Numerical Modelling of Medium Slurry Flow in a Vertical Pipeline

*Artur S. Bartosik*

### **Abstract**

The study deals with the modelling and experiments of vertical solid-liquid turbulent flow with narrowly sized solid particles of average diameters equal to 0.125 mm, 0.240 mm and 0.470 mm, and solid concentrations by volume from 10% to 40%, called medium slurry. The physical model assumes that the slurry with solid particles surrounded by water is flowing upward through a vertical pipeline with solid concentrations from 10–40% by volume. Experiments with such slurries clearly indicated enhanced damping of the turbulence, which depends on the diameter of the solid particles. The mathematical model constitutes conservative equations based on time averages for mass and momentum. The closure problem was solved by taking into account the Boussinesque hypothesis and a two-equation turbulence model together with an especially designed wall damping function. The wall damping function depends on the average diameter of the solid particles and the bulk concentration. The predictions' results were successfully compared with the measurements. The study demonstrates the importance of solid particle diameter and showed that using a standard wall damping function gives higher friction compared to measurements. The main objective of this study is to present a mathematical model for medium slurry flow in a vertical pipeline, including a specially designed wall damping function, and to demonstrate the influence of solid particle size on frictional head loss. The effect of mean particle diameter and solid concentration on frictional head loss has been discussed and conclusions were formulated.

**Keywords:** medium slurry flow, experiments on slurry flow, modelling of vertical slurry flow, modelling of slurry turbulence, numerical modelling

### **1. Introduction**

Solid-liquid turbulent flow is an interdisciplinary research area of great technological and commercial importance. It is widely used in mechanical and chemical engineering, power plants, food and mining industries, and in medicine. Solid-liquid flow appears mainly in pump transport in various pipeline systems [1–3]. Such transportation faces problems such as abrasion of the pump's elements, especially between the rotating impeller and the stationary throat bush, the rotating shaft sleeve, and the stationary packing inlet and outlet ducts. Due to the abrasive characteristics of the solid particles, the pumps and pipelines suffer during the work under these

conditions. Power, capacity, wear and breakdown resistance, and robustness are the essential keys for efficient pump dredging. The characteristics of dredging pumps and pipelines require a long working life in abrasive conditions, a limited influence of wear on pump performance and a low net positive suction head requirement. Abrasion in a pump or pipeline can be defined by the loss of weight per unit area or the loss of thickness under the dynamic action of solid particles that act on the solid wall [4–6]. In this process, the 'particle-wall' interaction, especially at the high particle size and density, high flow rate, and high solid concentration, plays a crucial role, and research on the determination of the 'particle wall'stress is much desired [7, 8]. Another particular application of solid-liquid flow is related to the drug delivery system in which magnetic targeting offers the ability to target a specific site, such as a tumour [9]. Solid-liquid flow can also exist with extremely high velocity to cut concrete, rock, glass, steel, ceramics, composites and plastics. The finished edge obtained by such a process often eliminates the need for post-machining to improve surface finish [10].

Solid-liquid flow could be classified as stationary bed, moving bed, heterogeneous and pseudo-homogeneous, or as settling or non-settling types [11]. Settling slurry is formed mainly by coarse particles. When predicting frictional head loss in slurry flow with coarse or medium particles, it is reasonable to assume the Newtonian model, as now one can measure the rheology of such slurries [5]. In coarse dispersive slurry flow, one should mention the basic research of Bagnold [7] and mathematical models, such as, for instance, [12, 13], which include solid-liquid and/or solid-solid interactions.

Non-settling slurries usually contain fine particles with an average diameter below 0.040 mm and can form a stable homogeneous mixture exhibiting an increased apparent viscosity. Such slurries usually exhibit yield stress and require a proper rheological model incorporated into the momentum equation. Fine particles demonstrate an increased viscous sublayer, which means that the damping of the turbulence appears in the near-wall region [14]. In this case, the mathematical model includes an apparent viscosity concept, a suitable rheological model [15] and a properly defined wall damping function [16].

Slurries with medium solid particles of average diameter between 0.10 mm and 0.50 mm are usually assumed to be a Newtonian solid-liquid mixture or as a mixture of two separated phases. If the bulk velocity of the slurry is sufficiently high, such slurry can exhibit a non-settling type. If the slurry flow occurs in a vertical pipeline, this flow can be treated as axially symmetric. It is important to emphasise that slurries with medium solid particles exhibit enhanced damping of turbulence because frictional head loss is relatively low compared with carrier liquid flow [17].

Mathematical modelling of solid-liquid flow usually requires reliable experimental data to validate the model. The most desirable measurements are those close to the wall of the pipe. Measurements at high concentrations of the dispersed phase are very difficult. As a result of these difficulties, most measurements regard gas-solid or gasliquid flow. Mathematical modelling of solid-liquid flow is far away from the knowledge gathered for Newtonian flows, and turbulent flow is still the main challenge of CFD. Analysing the predictions of turbulent solid-liquid flow, the mixture theory models are the most general and are based on rigorous fluid mechanics frameworks [18]. Researchers around the world have been working to develop accurate models for frictional head loss and velocity distribution in slurry flows. Frictional head loss is one of the most important technical parameters to be evaluated by the designers for designing a pipeline slurry transportation system, and the parameter that dictates the

selection of pump capacity. Determining the most efficient and economical way to pump any solids into a carrier liquid requires careful consideration and analysis of numerous factors, some of which can have a significant impact on performance and cost. Among them, there are an averaged solid particle diameter, solid concentration, particle density, deposition velocity, carrier liquid properties and properly matched characteristics of a pipeline with the characteristics of a pump.

The main objective of this study is to present a mathematical model for medium slurry flow in a vertical pipeline, including a specially designed wall damping function, and to demonstrate the influence of solid particle size on frictional head loss.

### **2. Literature review**

Analysing measurements with respect to the behaviour of particles suspended in a surrounding fluid, we recognise several techniques. For example, Roberts and Kennedy [19] used pulsed injections of salt water and tagged radioactive particles. Kowalewski [20] used the ultrasound technique, while Altobelli et al. [21] used nuclear magnetic resonance (NMR). Yianneskis and Whitelaw [22], Nouri et al. [23], Nouri and Whitelaw [24], Wildmanet et al. [25] and Chen and Kadambi [26, 27] used Laser-Doppler anemometer (LDA). The necessary condition for the measurement of LDA is that the dispersed phase consists of transparent solid particles, whose refractive index is close to the carrier phase [22–27]. In addition to the above techniques, it is worth mentioning the technique applied by Shook's research team. The authors used the original method to measure the lateral variation of solids concentration in vertical pipelines [28–30].

Yianneskis and Whitelaw [22] measured the local solid and liquid phase velocities in a fully developed pipe flow using the LDA technique. The dispersed phase in the carrier liquid consisted of transparent solid particles with a diameter of 0.270 mm. The most important conclusion from their research was the fact that the slip velocity between the solid and liquid phases clearly decreases with increasing concentration of the solid phase. Sumner et al. [17, 31] and Nasr-El-Din et al. [32] measured the solid concentration distribution in slurry with medium and coarse particles. They concluded that the averaged particle diameter has a crucial influence on friction losses and the profile of solid concentration across a pipe. If the solid particles are coarser, the solid concentration distribution decreases within the near wall region.

The turbulence of the solid particles in the region near a pipe wall was examined by several researchers, such as Nouri and Whitelaw [24], Chen and Kadambi [27], Kuboi et al. [33], Schreck and Kleis [34], and Gai et al. [35]. Researchers observed that the ejection-sweep cycle is affected strongly by particles. They concluded that the slip velocity decreases as the solid concentration increases. The authors emphasised that the presence of solid particles can induce attenuation or amplification of the turbulence. Modification of turbulence depends on the physical properties of the solid and liquid phase. Modulation of turbulence by a dispersed solid particle was also emphasised by Gore and Crowe [36], Sundaresan et al. [37], Jianren et al. [38], Eaton et al. [39], Fessler, J.R.; Eaton [40], and Li et al. [41].

Sundaresan et al. [37] outlined a number of scientific challenges that represent building blocks for a comprehensive understanding of disperse flows encountered in a variety of technologies and in nature. Researchers concluded that new experiments and/or analyses are needed to cast light on the important phenomena that cause

turbulence damping or generation. The authors suggested that the experiments should be conducted in simple turbulent flows, such as grid turbulence, fully developed pipe or channel flow, or simple axisymmetric flows. Regardless of geometry, experiments must include a wide range of particle parameters in a single fixed facility. Their conclusions are still outstanding in the mathematical modelling of slurry flow and especially for medium and coarse particles.

In conclusion, one can say that experiments on the influence of solid particles on turbulence are extremely difficult. Measurements at high solid concentrations of the dispersed phase are avoided because of the risk of damage or contamination of intrusive hot film or hot wire probes. With optical methods, the beam is attenuated by particles. As a result of these difficulties, most of the measurements in the literature refer to gas-liquid or gas-solid flows. Other methods of measurement, such as NMR, face problems with resolution, which is fundamental if velocity fluctuations are needed, especially at the pipe wall.

On the basis of the gathered knowledge, we can say that the averaged solid particle diameter has a significant influence on the frictional losses. We know that the difference between the solid and liquid phases is important; however, its importance decreases as the solid concentration increases. Small and medium solid particles can suppress, while large particles can enhance frictional losses. Of course, the phenomenon is more complex as the properties of the solid and liquid phase, and flow conditions, such as the velocity and geometry effect on the level of turbulence.
