂s| in the pump impeller is smaller than that of log₁₀|∂E/∂n|, especially at the initial starting stage of n_d/6.

The evolution characteristics of partial deviations can be further investigated by using the mass-weighted average of partial deviations, which is called mean partial deviations, as shown in Figure 11. Note that, with the increase of the rotational speed, the two mean partial deviations increase, including log₁₀|∂E/∂n| and log₁₀|∂E/∂s|. These magnitudes in the impeller passage are larger than that near the pump volute. The magnitude of the mean log₁₀|∂E/∂n| is larger than that of the mean log₁₀|∂E/∂s| either near the pump volute or in the impeller passage.

Although a similar trend can be observed in the mean partial derivation during RSP and under QSC, there are still several differences. First, the mean partial derivations for the former are larger than that for the latter in the pump impeller. However, an opposite phenomenon is observed in the region near the pump volute, except when the instantaneous rotational speed is close to its design value. Second, the growth rate during RSP is remarkably larger than that under QSC, except for the mean partial derivations of log₁₀|∂E/∂s| in the pump impeller.

4.3. Energy Change Field

Figure 12 shows the distributions of LEGF on the center plane of the centrifugal pump at different instantaneous rotational speeds during RSP. When the instantaneous rotational speed equals n_d/6, the region of log₁₀ K ≥ 0.3 is significantly smaller than that of log₁₀ K ≤ −0.3, especially in the pump outlet pipe and the impeller passage. With the increase of rotational speed, the regions of log₁₀ K ≤ −0.3 and log₁₀ K ≥ 0.3 are decreased and increased, respectively. When the instantaneous rotational speed is larger than 4n_d/6, the region of log₁₀ K ≤ −0.3 is smaller than that of log₁₀ K ≥ 0.3. It should be emphasized that the regions of log₁₀ K ≤ −0.3 and log₁₀ K ≥ 0.3 are presented because they can describe the dominant energy loss and dominant energy increase, respectively, as mentioned in Section 2. Therefore, the region of dominant energy loss is remarkably larger than that of dominant energy increase at the instantaneous rotational speed of n_d/6, especially in the volute outlet and the impeller passage. With the rotational speed increasing, the decrease and increase trends can be observed in the regions of DEI and DEL, respectively.

The evolution characteristics of LEGF can be further investigated by the mean LEGF, as shown in Figure 13. Note that the magnitudes of the mean LEGF increases with the increase of the impeller rotational speed, especially when the instantaneous rotational speed is less than 4n_d/6. The mean LEGF near the pump volute is less than that in the impeller passage. However, when the impeller speed reaches 4n_d/6, the mean LEGF near the pump volute exceeds that in the pump impeller.

For comparison, the distributions of LEGF and the mean LEGF under QSC are also shown in Figure 13 and Figure 14. Note that their profiles are similar for different rotational speeds. The mean LEGF is insensitive to the rotational speed, which is approximately 0.15 in the pump impeller and 0.45 near the pump volute, respectively. Therefore, these significant differences indicate that RSP has an important effect on the energy change field.

Figure 15 shows the relationship between mean LEGF, and time during RSP after the impeller rotational speed reaches its design value. Note that the LEGF’s fluctuation near the pump volute is larger than that in the pump impeller, whereas they are all relatively insensitive to the time and approaches to the corresponding value under QSC. Meanwhile, an extreme value is observed when the time is at 18t_d, especially near the pump volute.

5. Conclusions

We set up a centrifugal pump loop system, that includes a centrifugal pump, a valve, a cistern, and cylindrical pipes. A numerical simulation of the loop system with a constant valve opening of 0.569 and a linear rotational speed (i.e., rapid starting period, RSP) is performed to study the energy change characteristics in the centrifugal pump. The linear rotational speed is described as its increasing from 0 rev/min to 2900 rev/min (design rotational speed) in 0.248 s. The reliability of the numerical results is verified by comparing them with experimental data.

Compared with the results under quasi-steady conditions (QSC), the static pressure is higher and the streamline is more chaotic. With the increase in rotational speed, the flow rate lags behind the pump head.

The larger values of partial derivations of mechanical energy, including its normal component and tangential component, are mainly concentrated at the impeller outlet. The evolution trend of the partial derivations is similar, which are increasing with the rotational speed increased. The mean partial deviations in the pump impeller during RSP are larger than that under QSC. An opposite phenomenon is observed near the pump volute, except when the rotational speed is close to its design value. Moreover, the slope of the mean partial deviation during RSP is larger than that under QSC.

With regard to the energy change field, the region of dominant energy loss (DEL) is remarkably larger than the region of dominant energy increase (DEI) at the instantaneous rotational speed of n_d/6, especially in the impeller passage and volute outlet. As the impeller rotational speed increased, the decrease and increase trends can be observed in the regions of DEI and DEL, respectively, leading to a sharp increase in the mean energy gradient function. Meanwhile, although the function both in the pump impeller and pump volute is increased, the slope of the former is less than that of the latter. After reaching the design rotational speed, the mean function gradually approaches the corresponding value under QSC, which is basically maintained at 0.15 in the pump impeller and 0.45 near the pump volute.

Therefore, the evolution characteristics of the energy change field in the centrifugal pump during RSP are significantly different from that under QSC. In the future, we will not only study the valve opening effects on the energy change field, but also investigate the corresponding characteristics during RSP.

Author Contributions

Conceptualization, X.C.; methodology definition, X.C. and X.Z.; software, X.L.; Validation, X.Z.; formal analysis, X.C. and X.Z.; investigation, X.C. and X.Z.; writing—original draft preparation, X.C. and X.Z.; writing—review and editing, X.C., X.Z. and X.L.; supervision, X.C., X.Z. and X.L.; project administration, X.C. and X.L.; funding acquisition, X.C. and X.L. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

The Authors would like to acknowledge the precious contribution to this work given by Zuchao Zhu from the Zhejiang Sci-Tech University and Lulu Zheng from Henan Institute of Technology for their extensive experimental activity supporting the research of a centrifugal pump during the rapid starting period.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

RSP	rapid starting period
QSC	quasi-steady conditions
DEI	dominant energy increase
DEL	dominant energy loss
LEGF	logarithm of energy gradient function
cp	pressure coefficient (-)
D	impeller inlet diameter (mm)
E	total mechanical energy (Pa)
H	head (m)
K	energy gradient (-)
n	rotational speed (r/min)
ns	specific speed (-)
p	static pressure (Pa)
Q	flow rate (m3/h)
t	time (s)
u 1	circumferential velocity of the impeller (m/s)
Z	number of blades (-)
∂/∂n	partial derivations in the normal components (-)
∂/∂s	partial derivations in the tangential components (-)
ϕ	rate of energy dissipation (-)
μ	viscosity (Pa·s)
ρ	density (kg/m3)
η	efficiency (-)
d	design
ref	reference

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Figures and Tables

Figure 1. Geometric models of (a) whole loop system, (b) valve, and (c) centrifugal pump.

Figure 2. Relationship between rotational speed and time.

Figure 3. Meshes of the centrifugal pump loop system.

Figure 4. Pump head (left) and efficiency (right) curves for different flow rates under the QSC [34,35].

Figure 5. Pressure on the center plane at design load. (left) Pump loop system and (right) single pump [35].

Figure 6. Evolution of the pump head and flow rate during RSP (left) and under QSC (right).

Figure 7. Distributions of streamlines and static pressure on the center plane during RSP.

Figure 8. Distributions of streamline and static pressure on the center plane under QSC.

Figure 9. Distributions of log10|∂E/∂n| on the center plane during RSP.

Figure 9. Distributions of log10|∂E/∂n| on the center plane during RSP.

Figure 10. Distributions of log10|∂E/∂s| on the center plane during RSP.

Figure 11. Mean partial derivation of mechanical energy in the normal component (left) and tangential component (right).

Figure 12. Distributions of LEGF on the center plane during RSP.

Figure 12. Distributions of LEGF on the center plane during RSP.

Figure 13. Mean LEGF at different rotational speeds.

Figure 14. Distributions of LEGF on the center plane under QSC.

Figure 15. Relation between mean LEGF and time during the RSP after the rotational speed reaches its design value.

Table 1

Main parameters of the centrifugal pump.

Parameter	Notation	Value	Unit
Number of blades	Z	6	-
Impeller inlet diameter	D	86	mm
Design rotational speed	nd	2900	r/min
Design flow rate	Qd	45	m3/h
Specific speed	ns	90	-
Design head	Hd	30.9	m
Design efficiency	η	0.645	-

Table 2

Number of the system meshes.

Subdomain	Mesh Quantity (Ten Thousand)
Centrifugal pump	659
Valve	12
Cistern	298
Upstream pipeline of the centrifugal pump	177
Downstream pipeline of the centrifugal pump	265
Pipe between valve and cistern	32
Overall system	1443

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