Investigation on the power loss and radial force characteristics of pump as turbine under gas–liquid two-phase condition

Abstract

To study the power loss and radial force characteristics of pump as hydraulic turbine (PAT) under gas–liquid two-phase condition, a single-stage centrifugal pump in reverse mode as turbine was adopted. Based on the Navier–Stokes equation and standard k-epsilon turbulence model, CFD (computational fluid dynamics) technology was used to simulate the flow field in hydraulic turbine. The results show that the gas content has a serious effect on the hydraulic turbine performance. Under the gas–liquid two-phase condition, the fluid velocity distribution in turbine is uneven, and the power loss is not uniform enough when the gas content is less than 0.1. The radial force vector distribution all closed to ellipse shape for different gas content and flow rates, but near the volume tongue an obvious wave appeared. The more the flow rate is, the larger the radial force is. The radial force does not change as regular as that of pure liquid condition, serious fluctuation appeared. For different gas contents, the dominant frequency always equals the rotating frequency of impeller, and subordinate dominant frequency does not appear. Under low gas content and high flow rate, both the fluctuation amplitude and dominant frequency are the maximal.

Introduction

As a key energy saving technique, an energy recovery device for high pressure fluids, the reverse rotating centrifugal pump as hydraulic turbine (PAT) is widely applied in the industries.^1–5 When the pump as turbine recycles chemical fluids, a certain amount of gas mixing is inevitably contained in the high pressure fluids. It is important that how much the content of indissoluble gas contained in the liquid is, which impinges huge impacts on the recovery efficiency and operation stability. For instance, the gas content in high pressure fluid can reach a high level of 20% in the process of desulfuration and decarburization technique. ⁶ If the gas content is completely neglected, the predicted results of hydraulic performance are out of reality. Therefore, the factual situation in hydraulic turbine should be considered as gas–liquid two-phase flow to get more reliable and acceptable performance prediction. Up to now, overwhelming majority of the documents on PAT are obtained under pure liquid medium,^7–12 while the documents on pump as turbine for gas–liquid flow are relatively fewer and under development, which only found some investigations of inner field, steady hydraulic force, pressure fluctuations,^13,14 and so on, which are far from the study of gas–liquid hydraulic performance in PAT. While energy conversion characteristics and radial force are two important indicators for turbine research, which have a great impact on turbine efficiency and operation stability. In order to comprehensively investigate the hydraulic performance of gas–liquid flow in PAT, the CFD (computational fluid dynamics) techniques are employed to systematically analyze the inner field, power loss process, and transient radial force based on previous research productions on hydraulic turbine operating with pure liquids, which can provide helpful theoretic supports and engineering guidance for further study on gas–liquid two-phase flow in centrifugal pump as turbine machinery.

Research model

A single-stage centrifugal pump as turbine with 84.5 specific speed is selected as research model, and the design flow rate is 170 m³/h, the head is 32.5 m, the rotating speed is 1450 RPM. The main geometrical parameters are listed in Table 1. When the pump is used as turbine, the outlet of pump is the turbine’s inlet, and the inlet of pump is the turbine’s outlet. The three-dimensional geometry of PAT is built by Pro/Engineering. The three-dimensional modeling and meshing diagram of pump as turbine are shown in Figure 1. The grids are generated by ICEM CFD software. A grid independent test of turbine model’s performance was performed. It is found that when the mesh numbers are more than 1.1 million, the variation of efficiency is within 0.3% as is shown in Figure 2. The final total mesh numbers are determined as 1,140,000. The whole computational domain of PAT is constituted of volute, impeller, and draft tube. The outlet is extended to a certain extent for the fully developed flow in PAT.

Table 1.

Main geometric parameters of pump as turbine.

Impeller							Volute
D1 (mm)	D2 (mm)	${\mathrm{b}}_2$ (mm)	$\phi$ (°)	${\beta }_2$ (°)	${\beta }_1$ (°)	Z	D3 (mm)	${\mathrm{b}}_3$ (mm)	D (mm)
132	328	18	105	29.5	36	6	340	44	100

Figure 1.

Three-dimensional modeling and meshing diagram of pump as turbine: (a) Three-dimensional modeling; (b) Meshing diagram.

Figure 2.

Variation of hydraulic efficiency with mesh number.

Research methods

The basic assumptions for gas–liquid flow in PAT

There are strong interactions between phase interfaces in the process of gas–liquid flow. The factual flow in PAT is much complex turbulent flow because the gas–liquid flow pattern is still unclear and the complicated passage structure. Therefore, the basic assumptions have to be introduced when analyzing the two-phase flow characteristics in PAT.

1. The high-pressure mixture fluids are homogeneous, which means the gas is uniformly distributed in liquid medium.

2. The gas is ideal, which satisfies the state equation of ideal gas.

3. The duration of mixed fluids flow in PAT is very short, which means the heat exchange with outside can be neglected.

4. The change phase of gas expansion or condensation belongs to isothermal process, which means the inner energy variations are also neglected.

Numerical calculations of PAT

Calculation period and time step set-up

In order to capture the transient characteristics of gas–liquid flow in PAT, the unsteady calculations are employed herein with different gas fractions. The steady convergence results are set as initial value of unsteady calculation. The rotation speed of PAT is 1450 RPM. Then the period T for one cycle is 0.041379 s. The time step is selected as 0.0003448 s for the impeller rotating three degrees, namely, the calculation experienced 120 steps in one cycle.

Turbulent model and boundary condition set-up

The mixture multiphase model is chosen as two-phase flow model in PAT according to the numerical calculation references of gas–liquid flow in centrifugal pump. The time-averaged N-S equations are closed by the standard k-ε turbulence model. The momentum and continuity equations are decoupled and solved by SIMPLEC algorithm. The main forms of momentum and continuity equation are written as follows: ¹⁵

Continuity equations

$$\frac{\partial }{\partial t}\left({\rho }_m\right)+\nabla \cdot \left({\rho }_m{\stackrel{\rightharpoonup }{v}}_m\right)=0$$

(1)

where ${\stackrel{\rightharpoonup }{v}}_m$ is the mass-averaged velocity, which can be calculated as

$${\stackrel{\rightharpoonup }{v}}_m=\frac{\sum _{k=1}^n{\alpha }_k{\rho }_k{\stackrel{\rightharpoonup }{v}}_k}{{\rho }_m}$$

(2)

${\rho }_m$ represents the mixture density and is written as

$${\rho }_m=\sum _{k=1}^n{\alpha }_k{\rho }_k$$

(3)

${\alpha }_k$ represents the volume fraction of $k\mathrm{th}$ phase component.

Momentum equations

$$\begin{array}{l}\frac{\partial }{\partial t}\left({\rho }_m{\stackrel{\rightharpoonup }{v}}_m\right)+\nabla \cdot \left({\rho }_m{\stackrel{\rightharpoonup }{v}}_m{\stackrel{\rightharpoonup }{v}}_m\right)=-\nabla p+\nabla \cdot [{\mu }_m\left(\nabla {\stackrel{\rightharpoonup }{v}}_m+\nabla {\stackrel{\rightharpoonup }{v}}_m^T\right)]\\ +{\rho }_m\stackrel{\rightharpoonup }{\mathrm{g}}+F+\nabla \cdot \left(\sum _{k=1}^n{\alpha }_k{\rho }_k{\stackrel{\rightharpoonup }{v}}_{dr,k,}{\stackrel{\rightharpoonup }{v}}_{dr,k}\right)\end{array}$$

(4)

where n is the phase number, $\mathrm{F}$ is the body force. ${\mu }_m$ is the mixture viscosity and can be expressed as

$${\mu }_m=\sum _{k=1}^n{\alpha }_k{\mu }_k$$

(5)

The ${\stackrel{\rightharpoonup }{v}}_{dr,k}$ represents the slip velocity of second phase, the form of which is written as

$${\stackrel{\rightharpoonup }{v}}_{dr,k}={\stackrel{\rightharpoonup }{v}}_k-{\stackrel{\rightharpoonup }{v}}_m$$

(6)

The sliding mesh technology is employed to handle the RSI (rotor–stator interaction) coupled in PAT for the unsteady calculation. Pure water and ideal gas are selected as working medium in PAT. The mass flow boundary condition is given at the inlet of PAT, and it is assumed that the gas distributes uniformly. The gas density can be calculated according to the following state equation of ideal gas

$$\mathrm{p}=\rho {\mathrm{R}}_{\mathrm{g}}\mathrm{T}$$

(7)

where ${\mathrm{R}}_{\mathrm{g}}$ is Clapeyron constant. Basically, among high-pressure fluids, the maximum volume fraction of gas is 20% in the process of desulfuration and decarburization in chemical industry. Herein, the volume fraction of gas ${\alpha }_k$ is set as 0.05, 0.1, 0.15, and 0.2, respectively.

The mixture mass flow rate of water and ideal gas is adjusted by changing the gas volume fraction at the inlet of PAT. Usually, to prevent the hydraulic turbine from getting cavitated, the 0.4∼0.6 MPa pressure is reserved at the outlet of PAT, therefore, the pressure of 0.5 Mpa is given at the outlet of PAT. The no-slip wall is given as the wall boundary condition. And the standard wall function is adopted near the wall. The residual target of convergence is set as 10⁻⁵. The pure water is given as the main phase, and the ideal gas is the second phase. There is no process of phase change and mass transfer between pure water and ideal gas.

Results and discussion

The external characteristic of PAT

In order to validate the rationality and accuracy of numerical calculation, the external characteristic results of experiment and numerical calculation are compared through hydraulic performance test. The comparisons for the case of gas volume fraction ${\alpha }_k$  = 0.2 are shown in Figures 3 and 4.

Figure 3.

Test bench for pump as turbine.

Figure 4.

Comparison between test and numerical performance curves of pump as turbine.

The efficiency was calculated as follows

$$\eta =\frac{\mathrm{M}\omega }{{\rho }_m\mathrm{gQH}}$$

(8)

where $\mathrm{Q}$ and ${\rho }_m$ , respectively, represents the flow rate and density of mixed fluid, $\mathrm{H}$ is the head, $\mathrm{M}$ is the torque, and ω is the rotating angular velocity of impeller.

Obviously, it is shown that the predicted $\mathrm{H}$ and $\eta$ performs same trend as experimental counterpart. Admittedly, the simulation results overestimate the efficiency of PAT for the reasons of the neglect of energy loss caused by seal ring clearance, mechanical seal, and bearing fractions. The predicted head and efficiency error at best point is varied with 4.81% and 4.11%, respectively, which is an acceptable error range for the simulations. Therefore, it can be concluded that the numerical method is reliable and accurate to predict the hydraulic performance and inner flow structure in pump as turbine, which can effectively provide helpful guidance for the study of operation stability.

The velocity distribution of PAT

In order to study the internal flow state of PAT and provide initial conditions for unsteady simulation, the velocity distribution in pump as turbine was analyzed under different working conditions, and the velocity contours are shown in Figure 5.

Figure 5.

Speed contours of pump as turbine under different working conditions: (a) 0.8Q. (b) 1.2Q. (c) 1.4Q.

At the flow rate of $0.8\mathrm{Q}$ , the velocity in turbine is uneven, and the uneven area mainly concentrates on the static–dynamic coupling surface and inside impeller. The low velocity area at the gas volume fraction of ${\alpha }_k$  = 0.2 is obviously less than that of ${\alpha }_k$  = 0.05, 0.1, and 0.15.

At the flow rate of $1.2\mathrm{Q}$ , the velocity distribution of mixed fluid is more uniform; when the mixed fluid flows from volute outlet to the impeller inlet, huge mass of fluid velocity changes smoothly at the static–dynamic coupling surface, only a small quantity of high velocity area existed and the low velocity area mainly concentrated in the pressure side of impeller. Because the pressure surface of the blade is the resistance area, the pressure energy of mixed fluid was consumed when the fluid did work to the impeller, so the velocity decreased when the fluid flowed through it. For different gas volume fraction, the velocity distribution regularity almost is same, except for small differences, so the gas volume fraction has little influence on the velocity distribution at the flow rate of $1.2\mathrm{Q}$ .

At the flow rate of $1.4\mathrm{Q}$ , the high velocity area of fluid appeared near the statistic–dynamic coupling surface and impeller outlet, the low velocity area appeared both in the impeller suction side and pressure side. The fluid velocity at the entrance of impeller suddenly increased, the impeller blade number Z equals to 6, and the increased velocity area number also equals to 6. It is the result of statistic–dynamic coupling of volume and impeller, and here the flow area becomes small. After flowing out of the impeller, the fluid enters the draft tube. Although most of the pressure energy of the fluid has been consumed, the flow area greatly reduced at the coupling surface of the impeller and the draft tube, as well as high flow rate, so the fluid velocity was led to increase again. The low velocity area appeared in pressure side and suction side, which is not good for the flow field of PAT. In addition, the gas volume fraction has more influence on the velocity distribution, the higher the gas volume fraction is, the more serious of the high velocity fluid concentrates at the impeller outlet.

Compared comprehensively, the fluid velocity distribution at the flow rate of $1.2Q$ is better than that of $0.8\mathrm{Q}$ and $1.4\mathrm{Q}$ , and for a same gas volume fraction, the flow regime of $1.4\mathrm{Q}$ is the worst, which of $0.8\mathrm{Q}$ is the second, and the flow regime of $1.2\mathrm{Q}$ is the best.

Power loss characteristics

During the process of energy recycling, the pressure energies were converted into mechanical energies through the pump as turbine, and then pumps or fans were driven by these mechanical energies. This is the energy conversion principle in PAT. To the pressure energies that going through the impeller, only some of them were converted into mechanical energies, the rest were lost. According to the law of conservation and transformation of energy, the power loss equals to the difference of total pressure energy and net power, the formula is as follows

$$\Delta \mathrm{P}=\mathrm{Q}({\mathrm{p}}_1-{\mathrm{p}}_2)-\mathrm{M}\omega$$

(8)

Where $\Delta \mathrm{P}$ is the power loss, $\mathrm{Q}$ is the flow rate of mixed fluid, ${\mathrm{p}}_1-{\mathrm{p}}_2$ is the pressure difference of impeller inlet and outlet, $\mathrm{M}$ is the torque, and is the rotating angular velocity of impeller.

When the high-pressure fluids with gas flowed in PAT, the power loss is different for various gas contents. Therefore, it is necessary to analyze the influence of gas contents on it. The transient characteristic of power loss varied under different gas volume fractions is shown in Figure 6, the last calculation cycle is marked on the abscissa, and the ordinate represents power loss in impeller.

Figure 6.

Time-domain diagrams of power loss on turbine impeller under the variable working conditions: (a) 1.0Q. (b) 1.2Q. (c) 1.4Q. (d)1.6Q.

From Figure 6, the power loss in one cycle performs periodic variations and the wave numbers are the same as the blade numbers. Under the different gas content the uncertainties of power loss are caused by the uncertainties of total pressure energy and net power. The larger the flow rate is, the greater the power loss is. Meanwhile, for the same flow rate, the higher the gas fraction level is, the less the power loss is. These above two phenomena occurred for the cases of gas volume fraction of ${\alpha }_k$  = 0.1, 0.15, and 0.2. But at the gas volume fraction of ${\alpha }_k$  = 0.05, the distributions of power loss are not uniform enough for different flow rates. The mean power loss of ${\alpha }_k$  = 0.05 is higher than that of other three contents under the condition of 1.0Q and 1.4Q. However, for the cases of 1.2Q and 1.6Q, the mean power loss are less than that of ${\alpha }_k$  = 0.1 and ${\alpha }_k$  = 0.15 and greater than that of ${\alpha }_k$  = 0.2 case.

To summarize, the power loss distributions are disorder and not uniform enough when the gas contents are less than ${\alpha }_k$  = 0.1. Otherwise, for the cases of gas fraction more than ${\alpha }_k$  = 0.1, the distributions are inclined to regular and the power loss is decreasing with the rise of gas volume fraction.

Transient radial force analysis

Radial force vector distribution

The transient hydrodynamics of radial force is comprehensively studied, the unsteady numerical calculations are run for the cases of 1.0Q, 1.2Q, 1.4Q, and 1.6Q, respectively. The numerical values of radial force are integrated by the pressure and shear stress of blade suction side, pressure side, hub, and shroud. The distribution curves of radial force vector are obtained for different gas volume fractions in PAT in Figure 7. The z axis was chosen as a rotating axis, the radial force F_X and F_Y is the component force in the direction of x and y axis, respectively. The point represents the radial force on impeller at certain moment.

Figure 7.

Vector diagrams of radial force on impeller under variable working conditions: (a)1.0Q. (b) 1.2Q. (c) 1.4Q. (d)1.6Q.

It is shown that the radial force vectors on impeller are distributed within four quadrants in the rectangular coordinates. The curves are asymmetrical and the centers of vector are partially displaced to first and fourth quadrants, which performs same trend as that in pure liquid medium. In the phase of rotating cycle, the magnitude and direction of radial force vector both experienced evident changes, and the curves of radial force distributions all closed to ellipse shape for different gas content and flow rates, but near the volume tongue an obvious wave appeared.

The radial force vectors performed wave character and huge force differences impinging on the impeller under different gas contents, which all can be found in Figure 7(a). The circumferential component of radial force vectors are uniformly distributed except for the case of ${\alpha }_k$  = 0.2 volume fraction, which all perform decreasing trend with the increment of gas volume fraction as shown in Figure 7(b). Here the maximum radial force of ${\alpha }_k$  = 0.05 gas contents in the y direction is 1.05, 1.33, and 1.67 times higher than that for the cases of ${\alpha }_k$  = 0.1, 0.15, and 0.2, respectively. In Figure 7(c) and (d), the radial force both Fx and Fy are the maximum at the case of ${\alpha }_k$  = 0.1. Therefore, for different gas contents and flow rates, the radial force distributing at the flow rate of 1.2Q are more regular than the other cases.

Time domain distribution of radial force on impeller

The transient radial resultant forces on impeller for different gas contents and operation working are shown in Figure 8, in which the abscissa varied with time sequence marked on the ordinate corresponding to the last revolution. The abscissa represents time in one cycle (the sixth rotating cycle is 0.2069∼0.2483), the ordinate represents resultant force of transient radial force. The maximum radial force fluctuation differences under variable gas contents and working conditions are shown in Table 2. The maximum radial force fluctuation difference is defined as follows

$${\mathrm{F}}^{\prime }=|\mathrm{F}-\Delta \mathrm{F}|$$

(9)

Figure 8.

Time-domain spectrums of radial force on impeller under different working conditions: (a) 1.0Q. (b) 1.2Q. (c) 1.4Q. (d) 1.6Q.

Table 2.

The maximum fluctuation difference and dominant frequency of radial force under variable gas contents and working conditions.

Flow rate		Gas volume fraction ${\alpha }_k$
		0.05	0.1	0.15	0.2
1.0Q	Fluctuation amplitude (N)	95.42	66.49	107.84	93.6
	Dominant frequency (N)	34.54	9.65	22.55	19.95
1.2Q	Fluctuation amplitude (N)	112.44	108.70	103.26	117.32
	Dominant frequency (N)	25.45	22.69	26.35	22.15
1.4Q	Fluctuation amplitude (N)	140.14	172.04	164.23	169.20
	Dominant frequency (N)	50.68	22.84	30.72	26.86
1.6Q	Fluctuation amplitude (N)	221.83	183.53	148.77	150.49
	Dominant frequency (N)	53.95	40.26	33.22	33.54

where ${\mathrm{F}}^{\prime }$ is the maximum radial force fluctuation difference, $\mathrm{F}$ is the transient radial force, and $\Delta \mathrm{F}$ is the average radial force in one cycle.

In Figure 8, under different working conditions, the more the flow rate is, the larger the radial force is. In one circle, the change trend of radial force is consistent and the radial force changes periodically; but the radial force does not change as regular as that of pure liquid condition, as serious fluctuation appeared, especially for the gas volume of ${\alpha }_k$  = 0.05. About these except for the influence of rotor–stator interaction, the gas cannot be ignored.

In Table 2, for the same flow rate and different gas content, the maximum radial force difference changes irregularly, and considerable difference exists. But for the same gas content and different flow rate, the maximum radial force difference changes regularly, and the total change trend is the larger the flow rate, the larger the radial force fluctuation amplitude. At gas volume fraction of ${\alpha }_k$  = 0.1, the maximum radial force of $1.6\mathrm{Q}$ is 2.76 times, 1.69 times, and 1.07 times than that of $1.0\mathrm{Q}$ , $1.2\mathrm{Q}$ , and $1.4\mathrm{Q}$ , respectively. Among all the flow rates and gas contents, the maximum radial force fluctuation appeared at the gas volume fraction of ${\alpha }_k$  = 0.05 and the flow rate of 1.6Q; under this working condition, the radial force difference also is at the maximum level and $F^{\prime }$  = 221.83 N.

To sum up, the gas inclusion plays a major factor to influence the radial force on impeller. But, it can be concluded that there are not distinct patterns for the cases of large and small flow rates, the maximum radial force fluctuation usually appears at the lower volume fraction and larger flow rate.

Frequency domain distribution of radial force on impeller

According to the time sequence of radial force imparted on impeller, the frequency domain spectrums of radial force on impellers were obtained through Fast Fourier Transfer (FFT), and the frequency domain spectrums of radial force under different working conditions are listed in Figure 9.

Figure 9.

Frequency-domain spectrums of radial forces on the impeller under different working conditions: (a) 1.0Q. (b) 1.2Q.(c) 1.4Q. (d) 1.6Q.

The ordinate shows the frequency amplitude of radial force on the impeller, and the abscissa shows the ratio of the radial force frequency to the rotating frequency. For this pump as turbine, the rotating speed of impeller is $1450r/min$ , and the impeller blade number $\mathrm{z}=6$ , so the rotating frequency ${\mathrm{f}}_{\mathrm{n}}=n/60=1450/60=24.167$ , and the blade frequency ${\mathrm{f}}^{\prime }=\mathrm{f}\times 6=24.167\times 6=145$ . The dominant frequency amplitudes of radial force under variable working conditions are shown in Table 2.

From Figure 9 and Table 2, for all the different flow rates and gas volume fractions, the dominant frequency of radial force equals to rotating frequency, the subordinate dominant frequency does not appear. After the $\mathrm{f}/{\mathrm{f}}_{\mathrm{n}}$ is greater than 10, no obvious frequency appeared. Due to the influence of gas and rotor–stator interaction, the time domain distribution of radial force is not regular, which leads to the irregular distribution of radial force in frequency domain. For the flow rate of $1.0\mathrm{Q}$ , $1.4\mathrm{Q}$ , and $1.6\mathrm{Q}$ , the maximum amplitude appeared at the gas volume fraction of ${\alpha }_k$  = 0.05, but for the flow rate of $1.2\mathrm{Q}$ , the maximum amplitude appeared at the gas volume fraction of ${\alpha }_k$  = 0.15. But for all the gas volume fraction, all the dominant frequency amplitudes are the maximum at the flow rate of $1.6\mathrm{Q}$ . And the value equals to 53.95 N, which is 1.34 times, 1.62 times, and 1.61times than that of $1.0\mathrm{Q}$ , $1.2\mathrm{Q}$ , and $1.4\mathrm{Q}$ . Undoubtedly, the huge waves of radial force in time domain are correlated with the maximum amplitude of dominant frequency for the corresponding operation condition. So at the low gas volume fraction and larger flow rate both the fluctuation amplitude and dominant frequency amplitude are the maximum.

Conclusion

1. On the effect of rotor–stator interaction and gas content, the velocity distribution in pump as turbine is uneven, low pressure area appeared, which is not good for the steady running of turbine. The power loss distribution is in disordered manner and not uniform enough when the gas contents are less than ${\alpha }_k$  = 0.1. Otherwise, for the cases of gas fraction more than ${\alpha }_k$  = 0.1, the distributions are inclined to regular and the power loss is decreasing with the rise of gas volume fraction.

2. Under the gas–liquid two-phase condition, the radial force vector distribution all closed to ellipse shape for different gas content and flow rates, but near the volume tongue, an obvious wave appeared. The impeller transient radial force is affected by gas content. The more the flow rate is, the larger the radial force is. When the radial force does not change regularly as that of pure liquid condition, serious fluctuation appeared. There are not distinct patterns for the cases of large and small flow rates, the maximum radial force fluctuation usually appears at the lower volume fraction and larger flow rate. For different gas contents, the dominant frequency always equals the rotating frequency of impeller, and subordinate dominant frequency does not appear. As a whole, under low gas content and high flow rate, both the fluctuation amplitude and dominant frequency are the maximal. For this hydraulic turbine, the flow rate equals to $1.6\mathrm{Q}$ and ${\alpha }_k$ equals to 0.05.