## Abstract

The phenomenon of blocking pipe would appear for a trailing suction hopper dredger (TSHD) rake arm pipe in the construction process. Thus, based on the theory of particle flow mechanics, a transient three-dimensional two-phase hydrodynamic and sediment mixture model was established in this paper. Besides, different particle sizes of sediment concentration and velocity in the piping in the rake arm of the law were analyzed to solve the problem of blocking pipe and the prevention of dredging to provide theoretical support. Then, a model predictive controller was designed to regulate the mudflow in the pipeline by controlling the mud pump speed and compared with the proportional-integral (PI) controller. According to the results, the particle size of sediment affects concentration distribution in the pipeline under the same construction conditions for the TSHD. Besides, the larger the particle size of the sediment, the more significant the difference in the sediment concentration distribution in the pipeline. Similarly, the flow velocity is another influencing factor behind the change in concentration distribution in the pipeline, the increase in the flow velocity will improve the uniformity of concentration distribution in the pipeline, and the critical velocity will maintain the concentration distribution within a reasonable range. In terms of transportation control, the designed predictive model controller is capable of reducing overshoot and shorten the time required for adjustment. To sum up, the research result not only provides a valuable reference for the theoretical analysis and a controller design of the pipeline transportation system in the TSHD but also verifies the feasibility of the computational fluid dynamics model used in the study of pipeline transportation mechanisms.

## HIGHLIGHTS

A transient three-dimensional two-phase hydrodynamic and sediment mixture model was established.

Different particle sizes of sediment concentration and velocity in the piping in the rake arm of the law were analyzed to solve the problem of blocking the pipe and prevention of dredging.

A model predictive controller was designed to regulate the mudflow in the pipeline by controlling the mud pump speed.

## INTRODUCTION

A dredger plays an essential role in some fields such as land reclamation, waterway widening, and river pollution control. Among the dredgers, the trailing suction hopper dredger (TSHD) is a kind of ship that excavates the seabed sediments through the scraper head and loads them into the cabin for loading mud, as shown in Figure 1. It possesses high mobility and strong resistance to wind and waves, enabling it to adapt to different working environments. Moreover, it has become the main force in the dredger.

The overall dredging process model of a large TSHD can be divided into the following four parts: ship model, rake head model, pipeline transportation model, and mud tank model (Braaksma 2008). Among them, the pipeline transportation system is used to transport the sediment mixture excavated by rake head to mud tank through the mud pump and the pipeline. The mixture transportation process, which affects not only the work efficiency of the dredger but also the unit energy consumption, plays a crucial role in the dredging process of the TSHD.

In the pipeline transportation system, the phenomenon of pipe blockage is the last thing people want to see. For the phenomenon of pipe blockage, most researchers perform judgment using the vital parameter non-silting critical velocity. The dredger pipeline transport of the sediment belongs to the coarse particles. Regarding coarse particle materials, different calculation formulas present large differences. The scope of application is harsher. It is not suitable for pipeline transportation of TSHDs. At present, sediment concentration, density, particle size, and pipe diameter are considered the main influencing factors for the calculation of non-silting critical velocity. In the TSHD, the diameter of the pipe will not change, and the density of the sediment solid particles is fixed. Thus, the factors affecting the plugging of the TSHD can be simplified as concentration, particle size, and flow rate. The existing formulas of Durand (1952), Shook & Roco (1985), Mehmet & Mustafa (2001), and Turian *et al.* (1987) are not applicable to this working condition as the diameter of the rake arm pipe of a large rake suction dredger is as long as 1 m. Besides, the construction personnel wish to understand the movement state and flow pattern distribution of the sediment in the pipeline in order to operate the dredger more effectively. It is of great significance to investigate the change in sediment concentration and velocity under different working conditions to avoid the phenomenon of pipe blocking in pipeline transportation.

Concerning experimental research on concentration distribution, Karabelas (2010), Roco & Shook (1983), Kaushal & Tomita (2002), and Gillies *et al.* (2004) successively conducted experimental studies on various working conditions of particles with different pipe diameters and different particle sizes under different conveying conditions. Considering various factors, the high value of a large TSHD makes it difficult to make a real ship experiment. Furthermore, the establishment of the mathematical model of the process and computational fluid dynamics (CFD) simulation are two useful methods in addition to the experiment.

With the development of computer technology, the CFD model has become a powerful tool for research and has been favored by many researchers in the field of dredgers. Yang *et al.* (2013) used CFD software to simulate the loading process of the sediment based on the Reynolds-averaged turbulence model and the solid–liquid two-phase flow theory. Andrun *et al.* (2020) adopted CFD to simulate mud throwing of rake suction dredger, revealing that the influence of cargo power on ship lateral stability in the process of mud throwing could not be ignored.

In the field of pipeline transportation, many researchers have conducted a lot of research with CFD as a tool. Compared with mathematical models, CFD models can visualize physical fields and handle complex boundary problems. Ling *et al.* (2003) employed an algebraic slip mixture (ASM) model in Fluent to simulate and analyze low-concentration slurry in the complete suspension stage of single particle size. The calculated results were consistent with experimental data. Kaushal *et al.* (2012) simulated the horizontal pipeline transportation conditions by the Eulerian two-phase flow model and the mixture model in Fluent, discovering that the Eulerian two-phase flow model is superior to the mixture model. Besides, Kaushal *et al.* (2017) simulated silica sand and fly ash through the Eulerian two-phase flow model in Fluent to obtain the pressure drop and the particle size concentration distribution of slurry. Li *et al.* (2018) developed a universally applicable comprehensive model using Fluent based on the particle dynamics theory, which can accurately describe the dynamic characteristics of slurry transportation in pipelines. In this paper, the mixed flow characteristics of different particle sizes in the pipeline system are taken as the research object to solve the problem of pipe blockage encountered during the construction of the TSHD through the CFD numerical simulation method, analyze the influence of different particle sizes of sediment on the internal flow of a large trailing suction dredger transportation pipeline system, provide theoretical guidance to avoid pipe blockage, and improve the working efficiency of the TSHD.

From another perspective, a considerable number of researchers have explored flow control in pipeline transportation. Priyanka *et al.* (2016) set a PID controller to regulate the flow of oil by controlling the percentage of control valve opening in an oil pipeline. Then, Priyanka *et al.* (2018) established a fuzzy PID controller to regulate the flow of the oil pipeline by controlling different pressure points. Most of the existing methods to control pipeline flow focus on the transportation of uniform materials. Yun *et al.* (2021) proposed a model predictive control (MPC) method to control slurry flow in cutter suction dredger pipeline transportation. The results suggested that the MPC controller can control slurry flow in pipeline transportation more effectively than other PID controllers. In this paper, the MPC controller was designed and compared with the traditional PI controller, with the focus on the mud pipeline transportation system of the TSHD.

The present paper is organized as follows. Section 1 describes the physical model and numerical methods, and results of the transport characteristics are shown in Section 2. Section 3 introduces the theory and design method of the control scheme, and results of the performance of the controller are shown in Section 4. Conclusions are summarized in Section 5.

## PIPELINE MODELING AND NUMERICAL SIMULATION METHODS

### 3D model construction of the pipeline

In this paper, the ‘New Hai Hu 8’ TSHD from rake head to the mud tank section pipeline is taken as the research object. According to the pipeline structural parameters, a three-dimensional model of the flow field in the pipeline is established, as illustrated in Figure 2. The three-dimensional model is divided into structured grids. Additionally, 25 layers of the boundary layer are established in the model to improve the quality of calculation and ensure the convergence of the calculation. The height of the outermost grid is 3 mm, and the growth ratio is 1.2. The grid of the 3D model is exhibited in Figure 3.

### Material setting

The common dredging soil of rake suction dredgers mainly includes fine sand, medium sand, medium-coarse sand, and coarse sand. For example, the soil type of the Yangtze River estuary dredging project is fine sand, while the soil type of the Xiamen airport dredging project is medium-coarse sand. According to the classification of soil, the average particle size of the soil is taken, and its physical properties are presented in Table 1.

Soil classification . | Average particle diameter (mm) . | Soil particle density (t/m^{3})
. | Soil particle density (°) . |
---|---|---|---|

Silt | 0.1 | 2.65 | 30° |

Medium sand | 0.2 | 2.7 | 30° |

Medium-coarse sand | 0.4 | 2.7 | 30° |

Coarse sand | 0.6 | 2.7 | 30° |

Soil classification . | Average particle diameter (mm) . | Soil particle density (t/m^{3})
. | Soil particle density (°) . |
---|---|---|---|

Silt | 0.1 | 2.65 | 30° |

Medium sand | 0.2 | 2.7 | 30° |

Medium-coarse sand | 0.4 | 2.7 | 30° |

Coarse sand | 0.6 | 2.7 | 30° |

The liquid phase is water at room temperature. Its physical properties are as follows: density and dynamic viscosity .

Soil type . | Silt, clay . | silt . | Fine sand, medium sand . | Coarse sand, gravel . |
---|---|---|---|---|

1.10 | 1.20 | 1.25 | 1.30 |

Soil type . | Silt, clay . | silt . | Fine sand, medium sand . | Coarse sand, gravel . |
---|---|---|---|---|

1.10 | 1.20 | 1.25 | 1.30 |

### Governing equation

The Euler–Euler model, which can describe the flow characteristics of liquid–solid two-phase flow more accurately, is suitable for the simulation of high concentration mudflow and achieves a good balance between accuracy and calculation. Therefore, the Euler–Euler two-fluid model is employed in this paper based on particle flow mechanics theory.

- (a)
Volume fraction

is the velocity of the phase *s*;

is the mass transfer between phases.

is the acceleration *g* of gravity;

is the source phase of the phase *s*;

is the shared pressure of the phase *s*;

is the pulsating velocity of the phase *s*.

- (c)
is the phase density of solid particles;

is the diameter of solid particles;

represents the viscosity of the liquid phase;

is the drag force function;

expresses the relaxation time of solid particles.

The Gidaspow model (Gidaspow *et al*. 1992) is selected for numerical simulation. The model is optimized on the basis of Wen and Yu's model and Ergun's model, which can accurately describe the transportation of sediment in high concentration.

### Boundary condition

The inlet is set as the speed inlet. Specific values can be set for the concentration and speed of each item. According to the CFD simulation of the rake head of the TSHD (Sheng *et al.* 2021), the mudflow rate is 3.5 m/s when the inlet concentration is 30%. The sediment velocity should be less than the liquid velocity. The inlet turbulence state is described by the turbulence intensity and hydraulic diameter, which are set as 5% and 0.9 m, respectively. The inlet pressure is calculated by the digging depth and mud pump head.

The outlet is provided with a pressure outlet. The pressure can be measured by the ship pressure sensor. The turbulence state at the outlet is described by turbulence intensity and hydraulic diameter, which are the same as those at the inlet.

The wall surface has a no-slip condition. The roughness height and roughness of the wall surface are 3 mm and 0.2, respectively. The area near the wall surface is treated by the standard wall surface function.

### Solution process and convergence scheme

In this paper, the commercial CFD software Fluent is used to solve the above governing equations and boundary conditions. The RNG model is selected for turbulence model, root mean square residual is adopted, and convergence residual is set as 10e^{−4}. The stability and accuracy of the results are guaranteed by the SIMPLE algorithm with phase coupling, and convergence is obtained. The momentum equation and the other equation are solved by the second-order upwind method and the first-order upwind method, respectively. The pressure relaxation factor and the momentum relaxation factor are set to 0.2, the volume fraction is set to 0.3, and the default values for other factors are used.

## ANALYSIS OF NUMERICAL SIMULATION RESULTS

According to the actual construction history data of a dredger, the real construction state of the dredger is simulated with the digging depth and the inlet flow rate of 20 m and 3.5 m/s, respectively (the speed measured by the mud pump speed sensor is 120 rpm at this time). Under the same conditions, the mud movement state of silt with different particle sizes in pipeline transportation is simulated. Fluent is used to simulate the transport state of four kinds of sediment with different particle sizes (0.1, 0.2, 0.4, and 0.6 mm) under this condition.

### Rationality of the model

In order to conduct the grid sensitivity study, we adopted three different numbers of structural grids to divide the three-dimensional pipeline model, with the key variables as the criteria for verification. As suggested by the result, the three different numbers of grids make little difference to sediment velocity, despite some fluctuations observed. To ensure the accuracy of the simulation, the 700 w grid was chosen.

The rationality of the model should be analyzed first to ensure the accuracy of the simulation. A velocity sensor is installed 2 m below the outlet of the pipe in the real ship. A velocity monitoring surface is set at the position of the velocity sensor in the CFD simulation. Medium-coarse sand (0.4 mm sediment) is used to simulate the mudflow in the pipeline within 30 s. The mudflow rate diagram is illustrated in Figure 4, which is consistent with the average flow rate of 3.57 m/s that is measured by the velocity sensor of medium-coarse sand (0.4 mm sediment) in Xiamen Airport under this condition.

Besides, this model adopts the simplified model of Li Mingzhi's model (Priyanka *et al.* 2018). This model has been verified to be able to basically describe the dynamic characteristics of slurry transportation in pipelines. Hence, no further verification is required in this study.

### Effect of particle size on the particle concentration distribution

For the pipeline transportation system of the TSHD, the most easily blocked pipe location occurs in the rake arm pipeline. Therefore, the mudflow characteristics in the rake arm pipeline need to be stressed.

Figure 5 exhibits the concentration distribution of sediment with different particle sizes in the harrow arm pipeline under the same working conditions, with the parameters of 0.1, 0.2, 0.4, and 0.6 mm, respectively. As suggested in Figure 5, the concentration distribution is different at different heights. When height *Y* = 5 m, *Y* = 10 m, *Y* = 15 m, and *Y* = 20 m, the calculation results of the concentration distribution of sediment with different particle sizes are compared, as illustrated in Figure 6. In the figure, *Y*+ denotes the dimensionless vertical position (*Y* + =*Y*/*D*, i.e., the ratio of the vertical position in the pipe to the pipe diameter), and *Cv* represents the volume concentration of the sediment.

Figure 5 indicates that the maximum concentration of 0.1, 0.2, 0.4, and 0.6 mm sediments reaches 31.91, 36.35, 46.04, and 48.64%, respectively, reflecting that particle size is the main factor affecting the concentration distribution in the pipeline. This is consistent with reality. As the sediment particle size increases, the effective gravity of the sediment particles increases and it is easier to settle, demonstrating the rationality of the model from the side.

Figure 6 exhibits the uneven distribution of sediment concentration along the central vertical line, with a lower and higher concentration in the upper part and lower part of the pipe, respectively. According to Figure 6, when *Y* = 5 m, the lowest and highest concentration of 0.1 mm sediment is 28.16% below the top of the pipe and 31.91% above the bottom of the pipe, respectively. The lowest and highest concentration of 0.6 mm sediment is 0.36% below the top of the pipe and 48.64% above the bottom of the pipe, respectively. This indicates that the difference in sediment concentration distribution increases with the increase in particle size. Since the initial concentration is set at 30%, the sediment is deposited in the pipeline and the transport of sediment in this state will cause the phenomenon of pipe blocking when the difference of sediment concentration distribution is large. Moreover, the sediment concentration distribution is different at different heights because the rake arm pipe is inclined. It can be observed that the concentration distribution of 0.1 and 0.2 mm sediment at different heights is almost unchanged, and the maximum concentration appears at *Y* = 20 m. The concentration distribution of 0.4 and 0.6 mm sediment significantly varies at different heights.

### Effect of particle size on velocity distribution of particles

Figure 7 presents the velocity distribution of sediment with different particle sizes under the same working conditions in the harrow arm pipeline, namely 0.1, 0.2, 0.4, and 0.6 mm sediment. Figure 8 shows the comparison of calculation results of vertical velocity distribution in the sediment with four different particle sizes (0.1, 0.2, 0.3, and 0.4 mm) at different heights (*Y* = 5, 10, 15, and 20 m). In the figure, *Y*+ and *v* indicate the dimensionless vertical position and the horizontal coordinate of slurry velocity, respectively.

As presented in Figure 7, the flow rates of 0.1 and 0.2 mm sediment in the rake arm pipe are relatively uniform, while the flow rates of 0.4 and 0.6 mm sediment in the pipe are quite different. The maximum flow velocity of 0.6 mm sediment is 5.98 m/s above the center of the middle section of the harrow arm pipeline because the concentration at this position is the lowest concentration under the working condition. Under the condition of low concentration, the particle movement in the pipeline is mainly controlled by the drag force of water flow, and the influence of the drag force between the two phases and the friction resistance of the pipeline is weakened. Consequently, the velocity of sediment at this location increases.

Figure 8 suggests that the maximum flow rate of mud for 0.1 and 0.2 mm sediment under this condition is near the center of the pipeline. Under the conditions of 0.4 and 0.6 mm sediment, the maximum velocity point moves upward, reflecting that the particle distribution of 0.1 and 0.2 mm sediment is relatively uniform, while the particle distribution of 0.4 and 0.6 mm sediment is uneven due to the deposition phenomenon. Therefore, 0.4 and 0.6 mm sediment are prone to pipe blocking. At the bend of the rake arm pipe, the velocity distribution of sediment with different particle sizes is significantly different from that of other positions, and the velocity of 0.4 and 0.6 mm sediment decreases. Therefore, the pipe shape has a certain influence on the velocity. Furthermore, different particle sizes of the sediment have the same law. Specifically, the nearer the sediment to the pipe wall, the lower the velocity. The velocity near the pipe wall reaches 0 m/s owing to the friction and collision between the sediment and the pipe wall.

### Minimum practical flow rate

*C*represents the volume concentration of soil particles ();

*g*denotes the acceleration of gravity, = 9.81 (m/s

^{2});

*D*refers to the mud pipe diameter; is the deposition rate of particles in clean water (m/s); and (m) denotes the average particle size of sand.

is the practical minimum velocity coefficient;

is the critical flow velocity, as shown in Table 2.

According to the above formula, the no silting critical flow rate of 0.4 mm mud calculated is 4.5 m/s at a concentration of 30%. The particle concentration distribution of the rake arm pipe is illustrated in Figure 9, and the minimum practical flow rate of 0.4 mm mud calculated is 5.6 m/s at a concentration of 30%. The particle concentration distribution of the rake arm pipe is illustrated in Figure 10. It can be observed that the mud concentration distribution in the pipeline is relatively uniform under the lowest practical flow rate, and the mud concentration distribution can be improved by increasing the mudflow rate.

## PIPELINE VELOCITY MPC

The research results (Yun *et al.* 2021) demonstrated that the speed of the mud pump has a decisive influence on the mudflow rate. Hence, the control input is mud pump speed, and the output is the mudflow rate measured by the sensor. Particle size and mud concentration are measured as disturbances. Besides, a dynamic matrix control (MDC) controller is designed in this paper to suppress disturbance and achieve better tracking performance.

MPC is to predict the output of the future moment using a predictive model when the control process changes its input. MPC has different forms, among which MDC adopts the step response model of the object and has been widely used in practical processes because of its advantages such as easy model acquisition and effective solution to the delay process. The control structure of MDC is mainly composed of the prediction model, rolling optimization, feedback correction, and closed-loop control, as presented in Figure 11.

### Prediction model

*A*is the dynamic matrix:where

*m*,

*n*, and

*a*denote the control length, the maximum predicted length, and the step response coefficient.

### Rolling optimization

*Y*is replaced with

*Y*. Equation (13) is substituted into Equation (17). Without the consideration of constraints, the optimal control delta sequence has a theoretical expression aswhere denotes a coefficient, . Particularly, the optimal control increment sequence can be calculated theoretically once the predicted output vectors and coefficients are obtained.

### Feedback correction

### Analysis of predictive model control simulation results

In the simulation, the corresponding mudflow rate is 3.5 m/s when the initial mud pump speed is set at 120 rpm. According to the lowest practical flow rate under this working condition, the expected mudflow rate is set as 5.6 m/s, and the time range is set as 100 s. Then, the step response of the proposed DMC controller and the PI controller is evaluated through the derived process transfer function.

In the case of no disturbance, step response simulation results of the DMC controller and the PI controller are illustrated in Figure 11. The scale factor and the integration factor of the PI controller are 125 and 15, respectively. It can be observed that both controllers can reach the desired output value, and the proposed DMC control method reaches the desired output value in 18.9 s, which is twice that of the PI control method. Moreover, the change in the system input (mud pump speed) is also different, as shown in Figure 12. The maximum mud pump speed under the PI controller is 260 rpm, while the maximum mud pump speed under the DMC controller is 225 rpm. Under the control of DMC, the variation range of mud pump speed is small. All the results demonstrated that the DMC controller is superior to the PI controller for the case study.

## CONCLUSION

In this paper, the Euler two-phase flow model is used to establish a three-dimensional hydrodynamic model of pipe transportation in the rake arm section of the TSHD. The volume concentration distribution and velocity distribution of particle phase are obtained by simulating the slurry transportation state of a real ship under certain working conditions. Besides, the trends of these parameters are analyzed for different particle sizes. The results reveal that the average velocity under the monitoring surface is consistent with the velocity measured by the ship's velocity sensor.

The conclusions of this study are drawn as follows:

- (1)
Under the identical construction conditions for the TSHD, the particle size of sediment has an effect on concentration distribution in the pipeline. To be specific, the larger the particle size of the sediment, the more significant the difference in sediment concentration distribution in the pipeline. Similarly, the flow velocity is another influencing factor for the change in concentration distribution in the pipeline. Besides, with flow velocity increasing, the uniformity of concentration distribution in the pipeline will improve.

- (2)
In the rake arm pipeline of the rake suction dredger, the middle and lower sections of the rake arm pipeline should be considered first when the trend of pipe blockage occurs. The maintenance after the occurrence of pipe blockage should be emphasized.

- (3)
The critical velocity plays a role in keeping the concentration distribution of pipeline transportation within a reasonable range. During construction, the practical minimum flow velocity is referenced for the control imposed on mud velocity. Besides, increasing mud velocity is beneficial to improve the distribution of mud concentration.

Moreover, a dynamic matrix controller is designed under the assumption of no disturbance to regulate the mudflow rate in the pipeline by controlling the mud pump speed. Compared with the traditional PI controller, the proposed MPC controller has the advantages of short adjusting time and small overshoot. There are many constraints in the experimental study of pipeline transportation of TSHDs at present. Therefore, the mechanism analysis of the TSHD transportation system and the controller design in ANSYS FLUENT are combined in this paper to provide an effective method, so as to prevent and solve the pipe blockage in the transportation process.

## ACKNOWLEDGEMENTS

This research was funded by the Ministry of Industry and Information Technology Project (Document No. [2019]360).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST STATEMENT

The authors declare there is no conflict.

## REFERENCES

*Model-Based Control of Hopper Dredgers*

*PhD Thesis*

*The Hydraulic Transportation of Coal and Other Materials in Pipes*