## 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 particle-particle 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.

## 2. Governing equations and numerical method

The sketch of the computational flow domain is shown in Fig. 1, where *F*_{G} is gravity, *F*_{D} is the aerodynamic drag force, *F*_{LR} is the lift force that arises from particle rotation (the Magnus lift force), *ω*_{s} is the angular velocity of a particle.

It is assumed that the particulate phase is polydispersed and composed of several known mass fractions. These fractions can be of single material density and characterized by equivalent particle diameter of the fraction

### 2.1. Governing equations for the 2D RANS model

The model is based on the time averaged Navier-Stokes equations (RANS method), without any simplifications, such as the boundary layer simplifications. The vertical pipe flows are 2D unless the study of rotating flows.

A short presentation of the governing equations written for the axisymmetric channel case is as follows:

continuity equation for the gas phase:

where *u* and *v* are the longitudinal and radial velocity components of the gas phase.

longitudinal linear momentum equation for the gas phase:

where *ν*_{t} is calculated following the Boussinesq eddy-viscosity concept; *p* is pressure; *C*_{M} is calculated according to Crowe et al. (1998);

radial linear momentum equation for the gas phase:

turbulence kinetic energy equation for the gas phase:

where *k* and *k*_{s} are the turbulence kinetic energy of the gas- and particulate phases, respectively. The hybrid dissipation rate

continuity equation for the particulate phase:

where *x* and *r* directions

momentum equation in the longitudinal direction for the particulate phase:

where *g* is the gravitational acceleration.

momentum equation in the radial direction for the particulate phase:

where

angular momentum equation in the longitudinal direction for the particulate phase:

where

### 2.2. Boundary conditions for the RANS model

As inlet boundary conditions, it is assumed that particles enter the previously computed, fully developed flow domain of the single-phase flow, having the initial longitudinal velocity determined by the lag coefficient. The equilibrium outlet boundary conditions were set at the exit cross-section

where empirical constant

The wall boundary conditions for the particulate phase have taken into account the particle’s velocity lag determined through particles-wall interaction [19].

### 2.3. Numerical method

The control volume method was applied to solve mass and momentum equations of both phases by using the implicit lower and upper matrix decomposition method with flux-blending differed-correction and upwind-differencing schemes [31]. Calculations were performed in dimensional form for all flow regimes. The number of the control volumes was varied from 280000 to 1120000, corresponding to the increase in the pipe diameter from *D*=30.5 mm to *D*=61 mm, and their size remained constant across the pipe flow.

## 3. Numerical results

The numerical results presented in the figures have been obtained at a distance of *µ*m coal particles (physical density

The Reynolds number *Re* was assigned as the constant through all calculations and set equal to

The profiles of particles velocity *μ*m particles at the mass loading of

The following figures show the influence of various force factors on cross-sectional distributions of the velocity lag, particle mass concentration and turbulence modulation originated from the particles. Separately, there were analyzed the effect of the direct (turbulence) and indirect particle-turbulence interaction (no-coupling and coupling) and the inter-particle collisions.

The analysis of behavior of the normalized longitudinal velocity lag is shown in Figure 6 for various force factors for the 250 *μ*m coal particles at

If the motion of particles is exposed only by the viscous and gravitation forces (without the direct effect of turbulence, lift forces and coupling), the velocity lag between two phases approaches to the particles terminal velocity occurring in the steady-state flow domain, i.e. the ratio

On the face of it, the increase in the particle size should result in increase of absolute value of the velocity lag occurring for the given pipe diameter. However, the more detailed analysis shows that increase of the particles size results in reduce of the normalized longitudinal velocity lag (Figures 7 and 8). This effect is more pronounced with increase of the flow mass loading (cf. Figures 7 and 8).

Diminishing of the normalized longitudinal velocity lag observed for relatively dense flow at

In order to trace the effects of the particles size and mass loading on the turbulence modulation let us first examine the distribution of the particle mass concentration presented in Figure 9. As one can see, the growth of the particle size and flow mass loading makes profiles steeper [16, 32] with more pronounced tendency with respect of the particle size variation. The smaller particles are easier spread out of the pipe flow domain due to the higher value of turbulent diffusion coefficient, and the growth of the mass loading diminishes turbulence and its diffusion aligning process.

Figures 10 and 11 explicitly address to the coupling effect, which was observed for two flow mass loadings *µ*m coal particles. Obviously, the higher mass loading leads to the higher rate of the turbulence modulation, i.e. if there was turbulence attenuation occurred for the given particle size, then this process was intensified for the higher mass loading (cf. the corresponding curves plotted for the same particle sizes in Figures 10 and 11).

The next series of plots (Figures 12–18) show the effect of the pipe diameter for the constant Reynolds number on distribution of the normalized velocity lag, the particle mass concentration and the turbulence modulation.

Figures 12, 13 and 14 show the profiles of the normalized longitudinal velocity lag obtained for various pipe diameters for 250, 500 and 750 *µ*m coal particles. One can see that decrease of the pipe diameter results in the higher velocity lag and, as a result, in the stronger particles involvement into the turbulent motion. This fact is proved by the data of Figures 2 and 3 showing that the smaller pipe diameter corresponds to the higher level of the turbulence energy, and, sequentially to the higher rate of the particles involvement by the gas flow.

The effect of the particles collisions that may occur at the higher mass loading of

One can see that the effect of the pipe diameter has the same tendency as the effect of the particle size, i.e. the increase of the pipe diameter acts like the decrease of the particle size, straightening the profiles of the particle mass concentration (Figures 15 and 16). An accounting of the inter-particle collision effect intensifies the particle dispersion (s. Figure 16).

The turbulence modulation is shown in Figures 17 and 18 for the considered particles sizes in two marginal cases: *µ*m. It is evident that the increase of the particle size leads to decrease of the attenuation rate of turbulence. The effect of the inter-particle collisions (Figure 18) results in the enhancement of turbulence by particles in vicinity of the flow axis and its damping, that occurs in the region locating between the flow axis and the pipe wall.

## 4. Conclusions

2D RANS numerical method fitted with the appropriate closure equations was applied for the computational investigation of the upward turbulent particulate pipe flow at the distance of 100 calibers from the pipe entrance. The axial velocity lag, turbulent kinetic energy of gas and particles mass concentration, effected by the gravity, viscous drag, the particle-turbulence, particle-particle, particle-wall interactions as well as the Saffman and Magnus lift forces, were examined for various particle sizes and flow mass loadings at the same flow Reynolds number.

The obtained numerical results allow to draw the following conclusions pertaining to behavior of solid particles under the conditions of the upward turbulent pipe flow:

It is obvious that if the motion of particles is exposed only by the viscous and gravitational forces (without the direct effect of turbulence, lift forces and coupling), the absolute magnitude of the axial velocity lag approaches to the particles terminal velocity. However, simultaneous accounting of all force factors, effecting on the fine particles, results in substantial exceeding of their axial velocity lag as compared with their terminal velocity, that is due to intensification of influence of turbulence on a motion of the fine particles.

It was revealed that the effect of the particles size appears as follows:

the increase of the particles size results in reducing of the relative axial velocity lag. The absolute magnitude of the velocity lag approaches to the particles terminal velocity;

the fine particles spread more uniformly in the cross-section of the pipe as against the coarse ones, due to their higher coefficient of turbulent diffusion;

enlargement of the particles size gives the lower rate of the turbulence attenuation.

the increase of the flow mass loading causes the diminution of the relative velocity lag, and this is more pronounced for the fine particles. The same tendency also takes place when considering the inter-particle collisions for the large flow loading;

the increase of the flow loading results in the turbulence attenuation that is followed by the non-uniform cross-sectional distributions of the particles mass concentration, while the accounting of the inter-particle collisions causes the opposite trend, i.e. their flattening.