**3. Numerical results**

The numerical results presented in the figures have been obtained at a distance of *x* / *D*=100 from the pipe entrance. At this distance it was reasonable to stipulate that the steady flow conditions have been reached and there was no influence of the entrance conditions. The results presented here are mainly dimensionless, but some of them are given in dimensional form. 250-, 500 and 750 *μ*m coal particles (physical density *ρ<sup>p</sup>* =1600 kg/m<sup>3</sup> ) were used in investiga‐ tions. The flow mass loading was *m* \* =1 and 10 kg dust/kg air. The applied particles were light enough to respond to turbulent fluctuations of gas.

The Reynolds number *Re* was assigned as the constant through all calculations and set equal to 4.4×10<sup>4</sup> . The pipe diameter *D* was 30.5, 45.75 and 61 mm for the gas average velocities *u*¯ =21.6, 14.6 and 10.8 m/s, respectively. The average longitudinal velocity and turbulence energy radial distributions calculated for these three regimes are shown in Figures 2 and 3.

*v*\*

<sup>2</sup> <sup>=</sup>*τ<sup>w</sup>* / *<sup>ρ</sup>* <sup>=</sup>*cμ*

26 Computational and Numerical Simulations

are as follows:

{*u* = *τw ρ* 1 *æ*

*u* = *v*∗ 2 *y*

**2.3. Numerical method**

**3. Numerical results**

to 4.4×10<sup>4</sup>

ln(*<sup>y</sup>* +) <sup>+</sup> *<sup>C</sup>* <sup>=</sup>*v*<sup>∗</sup>

velocity lag determined through particles-wall interaction [19].

mm to *D*=61 mm, and their size remained constant across the pipe flow.

250-, 500 and 750 *μ*m coal particles (physical density *ρ<sup>p</sup>* =1600 kg/m<sup>3</sup>

enough to respond to turbulent fluctuations of gas.

0.5*k*. The computations near the wall were carried out at the half-width of the

*<sup>v</sup>*∗) 11.6<sup>≤</sup> *<sup>y</sup>* <sup>+</sup> <500

<sup>=</sup> (10)

, (9)

) were used in investiga‐

control volume off the wall. Then, for the longitudinal velocity of the gas phase and for the turbulence energy computed by means of its production *Pk* , the following boundary conditions

> ln(*<sup>E</sup> <sup>y</sup> ν*

1 *æ*

*<sup>ν</sup> <sup>y</sup>* <sup>+</sup> <11.6

*<sup>τ</sup> <sup>P</sup>*

r

where empirical constant *æ* =0.41; *y* =*Δ* / 2 (*Δ* is the width of the control volume).

*w k . . μ*

The wall boundary conditions for the particulate phase have taken into account the particle's

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 fluxblending 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

The numerical results presented in the figures have been obtained at a distance of *x* / *D*=100 from the pipe entrance. At this distance it was reasonable to stipulate that the steady flow conditions have been reached and there was no influence of the entrance conditions. The results presented here are mainly dimensionless, but some of them are given in dimensional form.

tions. The flow mass loading was *m* \* =1 and 10 kg dust/kg air. The applied particles were light

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

*u*¯ =21.6, 14.6 and 10.8 m/s, respectively. The average longitudinal velocity and turbulence energy radial distributions calculated for these three regimes are shown in Figures 2 and 3.

. The pipe diameter *D* was 30.5, 45.75 and 61 mm for the gas average velocities

*æc k y* 2 0 25 0 5 <sup>2</sup> ,

**Figure 2.** Profiles of the longitudinal gas velocity in the pipes *D*=30.5, 45.75 and 61 mm, Re=4.4×10<sup>4</sup> .

**Figure 3.** Profiles of the turbulence energy of gas in the pipes *D*=30.5, 45.75 and 61 mm, Re=4.4×10<sup>4</sup> .

The profiles of particles velocity *us* normalized to the longitudinal gas velocity, which was taken place at the pipe axis, and the particles mass concentration *α* normalized to its magnitude obtained at the pipe axis, are shown in Figure 4 for 250 *μ*m particles at the mass loading of *m* \* =1. The turbulence modulation *TM* determined as *TM* =(*k* / *k*<sup>0</sup> −1) ×100*%*, where *k* and *k*<sup>0</sup> are the turbulence energy of the gas phase for the particulate flow conditions and the gas flow

The following figures show the influence of various force factors on cross-sectional distribu‐ tions 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

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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 *m* \* =1. Here and below the longitudinal velocity lag is presented as the ratio of the longitudinal velocity slip between the gas and particulate phases to the terminal velocity of particles (*u* −*us*) / *vt*, where *vt* is the particle terminal velocity. One can see that larger particles have less magnitude of axial velocity lag then those of small particles with noticeable velocity difference. It looks like unexpected result,

**Figure 6.** Profiles of the normalized longitudinal velocity lag for 250 μm coal particles obtained for various flow condi‐

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 *ur* / *vt* converges to unity (the curve marked by triangles, Figure 6). However, as the numerical simulation shows, if the motion of particles is exposed by various force factors, then the normalized longitudinal velocity lag increases above the particles terminal velocity.

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

.

however, formation of velocity lag is multifold process.

tions, *m* \* =1, *D*=45.75 mm, Re=4.4×10<sup>4</sup>

collisions.

**Figure 4.** Profiles of the normalized longitudinal velocity of particulate phase and particle mass concentration of 250 μm coal particles in various cross-sections *x* / *D*, *m* \* =1, *D*=30.5 mm,Re=4.4×10<sup>4</sup> .

**Figure 5.** Profiles of the turbulence modulation by 250 μm coal particles in various cross-sections *x* / *D*, *m* \* =1, *D* =30.5 mm, Re=4.4×10<sup>4</sup> .

unladen with particles, respectively, is presented in Figure 5 for various exit cross-sections *x* / *D*=100, 180 and 260. Based on the results shown in Figures 4 and 5, one can conclude that for the saving of computation time, the exit cross-section *x* / *D*=100 can be considered as the steady-state two-phase pipe flow section.

The following figures show the influence of various force factors on cross-sectional distribu‐ tions 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 *m* \* =1. Here and below the longitudinal velocity lag is presented as the ratio of the longitudinal velocity slip between the gas and particulate phases to the terminal velocity of particles (*u* −*us*) / *vt*, where *vt* is the particle terminal velocity. One can see that larger particles have less magnitude of axial velocity lag then those of small particles with noticeable velocity difference. It looks like unexpected result, however, formation of velocity lag is multifold process.

**Figure 6.** Profiles of the normalized longitudinal velocity lag for 250 μm coal particles obtained for various flow condi‐ tions, *m* \* =1, *D*=45.75 mm, Re=4.4×10<sup>4</sup> .

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 *ur* / *vt* converges to unity (the curve marked by triangles, Figure 6). However, as the numerical simulation shows, if the motion of particles is exposed by various force factors, then the normalized longitudinal velocity lag increases above the particles terminal velocity.

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

unladen with particles, respectively, is presented in Figure 5 for various exit cross-sections *x* / *D*=100, 180 and 260. Based on the results shown in Figures 4 and 5, one can conclude that for the saving of computation time, the exit cross-section *x* / *D*=100 can be considered as the

**Figure 5.** Profiles of the turbulence modulation by 250 μm coal particles in various cross-sections *x* / *D*, *m* \* =1, *D*

**Figure 4.** Profiles of the normalized longitudinal velocity of particulate phase and particle mass concentration of 250

.

μm coal particles in various cross-sections *x* / *D*, *m* \* =1, *D*=30.5 mm,Re=4.4×10<sup>4</sup>

steady-state two-phase pipe flow section.

.

28 Computational and Numerical Simulations

=30.5 mm, Re=4.4×10<sup>4</sup>

**Figure 7.** Profiles of the normalized longitudinal velocity lag for 250, 500 and 750 μm coal particles, *m* \* =1, *D*=45.75 mm, Re=4.4×10<sup>4</sup> .

**Figure 9.** Profiles of the mass concentration of 250, 500 and 750 μm coal particles, *m* \* =1 and 10, *D*=45.75 mm,

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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

Figures 10 and 11 explicitly address to the coupling effect, which was observed for two flow mass loadings *m* \* =1 (Figure 10) and *m* \* =10 (Figure 11) for 250, 500 and 750 *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

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 concentra‐

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.

Re=4.4×10<sup>4</sup>

.

diffusion aligning process.

particle sizes in Figures 10 and 11).

tion and the turbulence modulation.

**Figure 8.** Effect of mass loading on longitudinal velocity lag, *m* \* =10; the flow conditions are the same as in Figure 7.

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 *m* \* =10 (s. Figure 8) clearly depicts the tendency of the turbulence attenuation by particles, or, in other words, decrease of direct effect of turbulence on the particles motion.

**Figure 9.** Profiles of the mass concentration of 250, 500 and 750 μm coal particles, *m* \* =1 and 10, *D*=45.75 mm, Re=4.4×10<sup>4</sup> .

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 *m* \* =1 (Figure 10) and *m* \* =10 (Figure 11) for 250, 500 and 750 *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 concentra‐ tion 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.

velocity lag (Figures 7 and 8). This effect is more pronounced with increase of the flow mass

**Figure 8.** Effect of mass loading on longitudinal velocity lag, *m* \* =10; the flow conditions are the same as in Figure 7.

**Figure 7.** Profiles of the normalized longitudinal velocity lag for 250, 500 and 750 μm coal particles, *m* \* =1, *D*=45.75

Diminishing of the normalized longitudinal velocity lag observed for relatively dense flow at *m* \* =10 (s. Figure 8) clearly depicts the tendency of the turbulence attenuation by particles, or,

in other words, decrease of direct effect of turbulence on the particles motion.

loading (cf. Figures 7 and 8).

mm, Re=4.4×10<sup>4</sup>

.

30 Computational and Numerical Simulations

**Figure 10.** Profiles of the turbulence modulation by 250, 500 and 750 μm coal particles, *m* \* =1, *D*=45.75 mm, Re=4.4×10<sup>4</sup> .

**Figure 12.** Profiles of the normalized longitudinal velocity lag for 250 μm coal particles in the pipes *D*=30.5, 45.75 and

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**Figure 13.** Profiles of the normalized longitudinal velocity lag for 500 μm coal particles in the pipes *D*=30.5, 45.75 and

61 mm, *m* \* =10, Re=4.4×10<sup>4</sup>

61 mm, *m* \* =10, Re=4.4×10<sup>4</sup>

.

.

**Figure 11.** Effect of the mass loading on the turbulence modulation by 250, 500 and 750 μm coal particles, *m* \* =10, *D*=45.75 mm, Re=4.4×10<sup>4</sup> .

**Figure 12.** Profiles of the normalized longitudinal velocity lag for 250 μm coal particles in the pipes *D*=30.5, 45.75 and 61 mm, *m* \* =10, Re=4.4×10<sup>4</sup> .

**Figure 10.** Profiles of the turbulence modulation by 250, 500 and 750 μm coal particles, *m* \* =1, *D*=45.75 mm,

**Figure 11.** Effect of the mass loading on the turbulence modulation by 250, 500 and 750 μm coal particles, *m* \* =10,

Re=4.4×10<sup>4</sup>

.

32 Computational and Numerical Simulations

*D*=45.75 mm, Re=4.4×10<sup>4</sup>

.

**Figure 13.** Profiles of the normalized longitudinal velocity lag for 500 μm coal particles in the pipes *D*=30.5, 45.75 and 61 mm, *m* \* =10, Re=4.4×10<sup>4</sup> .

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

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35

**Figure 16.** Profile of the normalized mass concentration for 750 μm coal particles in the pipes *D*=30.5, 45.75 and 61

**Figure 17.** Profiles of the turbulence modulation for 250 μm coal particles in the pipes *D*=30.5, 45.75 and 61 mm,

mm, *m* \* =10, Re=4.4×10<sup>4</sup>

*m* \* =10, Re=4.4×10<sup>4</sup>

.

.

of the inter-particle collision effect intensifies the particle dispersion (s. Figure 16).

**Figure 14.** Profiles of the normalized longitudinal velocity lag for 750 μm coal particles in the pipes *D*=30.5, 45.75 and 61 mm, *m* \* =10, Re=4.4×10<sup>4</sup> .

The effect of the particles collisions that may occur at the higher mass loading of *m* \* =10 (s. Figure 14) brings this process to slow down the particles motion. Therefore, the particles collisions result in the decrease of the normalized velocity slip as compared with the case of no collisions (cf. Figures 12, 13 and 14).

**Figure 15.** Profiles of the normalized mass concentration for 250 μm coal particles in the pipes *D*=30.5, 45.75 and 61 mm, *m* \* =10, Re=4.4×10<sup>4</sup> .

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).

**Figure 14.** Profiles of the normalized longitudinal velocity lag for 750 μm coal particles in the pipes *D*=30.5, 45.75 and

The effect of the particles collisions that may occur at the higher mass loading of *m* \* =10 (s. Figure 14) brings this process to slow down the particles motion. Therefore, the particles collisions result in the decrease of the normalized velocity slip as compared with the case of

**Figure 15.** Profiles of the normalized mass concentration for 250 μm coal particles in the pipes *D*=30.5, 45.75 and 61

61 mm, *m* \* =10, Re=4.4×10<sup>4</sup>

34 Computational and Numerical Simulations

mm, *m* \* =10, Re=4.4×10<sup>4</sup>

.

.

no collisions (cf. Figures 12, 13 and 14).

**Figure 16.** Profile of the normalized mass concentration for 750 μm coal particles in the pipes *D*=30.5, 45.75 and 61 mm, *m* \* =10, Re=4.4×10<sup>4</sup> .

**Figure 17.** Profiles of the turbulence modulation for 250 μm coal particles in the pipes *D*=30.5, 45.75 and 61 mm, *m* \* =10, Re=4.4×10<sup>4</sup> .

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

ones, due to their higher coefficient of turbulent diffusion;

considering the inter-particle collisions for the large flow loading;

way:

rate.

**Acknowledgements**

**Author details**

Alexander Kartushinsky\*

Technology, Tallinn, Estonia

mental and biomedical applications".

\*Address all correspondence to: aleksander.kartusinski@ttu.ee

**•** 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;

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37

**•** the fine particles spread more uniformly in the cross-section of the pipe as against the coarse

**3.** The given investigation shows that the effect of the flow mass loading acts in the following

**•** 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

**•** 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. **4.** The effect of the pipe diameter acts in the way that its increase: a) gives rise to decrease of the relative velocity lag, b) results in flattening of the cross-sectional distributions of the particles mass concentration and c) induces the decrease of the turbulence attenuation

The work was done within the frame of the target financing under the Project SF0140070s08 (Estonia) and supported by the ETF grant Project ETF9343 (Estonia). The authors are grateful for the technical support of Computational Biology Initiative High Performance Computing Center of University of Texas at San Antonio (USA) and Texas Advanced Computing Center in Austin (USA). This study is related to the activity of the European network action COST MP1106 "Smart and green interfaces - from single bubbles and drops to industrial, environ‐

Research Laboratory of Multiphase Media Physics, Faculty of Science, Tallinn University of

, Ylo Rudi, Igor Shcheglov, Sergei Tisler and Igor Krupenski

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

**Figure 18.** Profiles of the turbulence modulation for 750 μm coal particles in the pipes *D*=30.5, 45.75 and 61 mm, *m* \* =10, Re=4.4×10<sup>4</sup> .

The turbulence modulation is shown in Figures 17 and 18 for the considered particles sizes in two marginal cases: *δ*=250 and 750 *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.
