Pressure Study on Pipe Transportation Associated with Cemented Coal Gangue Fly-Ash Backfill Slurry

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This work simulates the Cemented Coal Gangue-fly Ash backfill (CGFB) slurry pipe transportation and provides the potential pressure loss equations that occurred in the pipe delivery. The conclusions can be used to predict the slurry resistance and guide the backfilling pipe design as well as the related pump optimization.

Abstract

Cemented coal gangue-fly ash backfill (CGFB) slurry has commonly been used to control subsidence damage caused by underground coal mining. This paper discusses the characteristics of CGFB slurry fluidity in its pipe transportation. A general description about the components of the CGFB is provided involving the percentage of composition, particle size distribution (PSD) and rheological performance. The CGFB flow characteristics of the slurry pipeline were simulated in a straight pipe and 90° elbow pipe, respectively, combined with the pressure loss and conveying velocity distribution. With the help of the commercial computational fluid dynamic (CFD) code FLUENT, the modeling was conducted with various slurry feeding velocities. These results showed the local resistance loss in a bending pipe is significantly higher than the resistance in a straight pipe under the same conditions associated with CGFB transportation. The velocity distribution of the slurry solid particles in the slurry’s movement forward is more decentralized as the hydraulic inlet velocity increases. Based on these simulation data, a correlation was developed to predict the resistance loss of the CGFB slurry as a function of the hydraulic inlet velocity, pipe diameter and CGFB slurry rheological characteristics.

1. Introduction

Longwall mining is the most efficient underground means of recovering coal resources and is widely used in China, the United States, Australia and Europe [1]. However, this type of mining method has produced serious surface subsidence issues, which cause surface structure deformation, groundwater discharge and plant recession due to the formation of subsidence basins [2]. Backfill technology has been used for years to mitigate the subsidence impacts. Especially in China, various backfill techniques have been invented and implemented to protect surface structures from damage [3].

The cemented gangue-fly ash backfill (CGFB) slurry is one type of novel backfill material which has been utilized in the backfill technology [4]. The CGFB is mixed on the surface and pumped underground through a pipeline to the working face. Here, it is injected into the gob that is left from the longwall mining extraction under the protection of specially designed hydraulic supports. After a given time, the slurry solidifies into a hard body and combines with the surrounding rock mass to support the overburden weight [5]. Thus, reduced upper strata movement is produced and the aim of mitigating the subsidence impacts is achieved.

The solid content of the fresh CGFB slurry is 78.05% by weight (with water accounting for the remaining 21.95%) and is composed of Portland cement (10%), fly ash (20%), coal gangue (48%) and additive (0.05%) [5]. The main ingredient of the additive is hydroxypropy methylcellulose.

Table 1. Particle size composition of the coal gangue use in the coal gangue-fly ash backfill (CGFB) slurry.

Element	D10 (µm)	D30 (µm)	D50 (µm)	D60 (µm)	D100 (µm)
Coal gangue	204.722	856.589	1952.857	2736.752	6634.312

list the compositions of the coal gangue particle sizes used in the CGFB slurry configuration and Figure 1 provides the fly-ash and cement particle size distribution.

The CGFB slurry was observed to exhibit non-Newtonian behavior [6]. A Bingham plastic model was found to be adequate for describing the slurry rheology at moderate shear rates [7,8]. The pilot plant loop test with a horizontal pipe was conducted during the CGFB pipeline transportation, in which the average loss of frictional resistance per unit length was observed as 3.80 kPa/m, while the average loss of local resistance was 4.11 kPa/m [9,10]. The maximum value which appeared in the monitor was 4.40 kPa/m and 4.78 kPa/m for the straight pipe transportation and the bend, respectively [9,10,11]. Under this level of transmission resistance, the CGFB slurry can be transported for a longer distance (over 3000 m) with a larger scale flow that satisfies the coal mine backfilling requirements [11].

In the slurry pipe conveying process, the friction between the slurry and the pipe wall, the interaction loss associated with the colloidal among the fine solid particles and the collision due to the slurry flow diffusion were concluded to be three aspects that compose the transmission resistance of high-concentration slurry [12]. In addition, the CGFB slurry pipe delivery length largely depends on the resistance loss that is produced during the process of the underground pipe transportation. Thus, understanding the slurry flow characteristics, predicting the slurry pressure loss and determining the reasonable transportation velocity is essential to ensure the safety of the CGFB slurry pipe transportation and reduce the risk of pipe blockage due to the high solid concentration.

The value of the high-concentration slurry flow resistance loss in the pipeline is most influenced by the slurry rheological characteristics and hydraulic conveying velocity [13]. The slurry yield stress and relative viscosity contributes to the slurry initial rheology but the conveying velocity’s impact on the resistance is not very clear. In the pipeline transportation process, an obvious difference is observed for the velocity distribution of the slurry flow along the slurry’s movement forward because of the limitation of the pipe wall and the influence of the slurry’s own rheological properties. The offset in the flow conveying velocity can cause the diffusion of the slurry. In addition, a large fraction of the drag stems from velocity fluctuations and globally 10% of all energy is used to overcome turbulent drag in one way or another [14]. Thus, the effect of the pressure drops and velocity distribution profiles needs to be studied thoroughly within the CGFB pipe transportation. According to this, this article has developed a study of the pipe transportation characteristics in the horizontal direction that is forced on the CGFB slurry with a solid concentration of 76.05% by weight.

Computational fluid dynamics (CFD) have been used to predict the flow characteristics for a long time and the effects show great agreement with the experimental tests. The literature review shows that the pressure drop characteristics in the slurry pipeline have been successfully predicted universally using commercial CFD tools. Krampa Morlu et al. [15] studied the flow in a vertical pipeline for slurry flows with coarser particles. Lin and Ebadian [16] investigated sand–water slurry flows in the entrance region of a horizontal pipeline using CFD with the ASM model. Kaushal et al [17] performed modeling of slurry pipeline flow shaving fine particles at higher solid concentrations using CFD. Kubiki and Simon, LO [18] compared the CFD modeling technique and the discrete element method (DEM) for solid distributions in horizontal pipelines. The CFD is universally accepted by investigators as solving complex flow problems.

As one type of novel backfilling material, currently, there is not much published research papers regarding the CFD simulation on the CGFB pipe transportation. Wu [19] uses experimental methods to research the pressure drop of CGFB in the pipe loop. Tariq S. Khan et.al [20] researches similar materials to the CGFB in a horizontal circular pipe in the lab. Based on the experimental study, Wu [19] reported a qualitative conclusion but without much quantitative data, which is not enough to conclude a function or equation to predict or guide the CGFB slurry pipe transportation further. Thus, employing the CFD simulation method to study CGFB slurry pipe transportation in detail is essential.

In the present work, an attempt has been made to reveal the flow movement features in the CGFB slurry pipeline using the commercial CDF code FLUENT in the aspects of pressure drop and velocity distribution. A simulation study was used to optimize the hydraulic conveying velocity and determine the CGFB slurry transportation’s crucial parameters in a straight and bending pipe.

2. Materials and Methods

The seize analysis was employed to classify the dimensions category of coal gangue while the standard hydrometer technology was use to obtain the particle size distribution (PSD) on fly-ash and cement by the MALVERN HYDRO 2000MU (Malvern Panalytical company, Malvern, UK) (Figure 1). The Scanning Electron Microscopy (SEM) test of CGFB was done to describe the internal fiber structures on the purpose to obtain the morphology of the sample under the help of HITACHI S-3500N machine (Hitachi Group, Tokyo, Japan). The rheometer (Figure 2) was used in the test to exam the rheological behavior of the CGFB. Figure 3 provides the rheological properties of CGFB on 76.05%, 78.05% and 80.05% solid concentration. The data showed the variation of the shear stress with shear rate at all concentration follows a straight-line behavior. It verifies the CGFB slurry presents a non-Newtonian behavior and can be represented by a Bingham plastic fluid.

In the tests, around 500 g slurry was prepared by mixing the required quantity of solid materials with tap water to obtain the desired mass concentration (Cm), an electronic balance with a deviation of +/− 10⁻⁴ g was used for weighting the solid materials accurately. Shear rate under the controlled rate ranges 0 to 120 r·s⁻¹ was employed to measure the corresponding slurry yield stress and viscosity as the 10 r·s⁻¹ increasing measuring intervals. All the measurement was repeated three times to minimize errors occurred in the tests. Further, at each concentration, measurements were done at three samples on the same solid concentration slurry to assess the extent of repeatability. The test was conducted at the constant temperature of 19.6 °C. The representative samples of CGFB were figure out at backfilling laboratory in China University of Mining and Technology, Beijing.

From Figure 3, it is observed that for all CGFB coal gangue suspensions, the shear stress value increase with the increase in value of strain rate. The CGFB slurry can be idealized as a homogeneous fluid of a single phase with modified properties because of the Bingham plastic body characteristics [21,22,23]. In addition, the chemical reaction products (C-S-H) among the cement, fly ash and water provide the cohesive materials. The fine particles including cement, fly ash and the C-S-H present homogeneous characteristics, which was reported by many researchers. The coal gangue narrows the gap among the particles in the slurry. In the process of the CGFB pipe transportation, the coal gangue particles dispersed and were suspended in the slurry under the pump pressure, which was observed in the experimental test. Thus, on the CFD simulation, the assumption that the CGFB slurry is a homogeneous fluid of a single phase, ignoring the coal gangue coarse particle size, was made.

Commonly, the Reynolds number was employed to classify the flow layer category to determine the laminar layer and turbulence layer. Typically, the figure of 2100 was used to watershed the laminar or turbulence. Turbulence is a characteristic state of the flow where Reynolds numbers are sufficiently large. Regarding the Bingham flow, the relative viscosity was introduced to calculate the Reynolds number [24].

$$Re=\frac{vD\rho }{\mu }$$

(1)

where Re is the Reynolds number, $v$ is the conveying velocity, m/s; $D$ is the diameter of the pipeline, m; $\mu$ is the slurry relative viscosity, Pa·s and $\rho$ is the slurry density, kg/m³.

The CGFB slurry workable test show the density of the CGFB ranges between 1920 to 2100 kg/m³ and the relative viscosity ranges from 1.64 to 1.78 Pa·s. Thus, the average density of 2000 kg/m³ and the average relative viscosity of 1.71 pa·s were employed to calculate the Reynolds number. The average value of the density and the CGFB slurry relative viscosity were obtained from the experimental study.

$$Re=\frac{vD2000}{1.71}<2100$$

(2)

$$D<\frac{1.8}{v}$$

(3)

Referring to Equation (3), awe assume the slurry pipeline diameter is 150 mm; under this condition, the CGFB slurry transportation sustains the laminar flow when the conveying velocity is up to 12 m/s, which means that most cases of CGFB slurry transportation belong to the category of the laminar layer. Based on the coal mine backfilling currently, the slurry conveying velocity ranges 1.6–2.0 m/s in site-practice. Thus, the study was made with the assumption that the CGFB conveying velocity ranges from 1.4 to 2.2 m/s. Within this range, the slurry definitely belongs to the laminar state. A simulation method was used to investigate the CGFB slurry conveying velocity impacts in a straight pipe and 90° elbow pipe based on the different inlet velocities. The CGFB flow characteristics in the pipe transportation of both the straight pipe and the bend were obtained at the various velocity ranges from 1.4 m/s to 2.2 m/s with increments of 0.2 m/s.

The structuration grids of the straight pipe and elbow pipe were established in the ICEM CFD (Ansys Corporation, Canonsburg, PA, USA) and then the finished mesh file was loaded into the ANASYS R15.0 software (Ansys corporation, Canonsburg, PA, USA). The CGFB slurry pipe transportation laboratory experimental test requires the minimum interval distance between the two monitors is 15,000 mm. The minimum distance is essential to reduce the data distribution because of the slurry movement. Thus, in the CFD simulation, the straight pipe was model with a length of 17,000 mm. I think the length is enough for the flow fully developed. The length and diameter of the straight pipe were considered as 17,000 mm and 150 mm, respectively. On the 90° elbow pipe construction, the radius was set as 600 mm and an extension for a length of 1,600 mm was developed on both the inlet and outlet parts. Figure 4 shows the geometry of the constructed straight pipeline and elbow pipeline. The material of pipeline was considered to be mild steel [17].

The validation of the model was obtained by simulating the steady laminar forced convection of water in a horizontal tube. The mesh independence was examined with different mesh sizes obtained by refining a coarser size until the results were unchanged.

3. Computational Models

The mathematical model mainly consists of mass, momentum and energy conservation equations in partial differential forms with associated boundary conditions. The CFD code FLUENT was used to estimate the pressure drop phenomenon in the slurry pipeline. In FLUENT, the Herschel–Bulkley model [25,26] was modified to represent the Bingham plastic model.

The CGFB slurry is mixed with multiple components including coal gangue, fly ash, cement and water as a Bingham plastic body under a high solid concentration of 78.05% as a single substance [13,25,27]. The slurry density is considered as the constant of 2000 kg/m³. The slurry viscosity was defined with the Herschel–Bulkley model. In the pipe transportation, the slurry move forwards with the plunger shape with higher yield stress and larger slurry relative viscosity like the entire [28]. We used the figure of 1.71 Pa·s as the consistency index and Figure 1 as the power-law index while setting the yield stress threshold as 78.91 Pa. The viscous-laminar model was applied to the high solid consideration of CGFB slurry to determine the flow layer characteristics of the mixture in the pipe.

The boundary conditions, namely velocity inlet, pressure outflow and no-slip conditions, were applied to the fluid flow domain. The inlet boundary condition was applied to the fluid flow at the velocities of 1.4 m/s, 1.6 m/s, 1.8 m/s, 2.0 m/s and 2.2 m/s, respectively and the volume fraction tended to generate a stabilized flow along the length of the pipe. The outlet boundary condition was applied at te outlet section. In order to simplify the analysis, the export pressure was provided with 0 Pa at the outlet section. The wall was treated as stationary with no slip. The average roughness height was calculated using water flow data of 0.5 that was mentioned in FLUENT for a uniform pipe surface roughness [21,29].

According to the actual delivery of the slurry, the accuracy of the simulation is set as 1 × 10⁻⁴; the number of iterations is set as 600. The data processing results mainly include the pressure loss contour of both the straight pipe and elbow pipe, the velocity distribution contour and velocity distribution profile.

4. Results and Discussion

In this present study, the aim is to predict the pressure loss by applying the Herschel–Bulkley model in a 17,000 mm long straight pipe and the 600 mm radius of the 90° elbow pipe with the same diameter of 150 mm. The simulations are conducted at feeding velocities of 1.4 m/s, 1.6 m/s, 1.8 m/s, 2.0 m/s and 2.2 m/s with a solid concentration of the CGFB slurry sustained at 76.05%. Unlike the straight pipe flows, where streamlines show an organized pattern, elbow pipe flow is characterized by apparently random motion of the fluid in the flow and its complicated physics need to be understood for better prediction and control of the flow in the slurry pipe transportation engineering.

4.1. Pressure Drop

The pressure loss contour of the straight pipe and 90° elbow pipe was analyzed. It was observed that as the feeding velocity increases, the highest pressure loss was found for all transportation case. On the straight pipe transportation example, the maximum predicted pressure loss up to 92,600 Pa that occurred at the case of feeding velocity of 2.2 m/s for the total length of 17,000 mm. At the same inlet velocity in the bending pipe, the maximum pressure loss was 31,700 Pa, in which the delivery length of the slurry was near 4142 mm.

Table 2. Predicted pipe transportation pressure loss based on the CFD simulation.

Transportation Pipe Feeding Velocity (m/s)	1.4	1.6	1.8	2.0	2.2
Predicted pressure loss on straight case (Pa)	65,600	72,200	78,900	85,700	92,600
Predicted pressure loss on elbow case (Pa)	19,800	22,600	25,500	28,500	31,700

Note: the total delivery length for predicted pressure loss on straight case is 17,000 mm; the total delivery length for predicted pressure loss on elbow case is 4142 mm.

provides the predicted maximum pressure loss from the CFD simulation calculation.

A pressure drop (Δp/l) is introduced to describe the pressure loss of the CGFB slurry in this paper, which was calculated in terms of the pressure loss per unit length through the pipeline [30]. Figure 5 shows the predicted pressure drop profile on the CGFB slurry suspension flow for the straight pipe and 90° elbow pipe at velocities of 1.4 m/s, 1.6 m/s, 1.8 m/s, 2.0 m/s and 2.2 m/s, respectively.

The profile clearly shows the increasing inlet velocity of the slurry, resistance and local resistance loss both increased but the local resistance increase is very significant. These results support the notion that the local resistance is significantly higher than the resistance at the same transportation conditions associated with high-concentration slurry pipeline transportation. But regarding the CGFB slurry, the difference on the transportation resistance between the straight case and elbow cases is not large because of the slurry adequate fluidity and the lower conveying velocity in pipe. It was realized that for the slurry transportation case with the feeding velocity of 1.4 m/s, the value of the pressure drop is noticed at 3860 Pa/m in the straight pipe and 4780 Pa/m in the 90° elbow pipe. The factor of the local resistance over the total resistance is 1.24. Similarly, for the velocity of 2.2 m/s, the magnitude of pressure drop was observed at 5450 Pa/m for the straight example, while it was 7650 Pa/m with the elbow pipe. In this situation, the factor extends to 1.40. Under this level of the magnitude of the pressure drop, the CGFB slurry can be delivery for a longer distance compared with the traditional pastes slurry.

This result agrees well with the pilot plant test results and it supports the conclusions that the CGFB slurry can satisfy the coal mine backfilling requirements with a large scale, large flow and high safety [11]. The CGFB backfilling technology is one of the potential methods to deliver the backfilling materials to 3000 m in the coal mine currently.

4.2. Velocity Distribution

The velocity distribution of the CGFB slurry flow in the pipe is shown in Figure 6 and Figure 7 based on the different respective feeding velocities. Figure 8 shows the velocity distribution profiles under the different transportation cases of the straight pipe and 90° elbow pipe.

The slurry velocity distribution contour on the pipe was conducted to analyze the CGFB slurry flow transportation stability characteristics. Transversal plots of the slurry flow were done in the middle section of the straight (at the position of 8500 mm away from the inlet) and the bent pipe (the position of the corner of the bending). The velocity distribution of the CGFB flow was provided on the cross-section surface based on the different inlet velocity cases of 1.4m/s, 1.6 m/s, 1.8 m/s, 2.0 m/s and 2.2 m/s, respectively.

The results from the straight pipe show (Figure 6) that the slurry moves forward with the shape of a plunger in the pipe due to the different velocity magnitudes at the different positions of slurry flow. A flow core-zone area and a non-flow core-zone area at the cross-section of the slurry flow were observed and were found to be in good agreement with the conclusions related to the flow-core zone distribution [31]. The slurry concentrates at the center region within a certain radius are located in an area with a relative high velocity (shown as the red zone in Figure 6). This area was defined as the flow-core zone while the remaining area in the cross-section of the pipe was defined as the non-flow core-zone. Beyond the flow-core zone area, within the region of the non-flow core-zone, from the center area to the pipe wall, the slurry conveying velocity presents a constant decreasing trend. At the position near to the pipe wall, the slurry conveying velocity is close to zero, which is considered as due to the “wall effects.” With suspension, there is always a slight decrease in particle concentration near the walls of the pipe, due to the steric effects, which result in the magnetized value of zero in the simulation.

The conveying velocity distribution contour is concentric in the cross-section of the slurry flow. The center is much higher while the side is lower. For the case wherein the feeding velocity is 1.4 m/s, the maximum predicted conveying velocity in the pipeline is 2.65 m/s, in which an exceeding increase of 1.25 m/s was found. For the case of 1.6 m/s, the maximum velocity in the pipeline observed is 3.04 m/s, which exceeds the corresponding feeding velocity up to 1.44 m/s. For the case of 1.8 m/s, the difference between the potential maximum conveying velocity and feeding velocity is 1.63 m/s. For the example of 2.0 m/s, the difference is 1.83 m/s and the magnitude rises to 4.22 m/s for the case of 2.2 m/s, which is more than double the feeding velocity. The increasing difference indicates that as the feeding velocity increases, the slurry tends to be more unstable and the slurry plunger shape is more obvious. The difference of the conveying velocity on the slurry flow determines the stable transportation characteristics of the CGFB. The larger the difference of the conveying velocity between the jointed points on the slurry, the more significantly unstable the transportation was. Thus, the feeding velocity of CGFB transportation in the pipe should be reasonably determined in order to maintain stable flow.

When the CGFB slurry was pumped into the pipe after configuration on the surface station, under the restriction of the pipe wall and the impacts of slurry rheological characteristic, the slurry flow deforms to the plunger shape to progress along the pipe extension. When the slurry goes through the bending section, due to the difference of the conveying velocity impacts on the slurry flow points, a new flow shape was formed to satisfy the pipe deformation. Thus, the flow-core zone (the higher-velocity distribution area) moves outside the pipe latter well while the non-flow core-zone (the lower velocity distribution area) concentrates at the pipe inner wall (shown in Figure 7). The conveying velocity distribution of the CGFB slurry in the bending section is diffused without the concentric circles. As the inlet velocity increases, the pressure loss increases but the area of the flow-core zone decreases, which indicates that a more significant unstable flow characteristic reaching.

The offset of the points of the slurry center and the pipe well in the conveying velocity is gradually decreased compared with the straight pipe. When the feeding velocity is 1.4 m/s, the potential maximum conveying velocity at the corner of the pipe is 2.57 m/s, while the velocity in the straight pipe is 2.65 m/s; when the feeding velocity is 1.8 m/s, the maximum conveying velocity in the bend is 3.24 m/s, while the maximum velocity in the straight pipe is 3.43 m/s; when the feeding velocity is 2.0 m/s, the maximum velocity in the bend is 3.56 m/s, while the maximum velocity in the straight pipe is 3.83 m/s; when the conveying velocity is 2.2 m/s, the monitored velocity in the bend is 3.87 m/s, while the responding velocity in the straight pipe is 4.22 m/s. The maximum conveying velocity of the straight pipe slurry is higher than the bending pipe conveying velocity, which shows that the restriction impacts of the bent pipe are more obvious than of the straight pipe.

Figure 8 provides the conveying velocity distribution profiles of the different conditions for the slurry through the straight pipe and 90° elbow pipe, respectively. The velocity profile of high-concentration slurry in a flow cannot be observed in the laboratory [32,33]. However, computational fluid dynamics (CFD) generate a vertical velocity profile which agrees well with the theoretical prediction. The measured points were set in a perpendicular direction with the slurry’s advancing direction. In Figure 8, the velocity distribution was plotted on the Y-axis, whereas the location in the pipe’s middle section from one side to another side was plotted on the X-axis. It was observed that a lower conveying velocity magnitude appears near to the region of the wall and the higher magnitude is distributed around the center. The value also indicates the stress contributed by particles would be constant in the velocity gradient and would be zeroed instantly once the flow is stopped. From Figure 8, the higher the inlet velocity of the slurry, the more dramatic the offset on the slurry conveying velocity at different positions in the pipe transportation, which causes worse stability for the CGFB slurry flow.

From the Figure 8, it was observed that asymmetric nature of particles about longitudinal axis in the straight pipe. It was also noticed that the higher conveying velocity concentration zone shifts away from the central location to the pipe wall with the deformation of the pipe. But the concentration zone maintains the same position because of the CGFB slurry well homogeneous characteristic as the Bingham flow. The slurry velocity distribution phrases the slurry rheological characteristics in further, On the CGFB slurry pipe transportation, the higher of the slurry inlet velocity leads the more drastic offset on the magnitude regarding the slurry conveying velocity. The higher slurry inlet velocity, the more decentralized the conveying velocity distribution occurring in the slurry is. The difference in the velocity significantly affects the slurry stability in pipe transportation. Thus, a reasonable inlet velocity is essential to consider before delivering the CGFB slurry in pipeline.

5. Mechanisms

The effect of the feeding velocity on the resistance loss of the CGFB slurry pipe transportation was analyzed and an attempt has been made to develop a correlation to estimate the resistance loss as a function associated with the conveying velocity in the straight pipe and 90° elbow pipe based on the mathematical analysis. The function can be used to predict the CGFB slurry pressure loss in the pipe transportation regarding the corresponding conveying velocity and provide the possibility of estimating the minimum pump power that delivers the slurry to the gob.

The CGFB slurry was investigated as a non-Newtonian fluid and which conforms to Bingham plastic body characteristics. The relation [34] of the shear stress to shear rate for the Bingham plastic body can be expressed as follows:

$$\tau ={\tau }_0+\gamma \mu$$

(4)

where τ₀ is the yield stress, Pa, $\mu$ is the viscosity coefficient, Pa·s and $\gamma$ is the typical shear rate, 1/s.

Based on the Bingham plastic body linen relationship, Buckingham created an equation that can be employed to determine the slurry transmission pipe dimeter [1]. The Buckingham equation was shown as the following:

$$\frac{8v}{D}=\frac{{\tau }_w}{\mu }[1-\frac{4}{3}\frac{{\tau }_0}{{\tau }_w}+\frac{1}{3}{\left(\frac{{\tau }_0}{{\tau }_w}\right)}^4]$$

(5)

where v is the slurry conveying velocity, m/s, D is the diameter of the transportation pipeline, m, ${\tau }_w$ is the shear stress at the wall, 1/s. Considering that the magnitude of the (${\tau }_0/{\tau }_w$)⁴ is so small that it can be ignored because of the CGFB slurry having a high solid concentration up to 76.05% by weight, Equation (5) was modified to Equation (6):

$$\frac{8v}{D}=\frac{{\tau }_w}{\mu }-\frac{4}{3}\frac{{\tau }_0}{\mu }$$

(6)

Figure 9 provides the overview of the slurry conveyance in the horizontal pipes (straight pipe) along the slurry’s advance direction. The slurry with a length of dL was intercepted as the research object to make the static analysis. The pressure loss $∆p$ at the length of dL is calculated. The static balance available for the objected slurry at the length of dL was shown as follows:

$$\pi R^2∆p=2\pi R{\tau }_{w}l$$

(7)

According to Equation (7), the shear stress occurring at the pipeline wall can be expressed as follows:

$${\tau }_w=\frac{R∆p}{2l}$$

(8)

In China’s engineering field, the resistance loss (i_s) in the straight pipe is introduced to describe the value of the pressure loss over unit length in practice [29]. i_s was defined as follows:

$$i_s=\frac{∆p}{l}$$

(9)

Combining Equations (6) and (8), Equation (9) is created as follows:

$$i_s=\frac{16}{3D}{\tau }_0+\frac{32v}{D^2}\mu$$

(10)

where l is the length of the fluid in the pipeline, $∆p$ is the delivery distance of the pressure drop and $i_s$ is the resistance in straight pipe.

Compared with the data from the CFD simulation result and the theoretical data calculated from Equation (10), Figure 10 was made. From the statistical analysis, the predicted data is well in agreement with the theoretical data.

Based on the CFD simulation data, the equation that can be applied in the 90° elbow pipe was derived in combination with Equation (10). Initially, a log–log graph was plotted for the local resistance in the 90° elbow pipe with the resistance loss of the straight pipe to obtain the function between them. Then, all the data from the variation of local resistance collected in the 90° elbow pipe was plotted against the slurry conveying velocity. Thus, the functional relationship of the pipe dimeter (D), the CGFB slurry yield stress (${\tau }_0$) and the CGFB slurry relative viscosity ($\mu$) is given in Equation (11) as

$$\begin{array}{l}{i}_s=\left(0.3v^2-0.7v+1.6\right)\times (\frac{16}{3D}{\tau }_0+\frac{32v}{D^2}\mu ) (1.4<v<1.8 m/s)\\ i_s=\left(0.6lnv+1\right)\times (\frac{16}{3D}{\tau }_0+\frac{32v}{D^2}\mu ) (1.8<v<2.2 m/s)\end{array}$$

(11)

Equation (11) evaluates the local resistance loss that can be used in the situation of the 90° elbow pipe. Equation (11) was employed to predict the local resistance loss in the 90° elbow pipe, developed based on Equation (10), which is used for the resistance loss of the straight pipe. In addition, the value that is used to develop the Equation (11) is from the CFD simulation data. Thus, Equation (11) is limited to evaluating the potential local resistance loss but the function provided the basis for developing a more reliable prediction result regarding the high solid mass slurry. This can be achieved by considering the qualitative conclusions around the relative viscosity and yield stress and applying the detailed configuration of similar slurry to upgrade the results. Further investigation on the 90° elbow pipe in the high-concentration slurry with different solid concentrations is necessary for the development of a comprehensive and practical prediction.

From Equations (10) and (11), the resistance loss was observed to depend on the pipe diameter, the inlet velocity of the slurry and the CGFB slurry rheological characteristic parameter (yield stress and relative viscosity). This verifies the report published by the Sun [35]. The yield stress and relative viscosity are CGFB properties determined by the particle size distribution (PSD), slurry solid concentration and the proportion of the components. Thus, the pipe diameter and the conveying velocity are the parameters that can be used to adjust the pressure loss in the CGFB slurry pipe transportation.

The equation shows the delivery pressure loss tends to be increased as the slurry conveying velocity increases or the pipe diameter decreases. The pipe diameter is related to the conveying velocity. As the pipe diameter increases, the conveying velocity decreases to deliver the slurry for the same length of distance because of the available pump power. The conveying velocity is an important parameter in the CGFB pipe transportation and the reasonable optimization of the conveying velocity can reduce the pipeline transportation resistance and enhance the level of transmission distance regarding the CGFB slurry.

6. Conclusions

This paper develops a study on the resistance loss regarding the CGFB slurry transportation in a straight pipe and 90° elbow pipe based on the various inlet velocities. From the above research, the following conclusions can be drawn:

The resistance loss along the length of the pipeline gradually increases with the rise of inlet velocity. The local resistance loss in the bend is significantly higher than the resistance in the straight pipe at the same conditions associated with CGFB transportation because of the pipe’s deformation impacts. The CFD predicted data show the maximum factor of the local resistance magnitude over resistance is 1.31 for the case with the inlet velocity of 2.2 m/s.

For the CGFB slurry pipe transportation, the higher the slurry inlet velocity, the more drastic offset regarding the slurry conveying velocity, which occurs on the slurry flow cross-section. The velocity difference among the adjacent points on the slurry flow causes the plunger shape of the CGFB flow. The decentralized conveying velocity distribution significantly affects the slurry stability in pipe transportation. Thus, a reasonable inlet velocity is essential to consider before delivering the CGFB slurry in the pipeline.

The core-flow zone and the non-core flow zone in the slurry flow cross-section were observed at the simulation of the CGFB pipe transportation. When the slurry is transported in a straight pipe, the concentric distributions are composed of the different conveying velocity regions found in the core-flow zone. With the increase of slurry inlet velocity, the conveying velocity of the flow core area also increases but the flow core area decreases. The deformation of the concentric circle was conducted as the CGFB slurry flows through the bending pipe under the restriction of the pipe wall, resulting in a great resistance loss.

Based on the simulation data, the equations were developed to predict the resistance loss of the CGFB slurry on the pipe transportation with the CFD results. The function, composed of the hydraulic inlet velocity, pipe diameter and CGFB slurry rheological characteristic, was intended to estimate the pressure loss regarding the CGFB slurry transportation for the straight pipe and bending pipe in a horizontal direction, which was used in other high-concentration slurry transportation instances with similar components.