# Influence of Tubular Turbine Runaway for Back Pressure Power Generation on the Stability of Circulating Cooling Water System

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Calculation Method

#### 2.1. CCWS 1D MOC Method

^{+}and C

^{−}, which are explained by Figure 2. Variable Δx is the length of the pipeline that the water hammer travels at time Δt.

^{+}equations:

^{−}equations:

_{0}are known, and the above equations can be simultaneously solved to determine the H

_{p}and v

_{p}of intersection point P of lines C

^{+}and C

^{−}at time t

_{0}+ Δt. Then, the calculation pipe section is divided into N sections, according to the calculation accuracy requirements. The total number of pipe nodes is N + 1. Each node is represented by i, the space step Δx = L/N, and the time step Δt = Δx/a. Let $B=a/gA$, and $R=f\Delta x/2gD{A}^{2}$. The characteristic equation can be expressed as the following form. Finally, a program is compiled to carry out numerical simulation calculations in combination with the initial conditions and boundary conditions of the boundary points in the pipeline system based on the characteristic equations.

_{1}, a

_{2}, and a

_{3}are constants related to the combination of the pumps.

^{−}equation of the outlet pressure head of the pumping station is as follows:

#### 2.2. 3D CFD Method for a Tubular Turbine

_{i}is the instantaneous velocity, p is the instantaneous pressure, ρ is the density, $\nu $ is the kinematic viscosity coefficient of the water molecules, and f

_{i}is the body force.

_{t}is defined as the turbulent kinetic energy k, and the Function of special loss rate ω, k, and ω are solved by using two transport equations.

_{t}is expressed as follows:

_{1}= 0.310, k is the turbulent kinetic energy, and ω is the turbulent frequency.

#### 2.3. 1D MOC–3D CFD Coupled Simulation Method

_{t}is the resultant moment acting on the runner, M

_{g}is the load torque, n is the runner rotational speed, and J is the moment of inertia. Friction is ignored in this work, because friction is relatively small, and ignoring it will not significantly affect the speed calculation results. The tubular turbine generator unit is small because of the moment of inertia. The total inertia moment of the unit is the sum of the additional inertia moment of generator (${J}_{1}$), water turbine (${J}_{2}$), and water body (${J}_{3}$). The calculation method of the total moment of inertia is shown as Equation (15).

^{2}.

_{g}= 0, the rotational angular velocity of the runner at any time is as follows:

_{t}

^{i}is the runner torque at the i moment; n

^{i}and n

^{i +}

^{1}are the rotational speed at the i moment and after a time step, respectively; and Δt is the time step.

## 3. Computational Model and Reliability Verification

#### 3.1. System Coupling Model

^{3}/h. Three fixed-blade tubular turbines were used on site to replace the original upper tower valve, and the extra back pressure of the system was used to generate electricity. Considering the investment cost, operation life and power generation of the hydraulic turbine, the annual income of the system after the optimization and transformation of residual pressure power generation is about 400,000 $. It can be seen that using the residual voltage of the system to generate electricity has high economic benefits. The specific parameters of the turbine are shown in Table 1.

^{−4}, 2.2 × 10

^{−4}, and 3.3 × 10

^{−4}s) were used to verify the time-step independence of the 3D model of the hydraulic turbine. The three time steps correspond to the time required for the runner to rotate 1°, 2°, and 3° at the design speed. The pressure calculation results of a certain measuring point in the runner domain under the three time steps are shown in Figure 6. Considering the calculation accuracy and calculation speed, the final time step is 2.2 × 10

^{−4}s. Where T

_{n}is the time required for the runner to rotate for one revolution.

_{in}and P

_{out}are the inlet and outlet pressure of the turbine respectively, which can be read through the pressure gauge; Δh is the height difference of inlet and outlet pressure gauges; N

_{m}is the instantaneous generating capacity of the turbine, which can be read by the electricity meter; η

_{m}is the generator efficiency, which can be obtained according to the parameters provided by the manufacturer.

_{BEP}) is 4100 m

^{3}/h. The numerical simulation and experimental water heads under the optimal working conditions are 19.6 and 18.6, and the efficiencies are 91.2% and 88.3%, respectively; thus, the errors of the water head and efficiency are 5.1% and 3.2%, respectively. The simulated values of the 3D model are in good agreement with the experimental values, thereby accurately reflecting the external characteristics of the turbine. In addition, it can be found that the test results are slightly smaller than the numerical results, mainly because only hydraulic efficiency is considered in the numerical simulation.

_{1}and V

_{2}shut off the runaway turbine. A 3D model is established for a hydraulic turbine, the other two parallel turbine branches belong to 1D simulation area, and the water supply network is simplified based on the hydraulic equivalence theory [28,30]. The coupled model is shown in Figure 8. The figure demonstrates that when the turbine unit is in the normal operation, valve V

_{1}is closed, and valve V

_{2}is opened. When the turbine unit is disconnected from the system, valve V

_{2}is closed, and valve V

_{1}is opened.

#### 3.2. Comparative Analysis of the Coupling Algorithm and 1D MOC Calculation Results

_{BEP}. During calculation, the static pressure of the sump is 19,600 Pa and the local atmospheric pressure is 101,325 Pa. The comparison results are shown in Figure 9. The results of the two methods show good consistency. Moreover, the flow rate, head, and rotational speed of the turbine are basically stable under the runaway state of the turbine.

## 4. Results and Discussion

_{BEP}and 1.1Q

_{BEP}. This work conducts a coupled simulation of the turbine runaway process when the initial steady state flow rates (abbreviated as initial flow rate) are 0.85Q

_{BEP}and 1.1Q

_{BEP}to comprehensively analyze the runaway and runaway shutdown process of the turbine.

#### 4.1. Analysis of the Influence of the Runaway Process of the Turbine on the Main Parameters of the System

_{out,rel}, Q

_{total,rel}, and Q

_{par,rel}, represent the ratio of the instantaneous value of each parameter to the initial value before escape, as shown in Equation (18)

_{BEP}working condition dropped by about 9%, and the 0.85Q

_{BEP}working condition was approximately 3% lower. The 1.1Q

_{BEP}working condition had a larger drop in the return water pressure compared with the 0.85Q

_{BEP}working condition.

_{BEP}condition and 0.5% under the 0.85Q

_{BEP}condition. The system flow rate increases even more under the 0.85Q

_{BEP}working condition than that under the 1.1Q

_{BEP}condition.

_{BEP}condition. Meanwhile, the parallel turbine flow rate decreased by about 4% under the 0.85Q

_{BEP}condition. The 1.1Q

_{BEP}condition reduced the flow rate of the parallel turbines by a larger proportion compared with the 0.85Q

_{BEP}condition.

#### 4.2. Analysis of the Pressure Pulsation of the Turbine and the Force Characteristics of the Runner during the Runaway Process

#### 4.2.1. Influence of the Initial Flow Rate on the Runaway Process

_{rel}, Q

_{rel}, and M

_{rel}, as shown in Equation (19):

_{BEP}condition, the runaway speed and flow rate of the turbine are 1.2 and 1.1 times those of the initial steady state conditions, respectively. When runaway occurs in the 1.1Q

_{BEP}condition, the turbine runaway speed and flow rate are 1.7 and 1.2 times those of the initial steady state condition, respectively.

#### 4.2.2. Analysis of the Pressure Pulsation Characteristics

_{BEP}and 1.1Q

_{BEP}. Under the two working conditions, the pressure change trends of the corresponding monitoring points are basically the same. In addition, monitoring point M1 is basically the same as the system return water pressure, and both gradually decrease and become stable with the development of the runaway process. The pressure at the M2 point gradually increases with the development of the runaway process and becomes stable under different initial conditions. Furthermore, the pressure at the M2 point at the start of the runaway has a relatively large pressure oscillation, and the operation of the unit is extremely dangerous under such a large pulsation condition. In the comparison of the two working conditions, the pressure oscillation phenomenon at the M2 point has a larger amplitude under the 1.1Q

_{BEP}working condition, but the duration is relatively short. The duration is about half of that under the 0.85Q

_{BEP}working condition. Point M3 is close to the runner, so its change is basically similar to that of point M2, showing an upward trend with the runaway process. Point M4 is close to the outlet of the draft tube, and its pressure variation law is similar to that of point M1, showing a downward trend with the development of the runaway process.

#### 4.2.3. Analysis of the Force Characteristics of the Runner

_{r}, F

_{x}, and F

_{y}, respectively, and the calculation Equation is as follows [33]:

_{1}represents the cross-sectional area of the runner inlet; V

_{r}is the radial velocity of the fluid particle; V

_{x}and V

_{y}represent the component velocities of the fluid particle in the x and y directions, respectively; ω is the rotational angular velocity of the runner; θ is the initial angle of the fluid particle; t is time; and P is pressure.

_{BEP}, the maximum radial force on the runner increases from 157 N in the initial steady state to 1290 N, an increase of 1133 N, and the radial force maximum value in the runaway state is 7 times that of the initial steady state. At the initial flow rate of 1.1Q

_{BEP}, the maximum radial force on the runner increased from 68 N in the initial steady state to 2510 N, an increase of 2442 N, and the maximum radial force is 37 times that of the initial steady state during the runaway state.

_{BEP}, 1.1Q

_{BEP}) is drawn to further analyze the variation law of the component forces of the radial force in the x and y directions during the runaway development process, as shown in Figure 16. In the process of runaway development, the component forces of the radial force in the x and y directions substantially increase, and the increase is consistent. In the middle stage of runaway development, the components of radial force in the x and y directions are significantly reduced. In particular, the component forces in both directions are close to zero in the small flow rate condition.

#### 4.3. Analysis on the Opening and Closing Law of the Control Valve during the Shutdown Process of the Turbine Runaway State

_{2}and V

_{1}. In each scheme, valves V

_{2}and V

_{1}are linearly closed or opened. The operating time of the two valves is used as the optimization variable. The optimization goal is to limit excessive positive and negative water hammer peaks and water hammer durations. The tubular turbine takes less than 3 s to run away from the start to the runaway state, and it generally takes several seconds for the actuator to operate from receiving the command. Therefore, the faulty turbine has been in the runaway state when valves V

_{1}and V

_{2}execute the shutdown command. In this paper, the initial time of runaway process is 0 s, and the valve starts to operate from the 3 s. According to the hydraulic characteristics of the valve provided by the manufacturer, the hydraulic opening of valve V

_{1}is equivalent to the turbine resistance when the hydraulic opening rates are 39% and 34% when the flow rates are 0.85Q

_{BEP}and 1.1Q

_{BEP}, respectively.

#### 4.3.1. Process Analysis of Separate Closure of Tandem Valves

_{1}remains closed, and valve V

_{2}considers three types of linear closing time. The specific scheme is shown in Table 5.

_{out,rel}under different valve closing schemes of valve V

_{2}. The three valve closing schemes under the working conditions of 0.85Q

_{BEP}and 1.1Q

_{BEP}will generate a larger positive water hammer. The peak pressure of the positive water hammer increases with the shortening of the valve closing time of V

_{2}, but the propagation time of the water hammer wave is short. Under the scheme 1, when the initial steady-state flow rate is 0.85Q

_{BEP}and 1.1Q

_{BEP}, the maximum P

_{out,rel}is 1.43 and 1.67 times of the initial steady-state condition, respectively. When the initial steady state flow rate before runaway is 0.85Q

_{BEP}, valve V

_{2}is closed alone, and the final system return water pressure is 37% higher than the target pressure. When the initial steady state flow rate before runaway is 1.1Q

_{BEP}, the valve V

_{2}is closed alone, and the final system return water pressure is 42% higher than the target pressure. Therefore, closing the system pressure of valve V

_{2}alone will affect the safety of system operation due to excessive positive water hammer.

#### 4.3.2. Analysis of the Separate Opening Process of the Parallel Valve

_{2}remains fully open, and valve V

_{1}is linearly opened to 49% of the opening degree considering three durations. The specific scheme is shown in Table 6.

_{out,rel}under different valve opening schemes of valve V

_{1}. The three valve opening schemes under 0.85Q

_{BEP}and 1.1Q

_{BEP}conditions will generate a larger negative water hammer. The valley value of the negative water hammer decreases with the shortening of the valve opening time of V

_{1}. However, the propagation time of the water hammer wave is short. Under the scheme 1, when the initial steady-state flow rate is 0.85Q

_{BEP}and 1.1Q

_{BEP}, the maximum P

_{out,rel}is 0.83 and 0.69 times of the initial steady-state condition, respectively.

_{BEP}, the final system return water pressure is 16% lower than the target pressure when valve V

_{1}is opened alone. When the initial steady state flow rate before runaway is 1.1Q

_{BEP}, the final system return water pressure is 28% lower than the target pressure when valve V

_{1}is opened alone. When valve V

_{1}is opened alone, the system pressure will affect the safety of the system operation due to excessive negative water hammer. Accordingly, the system pressure fluctuation increases with the increase in the initial steady-state flow rate before runaway.

#### 4.3.3. Synergistic Control Analysis of the Series and Parallel Valves

_{2}alone or opening V

_{1}alone will affect the safety of system operation. Therefore, V

_{2}and V

_{1}must be simultaneously operated. The specific scheme is shown in Table 7.

_{1}is the same as the valve closing time of V

_{2}, the positive pressure wave in the return water pressure will be large. When the operating time of valve V

_{1}is longer than the closing time of V

_{2}, the peak value of the positive pressure wave will increase. Meanwhile, when the operation time of valve V

_{1}is less than the valve closing time of V

_{2}, the peak value of the positive pressure wave gradually decreases as the operation time of valve V

_{1}shortens. However, an excessively short valve opening time will produce a smaller negative pressure valley. When the initial steady state flow rate before runaway is 0.85Q

_{BEP}, the final system return water pressure is 18% lower than the initial steady-state condition when adopt scheme 1, and the final system return water pressure is 38% higher than the initial steady-state condition when adopt scheme 6. When the initial steady state flow rate before runaway is 1.1Q

_{BEP}, the final system return water pressure is 31% lower than the initial steady-state condition when adopt scheme 1, and the final system return water pressure is 60% higher than the initial steady-state condition when adopt scheme 6. The comparison of the calculation results of the six schemes under the two initial flow rates before runaway shows that the operating time of valve V

_{1}should be less than the closing time of V

_{2}, and the reasonable coordinated operation of V

_{1}and V

_{2}can effectively suppress the fluctuation of return water pressure. When the valve opening time of V

_{1}is 60% of the valve closing time of V

_{2}(scheme 3), the system studied in this work is relatively optimal under large and small flow rate conditions.

_{1}must meet the automatic grid-connected operation conditions under different working conditions because the operating conditions of the hydraulic turbine vary. Accordingly, the setting value of the equivalent flow resistance opening of valve V

_{1}and the hydraulic turbine should be based on the hydraulic turbine under the steady state operating conditions. The running traffic is updated in real time.

## 5. Conclusions

- (1)
- After the tubular turbine entered the runaway process, it did not cause major water hammer problems to the system. Before and after the runaway of the fault turbine, when the initial steady-state flow rate of the turbine is 0.85Q
_{BEP}and 1.1Q_{BEP}, the return water pressure of the system decreases by 3% and 9%, the system flow rate increases by 0.5% and 1.8%, and the flow rate of the turbine in parallel with the fault turbine decreases by 4% and 8% respectively. - (2)
- When the tubular turbine enters the runaway process from the steady-state condition, the fault turbine shows the characteristics of increasing speed and flow rate. The change rates of speed and flow rate are great, and the time from the initial state to the runaway state is short with the increase in the initial flow rate. When the initial flow rate is 0.8QBEP, the runaway speed and flow rate of the turbine are 1.2 and 1.1 times the initial steady-state condition, respectively. When the initial flow rate is 1.1QBEP, the runaway speed and flow rate of the turbine are 1.7 and 1.2 times the initial steady-state condition, respectively. At the beginning of the runaway process, the pressure at the monitoring points in the runner area greatly fluctuated. The pressure oscillation amplitude of the monitoring points in the runner area increases with the increase in the initial steady-state flow rate, but the duration decreases. In addition, the radial force of the fault turbine runner in the runaway state greatly increases and shows violent oscillation compared with the initial steady-state condition. When the initial steady-state flow rate is 0.85QBEP and 1.1QBEP, the maximum radial force is 7 and 37 times the initial steady-state conditions, which will aggravate the shafting wear. Therefore, the runaway of hydraulic turbine in CCWS may lead to serious safety accidents, and it is necessary to shut it down as soon as possible.
- (3)
- During the shutdown of tubular turbine from runaway state, improper valve control scheme will cause severe positive and negative water hammer in the pipe network and seriously threaten the stable operation of the system. For the research system: Under different flow rate conditions, when the opening time of the parallel valve of the fault turbine is too short compared with the closing time of the series valve, the system return water pressure will produce a negative pressure wave, and the valley value of the negative pressure wave is about 69%~82% of the target pressure; On the contrary, when the opening time of the parallel valve is too long compared with the the closing time of the series valve, the return water pressure of the system will produce a positive pressure wave, and the peak value of the positive pressure wave is about 138%~160% of the target pressure; When the opening time of the parallel valve is 60% of the closing time of the series valve, the operating pressure of the system will return to the initial steady state smoothly.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Simplified schematic of the boundary conditions during a normal operation of the pumping station.

**Figure 5.**Coupling interface data transfer mechanism. (

**a**) Turbine inlet coupling interface. (

**b**) Turbine outlet coupling interface.

**Figure 10.**Influence of the turbine runaway on the system return water pressure, system flow rate, and parallel turbine flow rate. (

**a**) System return water pressure. (

**b**) System flow rate. (

**c**) Parallel turbine flow rate.

**Figure 13.**Change law of the pressure pulsation at the monitoring point. (

**a**) 0.85Q

_{BEP}. (

**b**) 1.1Q

_{BEP}.

**Figure 14.**Time–frequency analysis of the pressure signals at points M2 and M3. (

**a**) 0.85Q

_{BEP}—M2. (

**b**) 1.1Q

_{BEP}—M2. (

**c**) 0.85Q

_{BEP}—M3. (

**d**) 1.1Q

_{BEP}—M3.

**Figure 16.**Time domain diagram of the radial force components in the x and y directions during runaway. (

**a**) 0.85Q

_{BEP}. (

**b**) 1.1Q

_{BEP}.

**Figure 17.**Variation law of return water pressure under different valve closing schemes. (

**a**) 0.85Q

_{BEP}. (

**b**) 1.1Q

_{BEP}.

**Figure 18.**Variation law of the return water pressure under different valve opening schemes. (

**a**) 0.85Q

_{BEP}. (

**b**) 1.1Q

_{BEP}.

**Figure 19.**Variation law of the return water pressure under different valve cooperative operation schemes. (

**a**) 0.85Q

_{BEP}. (

**b**) 1.1Q

_{BEP}.

Name | Date | Unit |
---|---|---|

Propeller blade | 5 | Piece |

Vane | 12 | Piece |

Blade opening | 21 | ° |

Guide vane opening | 76 | ° |

Runner speed | 1500 | r/min |

Rotational inertia | 8 | Kg m^{2} |

Runner diameter | 430 | mm |

Bulb diameter | 450 | mm |

Inlet diameter | 900 | mm |

Outlet diameter | 800 | mm |

Number of Mesh (Million) | Head (m) | Efficiency (%) |
---|---|---|

298 | 19.18 | 91.09 |

382 | 19.35 | 91.14 |

456 | 19.36 | 91.15 |

533 | 19.36 | 91.15 |

Runaway State | Q (m^{3}/h) | H (m) | n (r/min) |
---|---|---|---|

1D MOC | 5492 | 23.4 | 2539 |

Coupling algorithm | 5893 | 22.1 | 2645 |

Error rate (%) | 7.3 | 5.6 | 4.2 |

Initial Steady State Condition Turbine Flow Rate | F_{r} Maximum (N) | |
---|---|---|

Initial Steady State Condition | Runaway State | |

0.85Q_{BEP} | 157 | 1290 |

1.1Q_{BEP} | 68 | 2510 |

Scheme | V_{1} Valve | V_{2} Valve |
---|---|---|

Scheme 1 | Fully closed state | Linear off, 2 s fully off |

Scheme 2 | Fully closed state | Linear off, 10 s fully off |

Scheme 3 | Fully closed state | Linear off, 20 s fully off |

Scheme | V_{1} Valve | V_{2} Valve |
---|---|---|

Scheme 1 | Linear opening, 2 s opening to resistance equivalent opening | Fully open |

Scheme 2 | Linear opening, 10 s opening to resistance equivalent opening | Fully open |

Scheme 3 | Linear opening, 20 s opening to resistance equivalent opening | Fully open |

Scheme | V_{1} Valve | V_{2} Valve |
---|---|---|

Scheme 1 | Linear opening, 2 s opening to resistance equivalent opening | Linear off, 10 s fully off |

Scheme 2 | Linear opening, 4 s opening to resistance equivalent opening | Linear off, 10 s fully off |

Scheme 3 | Linear opening, 6 s opening to resistance equivalent opening | Linear off, 10 s fully off |

Scheme 4 | Linear opening, 8 s opening to resistance equivalent opening | Linear off, 10 s fully off |

Scheme 5 | Linear opening, 10 s opening to resistance equivalent opening | Linear off, 10 s fully off |

Scheme 6 | Linear opening, 20 s opening to resistance equivalent opening | Linear off, 10 s fully off |

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**MDPI and ACS Style**

Wang, P.; Luo, X.; Lu, J.; Gao, J.; Cai, Q.
Influence of Tubular Turbine Runaway for Back Pressure Power Generation on the Stability of Circulating Cooling Water System. *Water* **2022**, *14*, 2294.
https://doi.org/10.3390/w14152294

**AMA Style**

Wang P, Luo X, Lu J, Gao J, Cai Q.
Influence of Tubular Turbine Runaway for Back Pressure Power Generation on the Stability of Circulating Cooling Water System. *Water*. 2022; 14(15):2294.
https://doi.org/10.3390/w14152294

**Chicago/Turabian Style**

Wang, Peng, Xingqi Luo, Jinling Lu, Jiawei Gao, and Qingsen Cai.
2022. "Influence of Tubular Turbine Runaway for Back Pressure Power Generation on the Stability of Circulating Cooling Water System" *Water* 14, no. 15: 2294.
https://doi.org/10.3390/w14152294