**1. Introduction**

Particulate flows in pipes have numerous engineering applications ranging from pneumatic conveying systems to coal gasifiers and chemical reactor design and are one of the most thoroughly investigated subjects in the area of multiphase flows. These flows are very complex and influenced by various physical phenomena, such as particle-turbulence and particleparticle interactions, deposition, by gravitational and viscous drag forces, particle rotation, and lift force.

Numerous theoretical and experimental researches, e.g., [1-20], studied various aspects of the behavior of gas and solid particles in particulate pipe flows.

The present study focuses on the effect of variation of the pipe diameter for a constant Reynolds number applied to vertical particulate turbulent pipe flows. The numerical investigation discussed here examined in detail the effects of direct and indirect particle-turbulence interaction (no-coupling and coupling) and gravity for various flow mass loadings of 250, 500 and 700 *μ*m coal particles. Additionally, the viscous drag force and the Magnus and Saffman lift forces are also taken into account. The behavior of the particulate phase was under consideration, both for the fine particles being liable to the turbulent fluctuations of gas and the larger particles, which have the lesser response to the flow turbulence.

The presented numerical model makes use of the two-fluid model, e.g., [21-25], and the Reynolds-averaged Navier-Stokes (RANS) approach [18, 19] applied to gas and solid particles.

Within the frame of the two-fluid model, the gas and the particles are considered as two coexisting phases that span the entire flow domain [18, 19]. Therefore, in order to describe the

flow of the particulate phase within the two-fluid model, the presented model implements the RANS approach. This approach is the most general and frequently used in modeling, its closure equations have been verified by numerous experiments, and the boundary conditions are easy to determine. The given modeling employs the model [14], which is the most relevant to account for mechanisms of a turbulence modulation caused by particles, since it includes both the turbulence enhancement and its attenuation by particles. The inter-particle collisions is another mechanism accounting for capture properties of turbulent particulate pipe flows, which has been modeled, e.g., in [16]. These two models enables comprehensive mathematical simulation of the two-phase upward pipe flow.

The presented model allows covering 100 and more calibers of a pipe flow. This is the main advantage over the numerical models based, for example, on direct numerical simulation (DNS) codes, (e.g., [26]), that handle usually with a short pipe length up to 10-20 calibers with imposing the upper limit for the flow Reynolds number.

The utilized two-fluid model with adoption of original collisional closure model [16] together with the applied numerical method has been verified and validated in our previous researches [18, 19] by comparison of numerical results with the experimental data [6]. In the given study, the effect of variation of the pipe diameter (or transport velocity) at a constant Reynolds number is numerically investigated in the particulate turbulent flow. This is a step forward for analyzing the external effect, namely, the flow configuration rather the internal effect with variation of the parameters of the flow.
