**1. Introduction**

Slurry pipeline transport is widely used in the food and chemical industry and in the transfer of minerals from a mine to the processing site or from the deep ocean to the surface level [1]. Pipeline transportation is more economical and environmentally friendly than other modes of transport, such as rail or trucking, when moving minerals over long distances. They also have a high safe delivery rate [2,3]. In the literature, we can find various approaches for the analysis of solid–liquid flow and its predictions [4–14]. Analyses are mainly dedicated to a proper class of flow, such as homogeneous, heterogeneous, and with steady or mowing beds. The proposed models are mainly algebraic. If predictions are considered, we use analytical or numerical methods [15–24]. Analytical methods are impractical and can be applied to very simple flow conditions. If numerical methods are considered, we recognize Direct Numerical Simulation (DNS), which is computationally costly, Large Eddy Simulation (LES), which is less expensive, Reynolds-averaged Navier– Stokes equations (RANS), which is the computationally cheapest, or a combination of the mentioned methods, called hybrid methods. The slurry flow phenomenon is very complex; therefore, mathematical models are strictly dedicated to a certain flow situation.

The heat transfer process in slurry transportation is essential, especially for long distances. Temperature affects the viscosity and rheological properties of the slurry and thus changes the energy loss during pipeline transport, which affects the operation of the pipeline system. Changes in the temperature of the transported slurry can be caused by heat transfer between the slurry pipeline and the environment and by the conversion of part of the mechanical energy into heat. In the case of pipeline operation in the Arctic Sea, for example, it could cause permafrost melting and instability of pipeline routes. In other cases, maintaining a constant temperature of the flowing medium is imperative, while in some other cases, the risk of freezing or loss of heat appears, which requires proper pipe insulation to avoid plug formation. Therefore, the ability to predict the flow of the slurry with heat exchange is a great challenge in CFD.

To predict the heat transfer between the flowing slurry and the environment, the first point is to set up a reliable physical model and then a mathematical model. The mathematical model should be able to predict the velocity and temperature fields of the slurry. When formulating a physical model for slurry flow with heat transfer, it is necessary to define the slurry properties and flow conditions like, for instance:


The study focuses on the modelling and simulation of convective heat transfer between a fine dispersive slurry and a pipe wall under a turbulent flow regime in a range of solid concentrations for commercial applications. The solid phase constitutes fine solid particles with an average diameter of about 20 μm, since small particles are responsible for increasing viscosity, non-Newtonian behavior, and damping of turbulence. Therefore, the relation between shear stress and shear rate is crucial to include in a mathematical model.

Solid particles are known to affect the flow structure and can enhance or suppress turbulence [25–33]. Generally, one can conclude that small particles can suppress, while large particles can increase turbulence. Of course, the phenomenon is more complex as the particle density, carrier liquid properties, and flow conditions, such as the velocity and geometry effect on the level of turbulence as well. Experimental data on frictional head loss for turbulent flow of fine dispersive slurry clearly demonstrate a lower frictional head loss than expected [6,25,34,35]. Wilson and Thomas hypothesized that in a turbulent flow of fine dispersive slurry, the viscous sublayer becomes thicker compared to a single-phase flow under the same flow conditions [25]. Therefore, the Wilson and Thomas hypothesis was taken into account to develop a mathematical model.

Considering the heat exchange between the transported slurry and the surrounding, we recognize methods and techniques focused on the enhancement of heat exchange, named passive or active. Passive methods, such as shape insert with dedicated perforation or mechanically deformed pipes, have been studied for several years and have become commercial solutions [36–41]. Active methods, such as air injection, bubble generation, or proper pulsation, can produce increases in the heat transfer process [42–45].

Heat transfer in the solid–liquid flow has been experimentally investigated by several studies [46–49]. However, experiments and studies on heat transfer in a turbulent flow with fine solid particles, especially for solid concentrations of commercial interest, above 20% by volume, are scattered in the literature. Existing experiments refer mainly to water-air or water-oil or nanofluids or slurries with low solid concentrations [50–53]. Research mainly includes solid particles made of metals, oxides, or carbides suspended in a liquid. Such

experiments and predictions focus on examining whether solid particles can increase or decrease the heat transfer process between the suspension and a pipe wall. The results show that such an increase depends mainly on the properties of the particles and the solid concentration [54,55].

Wang et al. [48] experimentally examined the thermal conductivity of Al2O3 and CuO nanoparticles mixed with water, vacuum pump liquid, engine oil and ethylene glycol. The average particle diameter of Al2O3 was 28 nm and the average particle diameter of CuO was 23 nm. All particles were loosely agglomerated in the chosen liquid. Experiments have shown that the heat transfer of the mixture increases with the volume fraction of Al2O3 particles [48].

Rozenblit et al. [56] conducted an experimental study on the heat transfer coefficient associated with solid–liquid transport in a horizontal pipe using an electro-resistance sensor and infrared imaging. They considered a flow of acetyl-water mixture on a moving bed with solid volume concentrations of 6%, 9%, 12%, and 15% by volume. They noted that the local heat transfer coefficient changes from its lowest value at the bottom of the pipe to its highest value above the carrier liquid at the upper heterogeneous layer. These experiments indicated that the heat transfer coefficient is strongly influenced by the cross-sectional distribution of the solid phase [56].

Ku et al. [49] experimentally examined the effect of solid particles on heat transfer. They used spherical fly ash particles with a mass median particle diameter of 4 to 78 μm and a particle density of 2270 kg/m3, and Reynolds numbers of 4000 to 11,000 in an 8 mm inner diameter pipe. The authors noted that the highest heat transfer coefficient was obtained for the dilute solution with C = 3%, and then gradually decreased with increasing solid volume concentration. They proposed a correlation for the Nusselt number. The correlation depends on Reynolds and Prandtl numbers, solid concentration, and the ratio of pipe diameter to average particle diameter. The correlation is limited to: Re = 3000–11,000; Pr = 3.8–5.0; D/dP = 102–615; C = (1–10)%. However, it is not clear whether such suspensions possess or do not possess the yield stress. Furthermore, the apparent viscosity of the slurry in their research was 1.75 higher compared to water if C = 30%. This contrasts with current studies, as the apparent viscosity of C = 30% is at least 13 times higher compared to water and increases as the wall shear stress decreases. Additionally, the Prandtl number in the current study is much higher compared to the limitation of Ku et al. 49]. In conclusion, we can say that it is impossible to compare the numerical predictions with the experimental data if the slurry properties differ much.

Bartosik performed simulations of the influence of the solid volume concentration [57] and the yield shear stress [58] on heat transfer in kaolin slurry. The author considered two solid concentrations of kaolin equal to 7.4% and 38.3%, which correspond to yield shear stresses of 4.92 Pa and 9 Pa, respectively. The author found that both solid concentration and yield stress strongly affect the Nusselt number and the Nusselt number decreases with increasing solid concentration or yield stress.

Analysis of the literature indicates that most of the research deals with heat transfer in a single phase, liquid-air flow, nanofluids, ice slurry, laminar flow, or with low solid concentrations. Experiments and modelling of heat transfer in non-Newtonian slurries containing minerals with fine particles, yield stress, and solid volume concentration at least 20% by volume are scattered. Reliable predictions require reliable measurements. Prediction difficulties arise when the solid concentration increases and the particle-liquid and/or particle–wall interaction occurs. A better approach to understanding the slurry flow mechanism and its correlation with the heat transfer process, especially at high solid concentrations, is needed, as it is important for better optimization design and operation control. The ability to simulate turbulent solid–liquid flow to predict frictional head loss, velocity, and temperature distributions remains one of the main challenges in CFD. However, from an engineering point of view, we need simple empirical or semiempirical correlations that allow the calculation of the heat transfer coefficient for known slurry. Therefore, proper correlations for the Nusselt number are still required.

The objective of the paper is to develop a new correlation of the Nusselt number for turbulent flow of fine dispersive slurry that exhibits yield stress and damping of turbulence. For such a slurry, a new correlation of the Nusselt number is proposed. The new correlation of the Nusselt number includes Reynolds and Prandtl numbers, solid volume concentration, and dimensionless yield shear stress.
