# Mathematical Model for the Movement of Two-Pipe Vehicles in a Straight Pipe Section

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Establishment of Mathematical Models

#### 2.1. Mathematical Model of Concentric Annular Gap Flow

_{x}, F

_{y}, F

_{z}are the mass forces along the X, Y, and Z axes, respectively; p is the dynamic water pressure; u

_{x}, u

_{y}, and u

_{z}are the fractional velocities of the annular gap flow along the X, Y, and Z axes, respectively; ν is the kinematic viscosity coefficient; and ρ is the liquid density.

_{r}, F

_{θ}, and F

_{z}are the mass forces along the radius, the angle of θ, and the Z axis, respectively; u

_{r}, u

_{θ}, and u

_{z}are the velocities along the radius, the angle of θ, and the Z axis, respectively; r = (x

^{2}+ y

^{2})

^{0.5}; and θ = arctan(y/x).

_{r}= F

_{θ}= F

_{z}= 0. It can also be assumed that the concentric annular gap flow is a univariate flow; that is, u

_{r}= u

_{θ}= 0, u

_{z}= u. Thus, Equation (2) can be simplified as follows:

^{2}u/∂θ

^{2}= 0, and ∂

^{2}u/∂z

^{2}= 0 can be derived from the continuity equation. Thus, Equation (3) can be simplified to:

_{c}. When z = r

_{c}, u = u

_{c}, r = r

_{p}, u = 0, then c

_{1}and c

_{2}can be obtained as follows:

#### 2.2. Mathematical Model of the Two-Pipe Vehicles

#### 2.2.1. Force Analysis of the Two-Pipe Vehicles

_{p}), the friction force on the support body (F

_{f}), and the shear force on the wall of the pipe vehicle (F

_{τ}). These six forces can be divided into two directions: horizontal and vertical. Those in the vertical direction include G, f, and N. These three forces are body forces, which are mainly related to the volume of pipe vehicle and the density of conveyed materials. Those in the horizontal direction include F

_{p}, F

_{f}, and F

_{τ}.

_{1}and G

_{2}represent the gravities of the rear and front vehicles, respectively; m

_{c}is the weight of a single vehicle when it is empty; and m

_{w}is the weight of the goods transported in a single vehicle.

_{c}is the volume of a single vehicle and ρ is the liquid density.

_{1}and Δp

_{2}denote the pressure differences between the front and rear of the two pipe vehicles, respectively, and A is the end face area of the pipe vehicle.

_{1}is the flow resistance coefficient, which is related to the fluid and the material of the pipe vehicle; v

_{a}is the average velocity of the annular gap flow; v

_{c}is the average movement speed of the two-pipe vehicles; and A’ is the sum of the side areas of the two pipe vehicles.

#### 2.2.2. Mechanical Equilibrium Equation for Stable Movement of the Two-Pipe Vehicles

_{1}is the wall shear stress of the pipe vehicle; τ

_{2}is the water flow shear stress at the interface of parts II and III; and Δp is the pressure difference between the upstream and downstream end faces of the pipe vehicle.

_{3}is the average shear stress at the pipe wall.

#### 2.2.3. Motion Equation for Stable Movement of the Two-Pipe Vehicles

_{c}/D

_{p}), then Equation (21) can be simplified to:

_{1}and τ

_{2}in Equation (18) can be expressed as follows:

_{1}is the flow resistance coefficient, which is related to the fluid and the material of the pipe vehicle; and λ

_{2}is the friction coefficient of the pipe wall.

_{p}, the diameter of the pipe vehicle D

_{c}, and the vehicle length l. This formula reflects the relationship between various factors and the movement speed of the two-pipe vehicles, which can be used to qualitatively analyze the internal relationship between the movement speed of the two-pipe vehicles and each factor, providing a certain theoretical reference for optimizing the structure of two-pipe vehicles.

## 3. Experimental Methods

#### 3.1. Experimental System and Procedures

- (1)
- The power device used in this experiment is a high-power centrifugal pump, and the adjustment device includes a control valve (gate valve) and an electromagnetic flowmeter.
- (2)
- The delivery device is made of trumpet-shaped stainless steel material, with a gate valve installed on the upper part, as shown in Figure 7a. The pipe vehicle starting device is installed 1.6 m downstream of the delivery device. When the pipe vehicle enters the pipeline, the starting device is in a closed state. When the pipe vehicle needs to be moved in the pipeline, the starting device is opened to release the pipe vehicle. The receiving device consists of three parts: A rectangular water tank, a baffle, and a plastic collection box, as shown in Figure 7b.

- (3)
- The experimental instrument mainly includes a two-pipe vehicles timing device and a flow field measurement device. The two-pipe vehicles timing device consists of an infrared probe and a display, as shown in Figure 8. Infrared probes are installed near the launching device and the outlet of the pipeline, respectively. When the two-pipe vehicles passes the first infrared probe, the timing starts, and the timing stops when it passes the second probe. According to the speed formula, the average movement speed of the two-pipe vehicles can be calculated. The flow field measurement device is a Laser Doppler Velocimeter (LDV). In order to reduce the refraction of the laser light by the pipe wall, a rectangular water jacket filled with water was added to the test pipe section.

- (4)
- The conveying pipe used in this test is composed of steel and plexiglass pipes. The lengths of the steel and plexiglass pipes are 8.6 m and 21.3 m, respectively, the thickness of the pipe wall is 5 mm, and the outer diameter of the pipe is 110 mm. The entire pipeline is formed by connecting multiple sections of plexiglass tubes through flanges, and a height-adjustable support body is installed at intervals below the plexiglass tubes in order to ensure that the plexiglass tubes are at the same level.

#### 3.2. Section Selection and Measuring Point Arrangement

_{1}= r

_{c}+ 1/5 B, r

_{2}= r

_{c}+ 2/5 B, r

_{3}= r

_{c}+ 1/2 B, r

_{4}= r

_{c}+ 3/5 B, and r

_{5}= r

_{c}+ 4/5 B (where B is the width of the annular gap). The intersection of the pipe radius and the measuring ring served as the layout measuring points, for a total of 60 measuring points, as shown in Figure 10.

#### 3.3. Experiment Plan

^{3}/h, 40 m

^{3}/h, 50 m

^{3}/h, or 60 m

^{3}/h; the model dimensions of the pipe vehicle l × D

_{c}were 150 mm × 60 mm, 150 mm × 70 mm, 150 mm × 80 mm, 100 mm × 60 mm, 100 mm × 70 mm, or 150 mm × 80 mm; the conveyed load m was 400 g, 600 g, 800 g, 1000 g, 1200 g, 1400 g, 1600 g, 1800 g, or 2000 g; and the vehicle spacing L was 7.5 cm, 15 cm, 30 cm, 45 cm, or 60 cm.

## 4. Results and Discussion

#### 4.1. Mathematical Model Verification

#### 4.2. Analysis of Influencing Factors for Two-Pipe Vehicles Movement Speed

- (1)
- It can be seen, from Figure 12a, that when the other variables remained unchanged, the average movement speed of the two-pipe vehicles presented a linear relationship with the flow rate. This is mainly due to the pressure difference between the front and rear faces being the main driving force for the two-pipe vehicles. With the continuous increase in the flow rate, the differential pressure force gradually increases. Although the resistance of the pipe vehicle also gradually increases with an increase in the flow rate, the increase in the resistance with the flow rate is less than the increase in the pressure, causing the total power of the two-pipe vehicles to increase. Therefore, the movement speed of the two-pipe vehicles increases gradually with an increase in the flow rate.
- (2)
- It can be seen from Figure 12b that when the length of the vehicle l is held constant, the average movement speed of the two-pipe vehicles increases gradually with an increase in the diameter of the pipe vehicle D
_{c}. This is mainly because, when the diameter D_{c}increases, the end face area, side wall area, and volume of the two-pipe vehicles increase accordingly. According to Equations (15)–(17), the pressure differential force and wall shear stress on the pipe vehicle gradually increase, while the frictional resistance decreases. The changes in differential pressure force, wall shear stress, and frictional resistance increase the driving force of the two-pipe vehicles, such that the movement speed of the two-pipe vehicles increases accordingly. When the diameter of the pipe vehicle is constant, the average movement speed of the two-pipe vehicles increases gradually with an increase in vehicle length. This is mainly because the longer the body length l, the larger the body side wall surface area and the body volume, such that the wall shear stress increases while the frictional resistance decreases and, so, the total driving force of the two-pipe vehicles increases. Therefore, the average movement speed of the two-pipe vehicles also increases. By comparing the experimental results, it can be seen that the average movement speed increment of the two-pipe vehicles caused by the change of diameter D_{c}is greater than that caused by the change in the vehicle length l. - (3)
- It can be seen from Figure 12c that with an increase in the conveyed load, the movement speed of the two-pipe vehicles shows a gradually decreasing trend. This is mainly due to the fact that when the model of the pipe vehicle is kept constant, the buoyancy of the two-pipe vehicles does not change; however, when the load increases, the support force of the pipe vehicle increases, and the frictional resistance increases accordingly. Therefore, the movement speed of the two-pipe vehicles is reduced.
- (4)
- It can be seen from Figure 12d that, with an increase in the vehicle spacing, the average movement speed of the two-pipe vehicles presents a gradually increasing trend. When the vehicle spacing is relatively small, the two-pipe vehicles can be regarded as a whole, and the pressure drop along the direction of the two-pipe vehicles is small; that is, the pressure difference between the front and rear surfaces of the two-pipe vehicles is small. With an increase in the vehicle spacing, the pressure difference between the front and rear surfaces of the pipe vehicle gradually increases and, thus, the movement speed of the two-pipe vehicles gradually increases. With a further increase in the distance between the vehicles, the influence between the two-pipe vehicles is almost negligible, and the movement speed of the two-pipe vehicles gradually changes to that of a single pipe vehicle moving in the pipeline. When the vehicle spacing reaches 300 mm, the movement of the two-pipe vehicles can basically be regarded as the movement of a single pipe vehicle. As the water flow goes through the flow process of “full pipe–annular gap–vehicle spacing–annular gap–full pipe”, when the two-pipe vehicles move in the pipeline, the energy loss is larger than that of a single pipe vehicle and the kinetic energy obtained by the two-pipe vehicles is smaller than that of a single pipe vehicle, such that the movement speed of the two-pipe vehicles is lower than that of a single pipe vehicle under the same working conditions. It can also be seen from the figure that when the spacing is between 50 mm and 150 mm, the growth rate of the average movement speed of the two-pipe vehicles is larger; that is, the slope of the curve is larger. When the spacing ratio is between 150 mm and 250 mm, the average movement speed of the two-pipe vehicles changes little with an increase in the spacing. When the spacing is greater than 250 mm, the growth rate of the average movement speed of the two-pipe vehicles becomes larger again.

_{1}, x

_{2}, …, x

_{m}, and we have a test sample size of n. Note that x

_{ik}represents the value of the independent variable x

_{i}in the kth trial, and Y

_{k}represents the result of the random variable Y in the kth trial. If there exists a linear relationship between Y and x

_{i}, the regression equation can be written as:

_{i}in Equation (33) are obtained using the following equations:

_{i}’ of Y to x

_{i}and the regression coefficient b

_{i}of Y to x

_{i}:

_{i}’ has nothing to do with the units of Y and x

_{i}, such that they can be directly compared. The larger the value of |b

_{i}’|, the greater the effect of x

_{i}on Y.

_{c}is 60 mm, 70 mm, or 80 mm; the pipe vehicle length l is 100 mm or 150 mm; the flow rate Q is 30 m

^{3}/h, 40 m

^{3}/h, or 50 m

^{3}/h; the conveyed load m is 400 g, 600 g, or 800 g; and the vehicle spacing L is 50 mm, 100 mm, or 150 mm. The five variables are orthogonally considered in pairs, such that the number of experiments was 162. The average movement speed of the two-pipe vehicles was obtained, according to the experimental data. Substituting the data into the corresponding equations, we obtain:

_{c}, the length of the pipe vehicle l, the conveyed load m, and the vehicle spacing L, in decreasing order. Therefore, in the process of transportation, it is necessary to comprehensively consider various factors to carry out reasonable matching such that the two-pipe vehicles can obtain a higher transportation movement speed.

#### 4.3. Research Implications and Limitations

## 5. Conclusions

_{c}and length l of the pipe vehicle, the movement speed of the two-pipe vehicles increased gradually. With increasing spacing L, the movement speed of the two-pipe vehicles gradually increased, moving closer to that of a single vehicle. The standard regression coefficient method in the multiple regression analysis was used to analyze the influencing factors, and it was found that the flow rate Q had the strongest influence on the movement speed to the two-pipe vehicles, followed by the diameter D

_{c}, vehicle length l, load m, and vehicle spacing L, in decreasing order.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Pipe side view. Note: 1. pipe vehicle; 2. Support body; 3. Pipeline; and 4. Concentric annular gap.

**Figure 2.**Three-dimensional rectangular coordinate system: (

**a**) Sectional view; and (

**b**) Side view. Note: 1. Pipe vehicle; 2. Pipe wall; 3. Concentric annular gap; l represents the length of the pipe vehicle, r

_{p}represents the pipe radius, r

_{c}represents the pipe vehicle radius, and B represents the width of the annular gap.

**Figure 3.**Structure diagram of the two-pipe vehicles: (

**a**) A single vehicle; and (

**b**) Two-pipe vehicle. Note: 1. Barrel; 2. Support body; 3. Pipeline; 4. Linker.

**Figure 4.**Force diagram of a two-pipe vehicles. Note: G

_{1}, G

_{2}are gravity; f

_{1}, f

_{2}are buoyancy; N

_{1}, N

_{2}are the force of the pipeline on the support body; F

_{p}

_{1}, F

_{p}

_{2}, F

_{p}

_{3}, F

_{p}

_{4}are the end pressure differential force; F

_{f}

_{1}, F

_{f}

_{2}are the friction force on the support body; F

_{τ}

_{1}, F

_{τ}

_{2}are shear force on the wall of the pipe vehicle; F

_{L}

_{1}, F

_{L}

_{2}are the spring force.

**Figure 5.**Schematic diagram of area division. Note: D

_{c}is the diameter of the vehicle, D

_{p}is the diameter of the pipe, v

_{a}is the velocity of the annular gap flow, v

_{c}is the movement speed of the two-pipe vehicles, and v

_{p}is the water flow velocity in the pipe.

**Figure 6.**Experimental system. Note: 1. Centrifugal pump. 2. Regulating valve. 3. Electromagnetic flowmeter. 4. Feeding device. 5. Brake device. 6. Infrared probe. 7. timer. 8. Pressure sensors. 9. Rectangular water jacket. 10. Laser Doppler velocimeter. 11. Laser probe. 12. Computer. 13. Water tank.

**Figure 9.**Layout of the measuring cross-sections. Note: 1. Rear pipe vehicle; 2. Front pipe vehicle.

**Figure 11.**Comparison of calculated and measured values: (

**a**) 2# cross-section; and (

**b**) 5# cross-section.

**Figure 12.**Two-pipe vehicles movement speed under different working conditions: (

**a**) Under different flow rates; (

**b**) under different vehicle models; (

**c**) under different loads; and (

**d**) under different vehicle spacings.

Q | l × D_{c} | L | m | Calculated Value | Measured Value | Relative Error |
---|---|---|---|---|---|---|

30 | 150 × 70 | 15 | 600 | 1.14 | 1.1 | 3.51 |

30 | 150 × 70 | 30 | 600 | 1.16 | 1.18 | −2.01 |

30 | 150 × 70 | 15 | 1000 | 1.05 | 0.98 | 6.24 |

30 | 150 × 70 | 30 | 1000 | 1.06 | 1.02 | 3.82 |

30 | 100 × 60 | 15 | 600 | 1.11 | 1.05 | 5.03 |

30 | 100 × 60 | 45 | 600 | 1.13 | 1.15 | −1.65 |

30 | 100 × 60 | 15 | 1000 | 1.01 | 0.96 | 5.29 |

30 | 100 × 60 | 45 | 1000 | 1.04 | 1.08 | −4.12 |

50 | 150 × 70 | 15 | 600 | 1.83 | 1.78 | 2.52 |

50 | 150 × 70 | 45 | 600 | 1.87 | 1.92 | −2.75 |

50 | 150 × 70 | 15 | 1000 | 1.67 | 1.68 | −0.35 |

50 | 150 × 70 | 45 | 1000 | 1.71 | 1.74 | −1.56 |

50 | 100 × 60 | 15 | 600 | 1.77 | 1.65 | 6.82 |

50 | 100 × 60 | 45 | 600 | 1.81 | 1.72 | 5.09 |

50 | 100 × 60 | 15 | 1000 | 1.62 | 1.51 | 6.99 |

50 | 100 × 60 | 45 | 1000 | 1.66 | 1.59 | 4.3 |

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**MDPI and ACS Style**

Jia, X.; Sun, X.; Li, Y. Mathematical Model for the Movement of Two-Pipe Vehicles in a Straight Pipe Section. *Water* **2022**, *14*, 2764.
https://doi.org/10.3390/w14172764

**AMA Style**

Jia X, Sun X, Li Y. Mathematical Model for the Movement of Two-Pipe Vehicles in a Straight Pipe Section. *Water*. 2022; 14(17):2764.
https://doi.org/10.3390/w14172764

**Chicago/Turabian Style**

Jia, Xiaomeng, Xihuan Sun, and Yongye Li. 2022. "Mathematical Model for the Movement of Two-Pipe Vehicles in a Straight Pipe Section" *Water* 14, no. 17: 2764.
https://doi.org/10.3390/w14172764