Computational Analysis of Pipe Roughness Influence on Slurry Flow Dynamics

Abstract

Slurry transportation is an essential process in numerous industrial applications, widely studied for its efficiency in material conveyance. Despite substantial research, the impact of pipe wall roughness on critical metrics such as pressure drop, specific energy consumption (SEC), and the Nusselt number remains relatively underexplored. This study provides a detailed analysis using a three-dimensional computational model of a slurry pipeline, with a 0.0549 m diameter and 3.8 m length. The model employs an Eulerian multiphase approach coupled with the RNG k-ε turbulence model, assessing slurry concentrations C_w = 40–60% (by weight). Simulations were conducted at flow velocities V_m = 1–5 m/s, with pipe roughness (R_h) ranging between 10 and 50 µm. Computational findings indicate that both pressure drop and SEC increase proportionally with roughness height, V_m, and C_w. Interestingly, the Nusselt number appears unaffected by roughness height, although it rises corresponds to V_m, and C_w. These insights offer a deeper understanding of slurry pipeline dynamics, informing strategies to enhance operational efficiency and performance across various industrial contexts.

1. Introduction

Slurry transportation through pipelines is a widely utilized method in the chemical, petrochemical, and mining industries, offering efficient transfer of solid particles suspended in a fluid medium, typically water [1]. This method is advantageous due to minimal contamination risk, low maintenance costs, high energy efficiency, and reduced material wastage, making it both cost-effective and sustainable for industrial applications [2]. Extensive research in slurry transportation has deepened our understanding of the complex dynamics and key characteristics involved in pipeline slurry transport.

For example, Kaushal et al. [3] experimentally examined key parameters, including pressure drop and solid concentration profiles for 125 µm, and 440 µm glass beads particles in a straight pipe at V_m = 1–5 m/s, and C_vf = 10–50%. Experimental work shows an increase in pressure drop corresponding with both rise in V_m, and C_vf, with 440 µm particles exhibiting a greater pressure drop than 125 µm particles. Similarly, Joshi et al. [4] investigated the effect of pipe roughness on pressure distribution, with results that pipe roughness marginally impacts pressure drop at low V_m. The optimal transport efficiency achieved by 125 µm particles flowing at V_m = 2 m/s and C_w = 40%. Further studies have expanded on the kinetic and thermodynamic parameters of slurry flow. Gopaliya et al. [5] analyzed various factors such as granular pressure, viscosity, and solid concentration within a straight pipe, highlighting that higher V_m (3.1 m/s), and C_vf (45%) correlate with increased transport dynamics. Wu et al. [6] introduced a model for predicting specific energy consumption (SEC) in low-viscosity Newtonian slurries. He also proposed a method to improve energy efficiency in Newtonian slurry transportation, revealing that energy consumption per unit mass increases with V_m. They also observed that SEC decreases with larger equipment dimensions, such as pipe diameter and tank size. Joshi et al. [7] employed Eulerian–Eulerian approaches in conjunction with the RNG k-ε turbulence model to predict the transportation and efficiency-related parameters of four industrial-oriented solid particles in a straight pipeline: silica sand, iron ore, glass beads, and bottom ash. The sizes for all particles range from 125 to 440 µm, with a velocity of V_m = 1–5 m/s and C_w = 40–60%. The computational investigation shows that smaller solid particles (125–275 µm) flowing at low-velocity V_m = 1–2 m/s and low efflux concentration C_w = 40% have a lower pressure drop and SEC for transportation, indicating that these constraints are highly efficient. Lately, Joshi et al. [8] conducted new research focused on the influence of carrier fluid density on SEC. In their study, an RNG k-ε turbulence multiphase model is used to predict various kinetic parameters, including pressure drop, granular pressure, granular temperature, wall shear stress, settling velocity, and SEC. All parameters are investigated for glass bead particles of five distinct sizes: 125–440 µm, considering different fluid densities and temperatures. Their findings indicated that a carrier fluid with lower density, and higher temperature is more energy-efficient for transportation. Moreover, they observed that the characteristics of slurry flow were influenced by particle size and solid concentration. Specifically, larger solid particles at higher solid concentrations were found to have higher settling velocities but lower transport efficiency.

Joshi et al. [9] used the Eulerian k-ε multiphase model in one of the early studies centered around a bi-modal flow in a straight pipe. The study is carried out for four distinct silica sand and fly ash mixture combinations, flowing at V_m = 2–5 m/s, and C_w = 40–60%. According to the findings, the 65:35 combination has the lowest pressure drop, which rises with V_m and C_vf. The results also show that slurry with a high percentage of catalyst particles, i.e., a 65:35 mix, improves slurry system transport efficiency. Moving forward to the heat transfer characteristics, Nayak et al. [10] investigated the heat transfer characteristics of the fly ash–water slurry mixture using the k-ε turbulence model, flowing through a horizontal pipeline with an inlet temperature of 300 K and wall temperature of 400 °K. The investigation uses five different particle sizes (4–78 µm), with a V_m = 1–5 m/s, and a C_vf = 10–50%. They observed that while the heat transfer coefficient increased with V_m, it decreased with higher C_vf. Additionally, larger particles yielded the highest heat transfer ratio. Moreover, the Nusselt number increases with a rise in Reynolds number.

The literature reveals that only a few studies are available on pipe roughness and their influence on pressure drop and concentration distribution. However, their effect on transport efficiency parameter (SEC) and heat transfer parameter (Nusselt number) has not yet been established. Therefore, the present research utilizes the RNG k-ε turbulence model and granular flow theory to evaluate the effects of pipe roughness on transport (pressure gradient and SEC) and thermal (Nusselt number) characteristics in a mono-dispersed glass bead slurry. Flow conditions include V_m = 1–5 m/s, and C_w = 40–60%. Computational results were validated against experimental data, providing insights for optimizing pipeline design in industries like petroleum, mineral processing, and chemicals reliant on slurry transportation.

2. Mathematical Formulation

2.1. Governing Equations

The numerical analysis employs the Eulerian multiphase model, which separately solves conservation equations for mass, momentum, and energy for each distinct phase in the system. This approach allows for an accurate representation of interactions between the solid and fluid phases. For comprehensive formulations of the continuity and momentum equations applied to both phases, a detailed description of the equations can be found in Joshi et al. [7] and Nayak et al. [10].

2.2. Heat Transfer Coefficient

The heat transfer rate, which takes place between multiple phases, can be calculated as a function of the temperature difference. The equation can be written as [10]:

$$Q_{sf}=h_{sf}\left(T_s-T_f\right)$$

(1)

where $h_{sf}=h_{fs}$ is the heat transfer coefficient that takes place between the solid and fluidic phases.

The correlation of the Nusselt number used by Gun et al. [11] for granular flow can be written as:

$$N{u}_s=\left(7-10{\alpha }_f+5{\alpha }_f^2\right)\left(1+0.7{\mathrm{Re}}_s^{0.2}{\mathrm{Pr}}^{0.33}\right)+\left(1.33-2.4{\alpha }_f+1.2{\alpha }_f^2\right){\mathrm{Re}}_s^{0.7}{\mathrm{Pr}}^{0.33}$$

(2)

Here, ${\alpha }_f$ is the volume fraction of the fluid phase, ${Re}_s$ and Pr are represented as the relative Reynolds number and fluid Prandtl number.

2.3. Specific Energy Consumption (SEC) Equation

SEC for the slurry transportation model is termed as the minimum amount of required energy for the solid particle transportation to the unit distance throughout the pipeline from inlet to outlet. The SEC is the function of pressure gradient, solid density, and volumetric concentration, which can be expressed as follows [6]:

$$SEC={\displaystyle \frac{\frac{\Delta P}{\Delta L}}{C_V{\rho }_S}}$$

(3)

3. Computational Modeling

Numerical modeling is a robust method to accurately forecast crucial multiphase factors. The primary focus of modeling orbits around solving the partial differential equations pertaining to momentum and energy. The commercial CFD software Ansys Fluent 2024R1 (ANSYS Inc., Canonsburg, PA, USA) is employed to analyze the kinetic and thermal characteristics of a slurry mixture flowing through a horizontal, straight pipeline [12,13,14]. This multiphase mono-dispersed model encompasses mainly two phases, (i) solid (glass beads), and (ii) liquid (water). Previous researchers have effectively used the Eulerian multiphase k-ε model to predict the kinetic and thermal behaviors of slurry flow. In present work, a straight pipe, oriented horizontally, measuring L = 3.8 m and D = 0.0549 m, serves as the conduit for transporting the mixture from inlet to the outlet. The chosen pipe geometry satisfies the criteria for fully developed flow, i.e., 50D. The schematic diagram of the computational model, which includes pipe geometry and boundary conditions, is depicted in Figure 1.

3.1. Meshed Model

Figure 2a displays the 3D meshed model of a straight pipe. In this meshing process, hexahedral elements are employed to create the mesh for the pipe model. The computational model exhibits an excellent orthogonal quality of 0.93417, which is highly desirable for CFD simulations, ensuring precise results [15]. Additionally, the skewness of the pipe is maintained at a low value of 0.0256, which adheres to recommended standards for CFD models. To accurately model the interaction between particles and the pipe wall, the “near-wall” mesh was suitably refined, ensuring a y+ value of 1 to account for boundary layer effects.

To maintain a consistent mesh quality and enhance computational efficiency, a mesh independence test was carried out. Multiple mesh configurations were generated, including element counts of 210 × 10³–661 × 10³, as depicted in Figure 2b,c. After careful examination, it was noted that the computational models with 472 × 10³ and 661 × 10³ elements produced virtually indistinguishable velocity distribution results, as shown in Figure 2b. In a similar way, analogous mesh elements are accounted for to obtain the pressure drop. The results show similar patterns, i.e., with 472 × 10³ and 661 × 10³ element mesh shows approximately similar results, see Figure 2c. As a result, the pipe geometry utilizing 472 × 10³ elements was chosen as the optimal setup for conducting the simulations.

3.2. Boundary Condition and Convergence Criteria

A slurry mixture consists of solid particles and water flows at a specific velocity (V_m = 1–5 m/s) and volumetric concentrations (C_w = 40–60%) from the pipe inlet section. The mean diameter and specific gravity of solid particles are considered as 125 µm and 2.470 Kg/m³, respectively. The no-slip condition is used for the pipeline wall [9] to obtain the kinetic and heat transfer characteristics [16]. An Eulerian technique combined with the RNG k-ε turbulent model is employed to simulate mono-dispersed slurry flow for a continuous fluid phase [17]. To achieve accurate and smooth characteristics in the slurry flow, a combination of a second-order upwind scheme and a SIMPLE algorithm with a convergence criterion set at 10⁻⁴ is employed.

3.3. Numerical Validation

A comparison of numerically obtained results and experimental results for 125 μm solid particles is illustrated in Figure 3. The comparison is carried out for glass bead particles flowing at V_m = 1–5 m/s and efflux concentration C_vf = 30%, and C_vf = 50%, see Figure 3a,b, respectively. From Figure 3, it can be seen that the computational outcome for solid particles is constant with the experimental work by Kaushal et al. [3].

4. Results and Discussion

4.1. Pressure Distribution

The pressure exerted by the slurry mixture results from the interactions between particles and the carrier fluid, causing a frictional pressure drop. Figure 4a–c depicts the normalized pressure drop for 125 µm solid particles as a function of pipe roughness (roughness height) while varying the C_w from 40% to 60%. The x-axis represents the normalized pressure drop, and the y-axis spans roughness heights of 10 to 50 µm.

As seen in Figure 4a, the findings demonstrate a direct relationship between pressure drop and roughness height. Slurry flowing through pipes with lower roughness heights exhibits less pressure drop, while the pressure drop is highest when the roughness height is 50 µm. Furthermore, V_m has a substantial impact on pressure drop, with an increase in slurry V_m leading to higher pressure drop. At lower V_m (1–2 m/s), the rise in pressure drop experienced due to increased pipe roughness is relatively moderate. However, at V_m exceeding the 2 m/s mark, a significant surge in pressure drop is experienced, especially noticeable in the V_m = 3–5 m/s. This surge in pressure drop is attributed to the elevated friction factors encountered at higher slurry flow V_m within the pipeline.

C_w also plays a noteworthy role in pressure drop, as shown in Figure 4b,c. The literature confirms that an increase in C_w results in a corresponding rise in pressure drop for various roughness heights and V_m. This can be explained by the fact that higher concentrations prevent particles from settling out prematurely, leading to increased interactions between solid particles and between solid and liquid components, consequently causing an increase in pressure drop.

4.2. Concentration Distribution

The figures depict the concentration distribution of 125 µm particles transported through a pipe with varying surface roughness heights (R_h = 10 µm, 30 µm, and 50 µm) at different V_m = 1 m/s (Figure 5a), 3 m/s (Figure 5b), and 5 m/s (Figure 5c). The color contour maps indicate particle concentration levels across the pipe cross-section. At low V_m (1 m/s), the particle concentration near the wall shows a more uneven distribution, with a visible impact of surface roughness on particle accumulation patterns, particularly for higher roughness heights. As V_m increases to 3 m/s, the distribution becomes more homogenized in the central region, although the influence of surface roughness persists near the wall. At the highest V_m = 5 m/s, particle concentration in the core flow region becomes nearly uniform, suggesting dominant inertial effects, while the wall regions still exhibit roughness-induced gradients. The roughness impact is mainly shown at lower V_m, whereas higher V_m reduces the relative impact of roughness due to stronger bulk flow and particle inertia.

Figure 6 illustrates the concentration distribution of 125 µm particles transported through a pipe at a velocity of V_m = 5 m/s, and R_h = 50 µm, with varying C_w = 40–60%. As the C_w increases, a pronounced rise in overall particle concentration is observed across the cross-section. At C_w = 40%, the particle distribution is relatively uniform, with slightly higher concentrations near the pipe’s lower regions. For C_w = 50%, the concentration becomes more prominent and evenly distributed across the core flow area, reflecting a moderate increase in particle density. At C_w = 60%, the particle concentration reaches its highest levels, as indicated by the dominance of orange and red shades, especially in the central and near-wall regions.

4.3. Specific Energy Consumption (SEC)

The efficiency of the slurry pipeline system’s material transportation is quantified by a metric known as SEC. Figure 7a–c displays the normalized SEC trends for 125 µm solid particles concerning pipe roughness (R_h) at C_w = 40–60%. The x-axis represents the normalized SEC, while the y-axis spans R_h from 10 to 50 µm.

For C_w = 40%, it is evident that energy consumption increases as R_h rises. This is attributed to the elevated R_h creating an additional frictional barrier, hindering slurry flow within the pipeline system. Consequently, pressure drop increases. The impact of V_m on SEC is noteworthy. The SEC of the slurry system escalates with a higher slurry V_m. At lower V_m (1–2 m/s), minimal deviation is observed in the pressure drop trendline. However, as V_m surpasses 2 m/s, a substantial rise in SEC becomes apparent, particularly in the V_m range of 3 to 5 m/s. This increase in SEC is linked to the elevated pressure drop encountered at higher V_m within the slurry pipeline.

C_w also significantly influences SEC, as depicted in Figure 7b,c. The literature confirms that an increase in C_w leads to a corresponding rise in SEC for various R_h and V_m. This is due to higher concentrations resulting in greater pressure drop, thereby requiring more power for transportation, consequently leading to an increase in SEC.

4.4. Nusselt Number

The Nusselt number, a dimensionless parameter commonly used to assess heat transfer between a moving fluid and a solid surface, is examined in Figure 8. This illustration displays the relationship between the Nusselt number and the average V_m, specifically for particles sized at 125 µm. The graph provides Nusselt number values for three distinct C_w = 40–60%, transporting at V_m = 1 to 5 m/s. The chart highlights the positive correlation between the Nusselt number and the mixture V_m. Slurry flowing at lower V_m exhibits lower Nusselt numbers, which progressively increase with the mixture V_m. This behavior is attributed to higher flow V_m enhancing heat transfer between the solid and liquid phases. Furthermore, it is noteworthy that slurry mixtures with lower C_w = 40% display the lowest Nusselt numbers, while those with C_w = 60% exhibit the highest values. This variation is due to the increased heat transferred by the higher-concentration slurry mixture.

5. Conclusions

This study investigates the influence of pipe roughness height (R_h) on the kinetic and thermo-fluidic transport characteristics of a solid–liquid slurry flowing through a horizontal straight pipe. An Eulerian multiphase model combined with the RNG k-ε turbulence model was employed to analyze the slurry’s kinetic and thermal behavior. Key findings from the numerical analysis include:

• The Eulerian model accurately predicts pressure drop for 125 µm particles in a horizontal straight pipe, with V_m = 1–5 m/s, and C_vf = 30%, showing strong agreement with experimental data.
• The normalized pressure gradient positively correlates with roughness height, which further varies with V_m: minimal at low V_m but increasing exponentially at higher V_m. Additionally, for a given V_m and roughness height, the pressure gradient increases proportionally with C_w.
• The surface roughness significantly affects particle concentration distribution at lower velocities, with roughness-induced gradients diminishing as V_m increases due to stronger inertial effects. Additionally, higher C_w results in a more uniform yet denser particle distribution across the pipe cross-section.
• Similarly to the pressure gradient, normalized Specific Energy Consumption (SEC) also rises with roughness height, V_m, and C_w, following a comparable trend.
• The non-dimensional Nusselt number remains independent of roughness height yet is significantly influenced by V_m and C_w. A lower V_m corresponds to reduced Nusselt numbers, which increase with arh rise in V_m and C_w. The maximum Nusselt number is observed at C_w = 60%.

Based on these findings, reducing pipe roughness height is recommended to enhance system efficiency and decrease additional power requirements.