Mathematical Model of the Piped Vehicle Motion in Piped Hydraulic Transportation of Tube-Contained Raw Material

Li, Yongye; Sun, Xihuan

doi:https://doi.org/10.1155/2019/3930691

Mathematical Problems in Engineering

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 3930691 | https://doi.org/10.1155/2019/3930691

Mathematical Model of the Piped Vehicle Motion in Piped Hydraulic Transportation of Tube-Contained Raw Material

Yongye Li¹and Xihuan Sun^1,2

Academic Editor: Mostafa S. Shadloo

Received08 Jul 2018

Revised22 Dec 2018

Accepted22 Jan 2019

Published18 Feb 2019

Abstract

The piped hydraulic transportation of tube-contained raw material is an emerging technique for transporting materials. In this technique, the piped vehicle is one of the core components, and its motion characteristics directly determine the transportation energy consumption and the transportation cost of this technique. To study the motion characteristics of the piped vehicle, the force of the piped vehicle was analyzed from the mechanical perspective in this paper. On the assumption that the piped vehicle moved steadily and it had sufficient stiffness, the mathematical model of the piped vehicle motion was established in the turbulent flow according to the stress characteristics of the piped vehicle and the factors influencing its motion characteristics, and then the mathematical model was tested by experiments. The findings show that the calculated values of the velocities of the piped vehicle were identical to the experimental values with changes in various influencing factors. When the flow discharge, the diameter or length of the piped vehicle increased, or the mass of transported material decreased, the velocity of the piped vehicle increased. The maximum relative error did not exceed 9.47%, which proved that the mathematical model of the piped vehicle motion was rational. The results can provide theoretical basis to improve the structure of the piped vehicle and the piped hydraulic transportation technique of tube-contained raw material.

1. Introduction

The four conventional transportation options are highway, railway, aviation, and shipping, which lead to problems such as environmental pollution [1] and energy shortage [2, 3]. As a means of transportation different from traditional ones, pipeline transportation can effectively alleviate environmental pollution and energy shortage to a certain extent. Pipeline transportation can be divided into two types: pipeline pneumatic conveying and pipeline hydraulic transportation [4]. Pipeline pneumatic transporting uses gas as transporting medium and granular or powdery substances are carried by high-flow gas, whereas pipeline hydraulic transportation utilizes water as the transporting medium and materials are transported by water flow [5]. At the present, there are mainly three types of pipeline hydraulic transportation: slurry pipeline hydraulic transportation, molding product pipeline hydraulic transportation, and piped hydraulic transportation of tube-contained raw material.

The slurry pipeline hydraulic transportation began in the middle of 19th century. At the beginning, the transportation of low-concentration solid materials in small-diameter and short-distance pipelines was studied in only a few countries and it was in the stage of small-scale experimental exploration. The transportation technique did not develop rapidly until the 1950s and gradually developed from short-distance and low-concentration transportation to long-distance and high-concentration transportation. To improve the transportation technique, many scholars studied the friction loss and resistance reduction measures [6, 7], flow state [8–10], concentration distribution [11, 12], pressure and velocity characteristics [13, 14], critical velocity [15], erosion wear [16, 17], and safety of pipeline operation [18, 19]. However, some drag reducing agents need to be added in the slurry pipeline hydraulic transportation and they can cause some pollution to the environment. Additionally, if the slurry hardens in the process of transportation, it will affect the safety and stability of the pipeline operation.

The molding product pipeline hydraulic transportation [20] was proposed by Henry Liu and Marrero at the Capsule pipeline Research Center of the University of Missouri Columbia and obtained a U.S. patent. In this technique, bulk material was firstly compacted into the cylindrical molding product, and then the molding product was transported by pipe over great distances [21, 22]. Liu et al. conducted many experimental studies, investigating hydrodynamics [23], molding product manufacturing [24], the effect of polymer additives [25], economics of molding product hydraulic pipeline transportation, and so on [26]. However, the molding product pipeline hydraulic transportation is only suitable for transporting molding solid materials and needs high velocity for delivery. Additionally, the movement of molding product in pipeline transportation is also unstable. Its front and rear sections are bumping up and down, colliding the pipe and affecting the life of the pipe.

As a type of piped hydraulic transportation, the piped hydraulic transportation of tube-contained raw material [27] is an emerging technique of transportation. It is developed on the basis of molding product pipeline hydraulic transportation. In this technique, the container of the cylindrical shape with six small cylindrical braces is firstly made, producing “the piped vehicle” which is waterproof and wear resistant. Then raw material is placed and sealed in the piped vehicle, which is transported by pipe over great distances under the action of water flow. Because raw material is placed and sealed in the piped vehicle during the transportation, no pollution is caused to the environment. Furthermore, the transporting medium of this technique is water, which can be mine water, reclaimed water, or other nondrinking water. And water can be recycled, so a large amount of water can be saved, considerably alleviating the increasing water resources tension. Therefore, the transportation technique is environmentally friendly and energy-efficient. Compared with conventional transportation options such as truck and rail, the piped hydraulic transportation technique of tube-contained raw material has several potential advantages [27]: it is less energy-intensive, less harmful to the environment, less dependent on weather, and more reliable. Increasing use of the piped hydraulic transportation of tube-contained raw material would reduce the number of heavy trucks on highways, which not only alleviates traffic congestion and accidents but also reduces noise and environmental pollution.

The technique of piped hydraulic transportation of tube-contained raw material was proposed by Sun Xihuan in 2007 [27]. Following that, much basic research work was conducted by Sun Xihuan and his coworkers to study hydraulic characteristics. Zhang Qiqi et al. [28] investigated the annular flow field around the piped vehicle, analyzed the variations of axial velocity and hydraulic loss with flow discharges and annular ratios, and concluded that the head loss was the least when the annular ratio was 0.4 and the flow discharge was 50m³/h. Zhang Xuelan et al. [29, 30] studied the wall-bounded flow around the piped vehicle at a high Reynolds numbers body in a determined computational domain and found that a reversing zone, appearing as a tongue zone with nested velocities higher than the surrounding area, existed behind the piped vehicle. Li Yongye et al. [31–33] analyzed the distribution features of the annular gap flow between the piped vehicle and the pipeline under different gap widths while the piped vehicle was moving and found that the annular gap flow velocity increased firstly and then decreased as the gap width enlarged. When the gap width was about 2 cm, the annular gap flow velocity reached a maximum. Meanwhile, the fluidal pressure characteristics in the pipe were analyzed under different transported loads. Zhang Chunjin et al. [34] established the mathematical models of the annular gap flow with the moving boundary under the different annular gap widths in the piped hydraulic transportation technique of tube-contained raw material according to the kinetic mechanism of the piped vehicle, solved by adopting the RNG k-ε turbulent model, the 6DOF coupling model, and the PISO algorithm of the commercial software Fluent, and obtained the effects of the annular gap width on the internal hydraulic characteristics of the flow velocity and the pressure in the annular gap flow with the moving boundary. Wu Jian et al. [35] studied the velocity characteristics of the concentric annular flow in the pipe with the piped vehicle’s diameters of 50, 60, 70, and 80mm at the flow discharge of 40 m³/ h and obtained the variation of the axial velocity of the annular gap flow with the diameter of the pipe vehicle. As described by them, the different flow discharges, masses of transported materials, and dimensions of the piped vehicle influenced the hydraulic characteristics of the piped hydraulic transportation of tube-contained raw material. With regard to the motion characteristics of the piped vehicle, the relationship between the velocity of the pipe vehicle and each influencing factor was determined by experiments. Pierre Chapelle [36] established the motion model of the tiny particles in a straight-pipeline pneumatic conveying by analyzing their stress characteristics. However, this model presented only the motion model of tiny particles in a straight pipeline and did not involve large bodies, bending pipeline, uphill pipeline, and downhill pipeline. Besides, this mathematical model used gas as the transporting medium.

In summary, sufficient information for the mathematical model of piped hydraulic transportation of tube-contained raw material was not provided by the above studies. However, establishing a reasonable mathematical model of the piped vehicle was a key problem in popularizing the technique of piped hydraulic transportation of tube-contained raw material. To solve this problem, the mathematical model of the piped vehicle motion was established by analyzing the force of the piped vehicle based on the continuity equation from the viewpoint of mechanics in this paper. The mathematical model would provide theoretical basis for the popularization and application of the technique of piped hydraulic transportation of tube-contained raw material.

2. The Establishment of Mathematical Model

2.1. The Force Analysis of the Piped Vehicle

The piped vehicle was taken as the research object. No matter how it moved in the pipe, the piped vehicle remained subject to gravity (), buoyancy (), support force of the pipe wall against the piped vehicle (), friction force (), pressure gradient force (), and shear force along the piped vehicle (), as shown in Figure 1.

The gravity (), buoyancy (), and support force of the pipe wall against the piped vehicle () belonged to the volume force, related to the density of the transported material and dimension of the piped vehicle.

The friction force () between the pipe wall and the piped vehicle prevented the movement of the piped vehicle, was resistance, and can be expressed aswhere was the friction coefficient.

The pressure gradient force (△P) was the force difference of water acting on the upstream and downstream ends of the piped vehicle, which was the driving force pushing the piped vehicle, and it can be expressed as follows:where was the pressure difference between the pipe flow acting on the head face and terminal face of the material tube; A was the face area of the piped vehicle expressed by .

The shear force along the piped vehicle () was the direct outcome of fluid viscosity, and it can be expressed as follows [37]:where was the density of the liquid, was the average rate of the annular gap flow, was the flow resistance coefficient related to the liquid and the material of the piped vehicle and was obtained from the Moody diagram according to the flow discharge and relative roughness of the piped vehicle surface, was the average velocity of the piped vehicle, and was the lateral area of the piped vehicle given by .

The pressure gradient force and shear force along the piped vehicle were the two main factors which decided the motion state of the piped vehicle.

2.2. Mathematical Model of the Piped Vehicle in Stable Motion

Figure 2 showed the mathematical model of the piped vehicle moving in the pressure pipe. When the piped vehicle moved in the pipe, the center of the piped vehicle and that of the pipe coincided; that is, they moved with the same axle because the ends of the piped vehicle were equipped with braces. Based on the continuity equation of the piped vehicle in motion, from the viewpoint of mechanics, the mathematic model of the piped vehicle in stable motion was established according to the stress characteristics of the piped vehicle.

When the piped vehicle moved steadily, the flow was characterized as steady flow in the coordinate system when the coordinate system was established on the piped vehicle. According to the continuity equation, mathematical model of the piped vehicle in stable motion can be described as follows:where was the average velocity of the flow in the pipe, was the diameter of the pipe, was the diameter of the piped vehicle, was the motion period, and the rest of the symbols were the same as described in the preceding paragraphs.

If we let , then the average rate of the annular gap flow can be expressed as follows.

When the piped vehicle moved steadily along the axle of the pipe, both the forces acting on the piped vehicle and the gap flow in the part where the piped vehicle was, should reach equilibrium.

Based on the force condition of the piped vehicle mentioned above, the shear force along the piped vehicle was connected with the average velocity of the annular gap flow and the average velocity of the piped vehicle. When the average velocity of the annular gap flow was faster than the average velocity of the piped vehicle, the direction of the shear force along the piped vehicle was consistent with the direction of the flow, and the shear force along the piped vehicle provided power for the piped vehicle. Otherwise, it would be resistant to the piped vehicle. If the shear force along the piped vehicle was power and the pipe section was an uphill section to the piped vehicle, then the balance equation of the force along the pipe axle can be expressed as follows.Therefore,where was the angle between the pipe axle and the horizontal plane.

The balance equation of the force along the pipe axle for the annular gap flow in the part where the piped vehicle was, can be expressed as follows:where was the pressure difference between the head face and terminal face of the annular gap flow, was the shear stress of the wall surface acting on the annular gap flow, and the rest of the symbols were the same as described in the preceding paragraphs.

Therefore,wherewhere was the friction coefficient of the pipe wall and was obtained from the Moody diagram according to the flow discharge and relative roughness of the pipe, was the length of the piped vehicle, and the rest of the symbols were the same as described in the preceding paragraphs.

Substituting (1), (3), and (5) into (7) yielded the following.Let .

Therefore,Substituting (3), (5), and (10) into (9) yielded the following.Let Therefore, where .

Therefore, the mathematical model of the piped vehicle motion was derived from (12) and (15) and it can be expressed as follows.

If the shear force along the piped vehicle was resistance, then (6) was given by and (8) was given by . Other derivation processes were the same as described and would not be discussed in the following sections. When the shear stress of the wall surface borne by the piped vehicle was resistance, the mathematical model of the piped vehicle motion was derived as follows:where ; .

If the pipe section was a downhill section, in (16) and (17) was replaced with. If the pipe section was a straight pipe, in (16) and (17) was replaced with 0.

Equations (16) and (17) represented the mathematical model of the piped vehicle motion. The mathematical model included all the main factors that influenced the velocity of the piped vehicle, such as the flow discharge, mass of transported material, and dimension of the piped vehicle. The mathematical model reflected the relationship between the velocity of the piped vehicle and the flow discharge, mass of transported material, and dimension of the piped vehicle. The mathematical model can be used to qualitatively analyze such relations and provided certain theoretical references to improve the piped vehicle structure.

3. Mathematical Model Verification

3.1. Test Materials and Method

As a container that carried raw material, the piped vehicle was the core component of the piped hydraulic transportation technique of tube-contained raw material. Its motion behaviors determined the capacity of transported material and directly influenced the energy consumption during the transportation process, thereby determining the cost and economic benefits of material transportation to a certain degree. Figure 3 presented the physical diagram of the piped vehicle. The body of the piped vehicle was cylindrical and made of plexiglass. Iron braces were placed on each end of the piped vehicle. The axis of the piped vehicle coincided with that of the pipe; that is, they kept moving with the same axis when the piped vehicles with different dimensions moved in the pipe because the iron braces always touched the pipe wall. Universal balls were installed at the end of the braces to reduce the friction between the braces and the pipe wall, and increased the service life of the piped vehicle.

The test device [38–40] mainly consisted of the water supply device, test pipe, delivering and receiving devices for the piped vehicle, turbine flowmeter that measured the flow discharge, and photoelectric sensor that measured the velocity of the piped vehicle. Figure 4 presented the experimental system device. The test pipe is a plexiglass pipe with inner diameter of 100 mm, wall thickness of 5 mm, and total length of approximately 44 m. The test pipeline consisted of straight, flat bending, oblique bending, and uphill and downhill sections. According to the test field, the turning radius of the flat bending sections and that of oblique bending sections were both determined to be 650 mm, and the slope of uphill sections and that of downhill sections were both 10°. Figure 5 showed the delivering device for the piped vehicle. The delivering device for the piped vehicle was a steel pipe. To enter the transportation pipeline smoothly, the steel pipe was trumpet-shaped, inclined at a certain angle, and connected to the transportation pipeline when equipped. The receiving device for the piped vehicle was a rectangular tank made of plexiglass plates. The tank contained opening baffle plates to separate the piped vehicle and the water.

(a) Schematic diagram

(b) Physical diagram

3.2. Mathematical Model Verification

The factors that influenced the motion characteristics of the piped vehicle mainly included the dimension of the piped vehicle, mass of the transported material, and flow discharge. These influencing factors were considered as the main controlling factors to investigate the motion characteristics of the piped vehicle under different working conditions. Experimental results from using photoelectric sensors indicated that the velocities of the piped vehicle were extremely close; that is, changes were only in the second decimal place. Therefore, each experiment was repeated thrice and the average value was used to meet the test requirements. To accommodate the size of the test pipe and ensure that the piped vehicle could go through the bending sections smoothly, we chose the following dimensions of the piped vehicle in the experiments: × was separately 150 mm × 50 mm, 150 mm × 60 mm, 150 mm × 70 mm, 100 mm × 70 mm, and 100 mm × 80 mm (L was the length of the piped vehicle and was the diameter of the piped vehicle). According to the power and capacity of the piped vehicle, the mass of the transported material ranged from 1150 g to 2000 g. In this paper, the masses of transported material were determined to be 1150 g, 1250 g, 1300g, and 1400g. According to the power of the pump, the flow discharges ranged from 30 m³/h to 90 m³/h. In this paper, the flow discharge values were determined to be 40 m³/h, 50 m³/h, 60 m³/h, and 70 m³/h. The motion velocity of the piped vehicle was derived from (16) or (17). All of the parameter values in the calculation were reported in Table 1, and the contrast between the calculated and experimental results was presented in Table 2 and Figure 6.

(a)

(b)

In Table 1, was ratio of the diameter of the piped vehicle and that of the pipeline, was the friction coefficient obtained via experiments, and and were obtained from the Moody diagram according to the flow discharge and relative roughness of pipe.

As shown in Table 2 and Figure 6, the calculated and experimental values of the average velocities of piped vehicles were both influenced by the flow discharge, dimension of the piped vehicle, and mass of transported material.

For the piped vehicle with the same dimension, its average velocity increased with the increase in the flow discharge under condition of the same mass of transported material. The main reason was that the power of the piped vehicle came from the flow discharge. The flow discharge increased, and the power which acted on the piped vehicle increased. Consequently, the average velocity of the piped vehicle increased.

For the same mass of transported material and flow discharge, the average velocity of the piped vehicle increased with the increase in its diameter or length. The main reason was that the dimension of the piped vehicle directly influenced the buoyancy and the pressure gradient force received by the piped vehicle, thereby influencing the frictional force received by the piped vehicle. According to the force analysis of the piped vehicle, the buoyancy received by the piped vehicle increased when either the diameter or length of the piped vehicle increased. Therefore, the frictional force decreased according to (1). Additionally the face area of the piped vehicle increased when the diameter of the piped vehicle increased. Thus, the pressure gradient force received by the piped vehicle increased according to (2). The pressure gradient force was the main power for the movement of the piped vehicle, while the frictional force was resistant to the movement of the piped vehicle. Therefore, the average velocity of the piped vehicle increased when the diameter or length of the piped vehicle increased.

For the piped vehicle with the same dimension and the same flow discharge, the average velocity of the piped vehicle decreased when the mass of transported material increased. The main reason was that the change in the mass of transported material only affected the frictional resistance received by the piped vehicle under the condition that other transporting conditions remained constant. According to (1), the frictional resistance received by the piped vehicle increased when the mass of transported material increased, consequently decreasing the average velocity of the piped vehicle.

As shown in Table 2, the maximum relative error between the calculated and experimental values of the velocity of the pipe vehicle did not exceed 9.47%, which would indicate that the mathematical model of the piped vehicle motion was not only correct in theory but also feasible in practice.

Under the conditions mentioned, if the flow discharge is between 30 and 90 m³/h and the mass of the transported material is between 1150 g and 2000 g, then the piped vehicle moves smoothly and steadily.

While the piped vehicle is moving in the pipe, different curvature radii of the bend pipes, different uphill and downhill pipe sections, and different shapes of the piped vehicles may all have an impact on the motion characteristics of the piped vehicle. Due to the limitations of the test site, this test just took one of the conditions into consideration. As for the rest of the conditions, we shall study them in detail in future.

4. Conclusions

The mathematical model of the piped vehicle motion in piped hydraulic transportation of tube-contained raw material was established according to the stress characteristics of the piped vehicle, and the motion characteristics of the piped vehicle were analyzed. The main conclusions are summarized as follows:

(1) The calculated results of the mathematical model of the piped vehicle motion in piped hydraulic transportation of tube-contained raw material are in good agreement with the experimental results. The maximum relative error does not exceed 9.47%, which proves that the mathematical model is rational.

(2) The velocity of the piped vehicle is related to the flow discharge, mass of transported material, and dimension of the piped vehicle. When the flow discharge and the diameter or length of the piped vehicle increases or the mass of transported material decreases, the velocity of the piped vehicle increases.

(3) The mathematical model includes the main factors influencing the velocity of the piped vehicle, such as the flow discharge, mass of transported material, and dimension of the piped vehicle. It can be used to qualitatively analyze and improve the piped vehicle structure.

(4) The established mathematical model can be applied not only to the test conditions mentioned but also to other test conditions where piped vehicles of different dimensions and different materials move smoothly and steadily in pipes with different diameters and materials.

In summary, the mathematical model established in this paper can be used to calculate the velocities of piped vehicles with different structures under various conveying conditions, thereby providing theoretical basis to improve the structure of the piped vehicle and offering references for the reasonable design of the piped hydraulic transportation technique of tube-contained raw material.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

We acknowledge the financial support from National Natural Science Foundation of China (Grant No. 51109155; Grant No. 51179116) and Shanxi Provincial Natural Science Foundation of China (No. 2015011067).

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Copyright © 2019 Yongye Li and Xihuan Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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