In order to analyze the response of a hydraulic turbine to a variation in the operating conditions, different laws of variation in time of the mass flow rate have been considered. After validating the overall numerical framework through comparison with relevant experiments, the performances of the considered turbine have been analyzed from a fluid-dynamic point of view. The results show that different time profiles of the mass flow rate (in this work, for simplicity, referred to as “transition functions”) have a varying influence on the transient behavior of the turbine. When a quadratic function is considered for the case of large flow, the transient head and torque increase gradually with time, the fluctuation amplitude of the transient hydraulic efficiency at the main frequency is the largest, and the fluctuation amplitude of the radial force is the smallest. For the small flow case, the time profile with exponential nature leads to the best results. The transient head and torque decrease gradually with time, the pulsation amplitude of the transient hydraulic efficiency is the largest at the main frequency, and the pulsation amplitude of the radial force is the smallest.

When the centrifugal pump is used as a hydraulic turbine in reverse, it can recover the residual pressure energy of the high-pressure fluid in the petrochemical industry, coal chemical industry, seawater desalination and other industrial processes and convert it into mechanical energy or electric energy for secondary utilization [

The transient transition process refers to the intermediate process that the fluid begins to transition to another stable condition when it is disturbed, in which the various performance parameters of the fluid will be changed dramatically in a short time [_{s}

To analyze the influence of different transition processes on the transient hydraulic performance of hydraulic turbines, this paper the numerical calculation of the non-stationary characteristics of the transition process from the optimum condition (

An IS80-50-315 low specific speed centrifugal pump (_{s}_{p}^{3}/h, head _{p}_{sp}_{t}_{p}

Parts | Parameter | Numeric value |
---|---|---|

Impeller | D_{1} |
315 |

D_{2} |
80 | |

d_{h} |
0 | |

b_{1} |
10 | |

β_{1} |
32 | |

Z | 6 | |

θ | 150 | |

Volute | D_{3} |
320 |

b_{3} |
24 | |

D | 50 |

According to the main geometric parameters of the impeller and volute casing of the hydraulic turbine in

The standard

In the transition process of variable flow rate, the mass flow rate is adopted for the inlet boundary condition, to prevent cavitation, the outlet boundary condition is set as the pressure outlet. And the pressure value is 0.5 MPa. In order to ensure the accuracy of the simulation results, the transition to a high or low flow rate after running 0.1 s at the optimum condition firstly, and to monitor the internal flow law of the turbine, the hydraulic turbine is transitioned to high and low flow rate conditions according to four basic functions in this paper, and these description functions are expressed as ANSYS CEL custom function language, as shown in

Linear function:

Large flow transition:

Small flow transition:

Quadratic function:

Large flow transition:

Small flow transition:

Exponential function type I:

Large flow transition:

Small flow transition:

Exponential function type II:

Large flow transition:

Small flow transition:

where, _{i}

For the non-constant numerical calculation, the impeller rotates 4° in one time step, then one time step is 4.6 × 10^{−4} s, so the total calculation time is 1.1 s. The internal flow is described by the Navier-Stokes (N-S) equation with Reynolds time uniform non-pressure, and the standard k-ε turbulence model is used, and the momentum and the continuous equations are solved by SIMPLE-C algorithm jointly. The interface between the dynamic and static calculation domains of the hydraulic turbine is set as transient dynamic-static rotor model, the standard wall function is used near the surface of the wall, the boundary condition of the wall is set to an adiabatic no-slip wall, the working medium is clear water at room temperature, and the convergence accuracy is set to 10^{−6}.

In order to verify the feasibility and accuracy of the numerical calculation method, the numerical calculation result of the external characteristics of the IS 80-50-315 centrifugal pump reversed as a turbine are compared with the test values, and the schematic diagram of the hydraulic turbine experiment is shown in ^{3}/h, the accuracy is ±0.1% [

CFX is used to define the function of transient head H and transient torque M changing with time and monitor their changing rules. The time-domain diagrams of transient head and torque transitioning to large flow rate and small flow rate, respectively, can be obtained as shown in

where, Ω_{J}, Ω_{M}, —blade influence coefficient; the first term refers to the formula for calculating the head under stable conditions, and the last two refer to the additional head in the transition process.

For

Obviously, when transitioning to a large flow rate, the exponential function type I transition function has the largest transient head compared with the other three functions, while the quadratic transition function has the smallest transient head. In the case of transition to low flow, the changing trend is just the opposite, that is, the quadratic function transition function has the largest transient head, and the exponential function type I transition function has the smallest transient head.

During the operation of the hydraulic turbine, the high pressure fluid from the volute acting on the impeller blades produces a non-constant torsional torque on the impeller, which converts the pressure energy of the fluid into the mechanical energy of the impeller rotation, and its magnitude directly affects the magnitude of the turbine’s work capacity, and the magnitude of this non-constant torque will change with the changes of the hydraulic turbine operating conditions [

The equation for the hydraulic efficiency of the hydraulic turbine is:

where, ^{3}/s;

There are three variables in

In conclusion, during the transient transition process from the optimal working condition to large flow conditions, compared with the transient hydraulic efficiency in stable conditions, the hydraulic efficiency wave peak of the hydraulic turbine has a significant downward trend, while the size of the wave trough does not change much; while during the transient transition process from optimal condition to the small flow condition, in contrary to the large flow transition, the hydraulic efficiency wave trough of the hydraulic turbine has a significant downward trend, while the size of the wave peak size does not change much.

In order to further study the influence of different transition processes on the transient hydraulic efficiency of the hydraulic turbine, the hydraulic efficiency within 0 to 1.1 s was selected for analysis, and the fast Fourier transform (FFT) was used to obtain the frequency domain diagram of the transient hydraulic efficiency during the transition process of variable flow, as shown in

Linear | Quadratic function type | Exponential function type I | Exponential function type II | |
---|---|---|---|---|

Transition to large flow rate | 4.16 | 4.45 | 4.02 | 4.30 |

Transition to small flow rate | 6.00 | 5.57 | 6.24 | 5.80 |

So in the transition process of variable flow, the hydraulic performance of the hydraulic turbine is not only affected by physical parameters such as pressure and flow, but also by the transition process characteristics, which are significantly different from the transient characteristics under stable working conditions, showing obvious transient characteristics of the transition process.

The transient radial force is the main cause of hydraulic vibration and noise in hydraulic turbines. In order to study the effect of different transition functions on the radial force of the hydraulic turbine, the data within the whole time period of 1.1 s were selected.

By applying fast Fourier transform (FFT) in the time domain of the transient radial force of the above transition process, the radial force frequency domain diagram of the whole transition process can be obtained, as shown in _{n}

Linear | Quadratic function type | Exponential function type I | Exponential function type II | |
---|---|---|---|---|

Large flow transition | 53.78 | 49.62 | 59.96 | 55.81 |

Small flow transition | 40.52 | 42.95 | 39.42 | 41.77 |

By comparing the transient hydraulic performance and radial force characteristics of different transition functions, it is found that: the use of different transition functions has a significant impact on the transient performance of the hydraulic turbine, in the transition process to large flow rate, the use of quadratic function type is better than the other three transition functions, while in the transition process to small flow rate, the use of exponential function type I is better than the other three transition functions.

In this paper, by means of numerical calculation, the performance variation of hydraulic turbine under different functional functions is analyzed, which provides the basic theoretical support for the study of the operating stability of hydraulic turbines, and has certain guiding significance for the actual transition process of hydraulic turbines. The conclusions are as follows:

Different transition functions have different effects on the instantaneous hydraulic performance of the turbine. When the quadratic function is used, the transition to large flow rate is the best, the pulsation amplitude of instantaneous hydraulic efficiency is the largest, and the radial force amplitude is the smallest. The main frequency fluctuation of type I exponential function is 20.83% larger than that of quadratic function. Using type I exponential function, the small flow transition is the best, the main frequency instantaneous hydraulic efficiency pulsation amplitude is the largest, the radial force pulsation amplitude is the smallest, and the main frequency fluctuation amplitude of quadratic function is 8.95% larger than that of type I exponential function.

In the process of transition to large flow, instantaneous head value and instantaneous torque value change with the change of time, and increase with the positive slope curve of approximate primary function. In the process of transition to low flow, the instantaneous head value and the instantaneous torque value change with time and decrease with the negative slope curve of the approximate first function. Additional head occurs when the instantaneous head value increases or decreases. At the same time, the fluctuation of instantaneous radial net force increases when the flow is transferred to a large flow. The fluctuation range of instantaneous hydraulic efficiency decreases. The fluctuation range of instantaneous radial resultant force decreases when the flow rate is low. The fluctuation range of instantaneous hydraulic efficiency increases.

_{1}

Impeller inlet diameter [mm]

_{2}

Impeller outlet diameter [mm]

_{h}

Hub diameter [mm]

_{1}

Inlet width [mm]

Impeller outlet diameter [mm]

_{3}

Volute base circle diameter [mm]

_{3}

Volute outlet width [mm]

Volute inlet diameter [mm]

Inlet placement corner [

Blade wrapping angle [

This work is financially supported by Gansu Province Key Research and Development Plan Projects (20YF3GA019), Gansu Province Science and Technology Project (20JR5RA447, 20JR10RA174, 20JR10RA203), Gansu Province Colleges and Universities Industrial Support Program Projects (2020C-20) and Key Laboratory of Fluid and Power Machinery, Ministry of Education, Xihua University (szjj2019-016, LTDL2020-007).

The authors declare that they have no conflicts of interest to report regarding the present study.