**2. Governing equations and numerical method**

The sketch of the computational flow domain is shown in Fig. 1, where *u*¯ is the gas average velocity, *FG* is gravity, *FD* is the aerodynamic drag force, *FLR* is the lift force that arises from particle rotation (the Magnus lift force), *ωs* is the angular velocity of a particle.

continuity equation for the gas phase:

**Figure 1.** Upward turbulent particulate flow in a pipe.

2 n

while *ν<sup>t</sup>*

*u* (*rv*) *x rr* 0, ¶ ¶ + =

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

r

¶ ¶¶ ¶ æ öæ ö ¶ ¶¶ ¶ ¶ <sup>æ</sup> <sup>ö</sup> ç ÷ç ÷ - + - =- + + - + W <sup>ç</sup> <sup>÷</sup> ¶ ¶ ¶ ¶ ¶¶¶ ¶ ¶ ¢ è øè ø <sup>è</sup> <sup>ø</sup>

*tt t t M r u u p u v u u r uv r C v*

where *ν*˜*<sup>t</sup>* =*ν<sup>t</sup>* + *ν* is effective viscosity, which is the sum of turbulent and laminar viscosities,

mass concentration of particles; *ur* =*u* −*us* and *vr* =*v* −*vs* are the relative velocities of particles

n

is calculated following the Boussinesq eddy-viscosity concept; *p* is pressure; *α* is the

% % % % (2)

longitudinal linear momentum equation for the gas phase:

n

*x x rr r x x x rr x*

¶ ¶ (1)

 na

RANS Numerical Simulation of Turbulent Particulate Pipe Flow for Fixed Reynolds Number

http://dx.doi.org/10.5772/57216

23

*r*

t

,

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 *δ*. According to [16], in the given formulation of the governing equations that follows, three solid fractions are assumed to be present. It is assumed that the aerodynamic forces, such as the drag, lift forces and gravity, act on all the particulate fractions.

### **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:

RANS Numerical Simulation of Turbulent Particulate Pipe Flow for Fixed Reynolds Number http://dx.doi.org/10.5772/57216 23

**Figure 1.** Upward turbulent particulate flow in a pipe.

continuity equation for the gas phase:

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

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

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

The sketch of the computational flow domain is shown in Fig. 1, where *u*¯ is the gas average velocity, *FG* is gravity, *FD* is the aerodynamic drag force, *FLR* is the lift force that arises from

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 *δ*. According to [16], in the given formulation of the governing equations that follows, three solid fractions are assumed to be present. It is assumed that the aerodynamic forces, such as the drag, lift forces and gravity, act on all

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

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

particle rotation (the Magnus lift force), *ωs* is the angular velocity of a particle.

simulation of the two-phase upward pipe flow.

22 Computational and Numerical Simulations

variation of the parameters of the flow.

the particulate fractions.

as follows:

unless the study of rotating flows.

imposing the upper limit for the flow Reynolds number.

**2. Governing equations and numerical method**

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

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial \left(r v\right)}{r \partial r} = 0,\tag{1}$$

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

longitudinal linear momentum equation for the gas phase:

$$\frac{\partial}{\partial \mathbf{x}} \left( u^2 - \tilde{\nu}\_t \frac{\partial u}{\partial \mathbf{x}} \right) + \frac{\partial}{r \partial r} r \left( u \mathbf{v} - \tilde{\nu}\_t \frac{\partial u}{\partial r} \right) = -\frac{\partial p}{\rho \mathbf{\hat{x}}} + \frac{\partial}{\partial \mathbf{x}} \tilde{\nu}\_t \frac{\partial u}{\partial \mathbf{x}} + \frac{\partial}{r \partial r} r \tilde{\nu}\_t \frac{\partial v}{\partial \mathbf{x}} - a \left( \frac{u}{\mathbf{r}'} + \mathbb{C}\_M \Omega v\_r \right) \tag{2}$$

where *ν*˜*<sup>t</sup>* =*ν<sup>t</sup>* + *ν* is effective viscosity, which is the sum of turbulent and laminar viscosities, while *ν<sup>t</sup>* is calculated following the Boussinesq eddy-viscosity concept; *p* is pressure; *α* is the mass concentration of particles; *ur* =*u* −*us* and *vr* =*v* −*vs* are the relative velocities of particles along the longitudinal and radial directions, respectively. Here *τ* ′ =*τ* / *C* ′ *<sup>D</sup>* is the particle response time that specifies the drag, defined by the expression *C* ′ *<sup>D</sup>* =1 + 0.15Re*<sup>s</sup>* 0.687 for the non-Stokesian regime [27]. The particle Reynolds number and Stokesian particle response time are defined as Re*<sup>s</sup>* =*δ* |*V* → *<sup>r</sup>* | / *ν* =*δ ur* <sup>2</sup> <sup>+</sup> *vr* <sup>2</sup> / *<sup>ν</sup>* and *<sup>τ</sup>* <sup>=</sup>*ρp<sup>δ</sup>* <sup>2</sup> / (18*ρν*), respectively. *Ω* =*ω<sup>s</sup>* −0.5(∂*v* / ∂ *x* −∂*u* / ∂*r*) is the angular velocity slip, with *ωs* being the angular velocity of the given particle fraction. The coefficient of the Magnus lift force *CM* is calculated according to Crowe et al. (1998); *ρ* and *ρp* are the physical densities of air and the particle material, respectively.

radial linear momentum equation for the gas phase:

$$\begin{split} & \frac{\partial}{\partial \mathbf{x}} \Big( \mu \upsilon - \tilde{\nu}\_t \frac{\partial \upsilon}{\partial \mathbf{x}} \Big) + \frac{\partial}{r \partial r} r \Big( \upsilon^2 - \tilde{\nu}\_t \frac{\partial \upsilon}{\partial r} \Big) = -\frac{\partial p}{\rho \hat{\upsilon} r} + \frac{\partial}{\partial \mathbf{x}} \tilde{\nu}\_t \frac{\partial \upsilon}{\partial r} + \\ & + \frac{\partial}{r \partial r} r \tilde{\nu}\_t \frac{\partial \upsilon}{\partial r} - \frac{2 \tilde{\nu}\_t \upsilon}{r^2} - \alpha \left( \frac{\upsilon\_r}{\tau'} - \left( \mathbb{C}\_M \Omega + F\_s \right) \mu\_r \right). \end{split} \tag{3}$$

[29]. The pseudoviscosity diffusion coefficients along *x* and *r* directions *Dc*

momentum equation in the longitudinal direction for the particulate phase:

*s s s s s s s M r*

*<sup>u</sup> uu r uv u r uv C v g*

é æ öù ¶ ¶ ¶ ¶

¶ ¶ ¶ ¶ <sup>ê</sup> ¢ ç ÷ú <sup>ë</sup> è øû

( ) ( ) ( ) ( ) ( ) *<sup>r</sup> s s s s s s s M sr*

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

( *s s u r v u rv C* ) ( *s s* ) ( *s s* ) ( *s s* ) *x rr x rr* ,

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 *x* =100*D*, i.e. the non-gradient derivatives from all velocities of all phases, turbulence kinetic energy and mass concentration over longitudinal coordinate were written according to [19]. Since the particulate flow in the vertical pipe is considered as axisymmetrical, the non-gradient boundary conditions were set at the pipe axis for the longitudinal velocity components of gas and particles, the turbulent energy and particle mass concentration. The boundary conditions were set zero at the pipe axis for the radial velocities of both phases and the particle angular velocity. The concept of "wall functions" [30] has been applied to set the boundary conditions at the wall. While applying the balance of the production and dissipation rate of kinetic energy "near the wall" with using the eddy-viscosity concept [31], it can link the friction velocity *v*∗ and shear stress *τw* through the turbulence kinetic energy as

 a w

¶ ¶ ¶ ¶ W

¶ ¶ ¶ ¶ é ù <sup>+</sup> =- - ¢¢ ¢ + - W+ <sup>ê</sup> <sup>ú</sup> ¶ ¶ ¶ ¶ ¢ <sup>ë</sup> <sup>û</sup>

a

tum swap in the longitudinal and radial motions of the given fraction [16].

*vu r vv uv r v C Fu*

+ =- - ¢ ¢ ¢ + + W- - ê ç ÷ú

aa

% % (6)

( ) ( ) ( ) ( ) *<sup>r</sup>*

 a

momentum equation in the radial direction for the particulate phase:

*x rr x rr*

*x rr x rr*

 aw

a

 a

where *g* is the gravitational acceleration.

particle collisions [16].

a

a

¯

, *v* ′ *s* ¯2

aw

and *v* ′

*<sup>s</sup>ω* ′ *s*

particle collisions calculated according to [16].

**2.2. Boundary conditions for the RANS model**

¯

*s* ¯2 , *u* ′ *sv* ′ *s*

where *u* ′

where *u* ′

*<sup>s</sup>ω* ′ *s*

¯

*x*,*r*

http://dx.doi.org/10.5772/57216

*p*

 r

r

<sup>2</sup> 1 ,

*v*

t

,

2

% % (7)

 a

are the velocity correlations due to particle collisions and induce momen‐

 a w a w t

are the linear-angular velocity correlations of particles due to inter-

<sup>+</sup> =- - ¢¢ ¢¢ - ¶ ¶ ¶ ¶ % % (8)

a

t

RANS Numerical Simulation of Turbulent Particulate Pipe Flow for Fixed Reynolds Number

stem from the

25

*Fs* is the coefficient for the Saffman lift force, which is due to the local shear of the flow; it is given for finite values of the particle Reynolds numbers by the correction [28].

turbulence kinetic energy equation for the gas phase:

$$\begin{split} &\frac{\partial}{\partial \mathbf{x}} \left( \imath \boldsymbol{k} - \tilde{\nu}\_t \frac{\partial \mathbf{k}}{\partial \mathbf{x}} \right) + \frac{\partial}{r \partial r} r \bigg( \imath \boldsymbol{v} - \tilde{\nu}\_t \frac{\partial \mathbf{k}}{\partial r} \bigg) = \\ &= 2\nu\_t \left\{ \left( \frac{\partial \mathbf{u}}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial r \mathbf{v}}{r \partial r} \right)^2 + \frac{1}{2} \left( \frac{\partial \mathbf{u}}{\partial r} + \frac{\partial \mathbf{v}}{\partial \mathbf{x}} \right)^2 \right\} + \frac{\alpha}{\tau} \left( \boldsymbol{u}\_r^2 + \boldsymbol{v}\_r^2 + \boldsymbol{k}\_s \right) - \boldsymbol{\varepsilon}\_{\mathbf{h},r} \end{split} \tag{4}$$

where *k* and *ks* are the turbulence kinetic energy of the gas- and particulate phases, respectively. The hybrid dissipation rate *εh* is calculated for the two-phase flow via hybrid turbulence length scale defined as harmonic average of the integral length scale of single-phase flow and interparticle spacing [14].

continuity equation for the particulate phase:

$$\frac{\partial}{\partial \mathbf{x}} \left( a \tilde{u}\_s \right) + \frac{\partial}{r \partial r} r \left( a \tilde{v}\_s \right) = 0,\tag{5}$$

where *u*˜*s* and *v*˜*s* are the longitudinal and radial components of the drift particle velocity of the given fraction, given by expressions *u*˜*<sup>s</sup>* <sup>=</sup>*us* <sup>−</sup>(*Dt* <sup>+</sup> *Dc <sup>x</sup>*)∂ln*<sup>α</sup>* / <sup>∂</sup> *<sup>x</sup>*, *v*˜*<sup>s</sup>* <sup>=</sup>*vs* <sup>−</sup>(*Dt* <sup>+</sup> *Dc r* )∂ln*α* / ∂*r*. Here *Dt* is the coefficient of turbulent diffusion of particles, which is calculated by the model [29]. The pseudoviscosity diffusion coefficients along *x* and *r* directions *Dc x*,*r* stem from the particle collisions [16].

momentum equation in the longitudinal direction for the particulate phase:

$$\frac{\partial}{\partial \mathbf{x}} \Big( a u\_s \tilde{\boldsymbol{u}}\_s \Big) + \frac{\partial}{r \partial r} \Big( r a u\_s \tilde{\boldsymbol{v}}\_s \Big) = -\frac{\partial}{\partial \mathbf{x}} \Big( a u \overline{u\_s'^2} \Big) - \frac{\partial}{r \partial r} \Big( r a u \overline{u\_s' v\_s'} \Big) + a \left| \frac{u\_r}{\tau'} + \mathbb{C}\_M \Omega \boldsymbol{v}\_r - g \left( 1 - \frac{\rho}{\rho\_p} \right) \Big|\_{\mathbf{y}} \Big| \Big) \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big| \Big|$$

where *g* is the gravitational acceleration.

along the longitudinal and radial directions, respectively. Here *τ* ′

*<sup>r</sup>* | / *ν* =*δ ur*

Stokesian regime [27]. The particle Reynolds number and Stokesian particle response time are

*Ω* =*ω<sup>s</sup>* −0.5(∂*v* / ∂ *x* −∂*u* / ∂*r*) is the angular velocity slip, with *ωs* being the angular velocity of the given particle fraction. The coefficient of the Magnus lift force *CM* is calculated according to Crowe et al. (1998); *ρ* and *ρp* are the physical densities of air and the particle material,

( )

r

*t t t*

n

%%%

¶ ¶ ¶ ¶ ¶¶ æ öæ ö ¶ ç ÷ç ÷ - + - =- + + ¶ ¶ ¶ ¶ ¶¶ ¶ è øè ø

<sup>2</sup> .

*Fs* is the coefficient for the Saffman lift force, which is due to the local shear of the flow; it is

*t rrs h*

where *k* and *ks* are the turbulence kinetic energy of the gas- and particulate phases, respectively. The hybrid dissipation rate *εh* is calculated for the two-phase flow via hybrid turbulence length scale defined as harmonic average of the integral length scale of single-phase flow and inter-

*u rv u v uvk*

2 2 <sup>2</sup> <sup>1</sup> <sup>2</sup> , <sup>2</sup>

*x x rr r r x r*

2

*v v p u*

*t r t M sr*

a

¶ ¶ æ ö + - - - W+ ç ÷ ¶ ¶ ¢ è ø

*r C Fu*

given for finite values of the particle Reynolds numbers by the correction [28].

n

= + + + + ++ - í ý ç÷ç ÷ ç ÷

( ) ( ) *<sup>s</sup> <sup>s</sup> u rv x rr* a

 a 0, ¶ ¶ + =

where *u*˜*s* and *v*˜*s* are the longitudinal and radial components of the drift particle velocity of the

Here *Dt* is the coefficient of turbulent diffusion of particles, which is calculated by the model

t

*v v v*

2

*t t*

% %

*x rr r x*

èøè ø è ø ¶ ¶ ¶¶ ï ï î þ

ì ü ï ï æöæ ö æ ö ¶ ¶ ¶¶

*<sup>k</sup> <sup>k</sup> uk r vk x x rr r*

¶ ¶¶ ¶ æ öæ ö ç ÷ç ÷ -+ -= ¶ ¶¶ ¶ è øè ø

n

<sup>2</sup> <sup>+</sup> *vr*

response time that specifies the drag, defined by the expression *C* ′

→

radial linear momentum equation for the gas phase:

n

*rr r r*

turbulence kinetic energy equation for the gas phase:

n

continuity equation for the particulate phase:

given fraction, given by expressions *u*˜*<sup>s</sup>* <sup>=</sup>*us* <sup>−</sup>(*Dt* <sup>+</sup> *Dc*

n

particle spacing [14].

% %

n

*uv r v*

defined as Re*<sup>s</sup>* =*δ* |*V*

24 Computational and Numerical Simulations

respectively.

=*τ* / *C* ′

*<sup>D</sup>* =1 + 0.15Re*<sup>s</sup>*

<sup>2</sup> / *<sup>ν</sup>* and *<sup>τ</sup>* <sup>=</sup>*ρp<sup>δ</sup>* <sup>2</sup> / (18*ρν*), respectively.

n

( )

¶ ¶ % % (5)

*<sup>x</sup>*)∂ln*<sup>α</sup>* / <sup>∂</sup> *<sup>x</sup>*, *v*˜*<sup>s</sup>* <sup>=</sup>*vs* <sup>−</sup>(*Dt* <sup>+</sup> *Dc*

*r*

)∂ln*α* / ∂*r*.

e

2 2

a

t

*<sup>D</sup>* is the particle

0.687 for the non-

(3)

(4)

momentum equation in the radial direction for the particulate phase:

$$\frac{\partial}{\partial \mathbf{x}} \left( a v\_s \tilde{u}\_s \right) + \frac{\partial}{r \partial r} \left( r a v\_s \tilde{v}\_s \right) = -\frac{\partial}{\partial \mathbf{x}} \left( a a \overline{u\_s' v\_s'} \right) - \frac{\partial}{r \partial r} \left( r a \overline{v\_s'^2} \right) + a \left[ \frac{v\_r}{\tau'} - \left( \mathcal{C}\_M \Omega + \mathcal{F}\_s \right) u\_r \right], \tag{7}$$
 e  $\overline{u\_s'}$   $\overline{u\_s' v\_s'}$   $\overline{v\_s'}$  are the velocity correlations due to particle collisions and induce momentum in the longitudinal and radial motions of the given fraction [11].

where *u* ′ *s* ¯2 , *u* ′ *sv* ′ *s* , *v* ′ *s* ¯2 are the velocity correlations due to particle collisions and induce momen‐ tum swap in the longitudinal and radial motions of the given fraction [16].

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

$$\frac{\partial}{\partial \mathbf{x}} \left( a a \boldsymbol{o}\_s \boldsymbol{\tilde{u}}\_s \right) + \frac{\partial}{r \partial r} \left( r a \boldsymbol{o}\_s \boldsymbol{\tilde{v}}\_s \right) = -\frac{\partial}{\partial \mathbf{x}} \left( a \boldsymbol{u}\_s' \boldsymbol{o}\_s' \right) - \frac{\partial}{r \partial r} \left( r a \boldsymbol{u} \overline{\boldsymbol{v}\_s' \boldsymbol{o}\_s'} \right) - a \boldsymbol{\mathcal{C}}\_{\boldsymbol{o}} \frac{\boldsymbol{\Omega}}{r}, \tag{8}$$
 for  $\overline{\boldsymbol{v}\_s' \boldsymbol{u}'}$ , are the linear-angular velocity correlations of particles due to interactions calculated according to I.

where *u* ′ *<sup>s</sup>ω* ′ *s* ¯ and *v* ′ *<sup>s</sup>ω* ′ *s* are the linear-angular velocity correlations of particles due to interparticle collisions calculated according to [16].

#### **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 *x* =100*D*, i.e. the non-gradient derivatives from all velocities of all phases, turbulence kinetic energy and mass concentration over longitudinal coordinate were written according to [19]. Since the particulate flow in the vertical pipe is considered as axisymmetrical, the non-gradient boundary conditions were set at the pipe axis for the longitudinal velocity components of gas and particles, the turbulent energy and particle mass concentration. The boundary conditions were set zero at the pipe axis for the radial velocities of both phases and the particle angular velocity. The concept of "wall functions" [30] has been applied to set the boundary conditions at the wall. While applying the balance of the production and dissipation rate of kinetic energy "near the wall" with using the eddy-viscosity concept [31], it can link the friction velocity *v*∗ and shear stress *τw* through the turbulence kinetic energy as *v*\* <sup>2</sup> <sup>=</sup>*τ<sup>w</sup>* / *<sup>ρ</sup>* <sup>=</sup>*cμ* 0.5*k*. The computations near the wall were carried out at the half-width of the 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 are as follows:

$$\begin{cases} \mu = \sqrt{\frac{\tau\_w}{\rho}} \frac{1}{\overline{x}} \ln(y^+) + C = \upsilon\_\* \frac{1}{\overline{x}} \ln\left(E \frac{y}{\nu} \upsilon\_\*\right) & \text{11.6} \le y^+ \le 500\\ \mu = \frac{\upsilon\_\*^2 y}{\nu} & y^+ \le 11.6 \end{cases} \tag{9}$$

$$P\_k = \frac{2\pi\_w^2}{\text{ar}\rho c\_\mu^{0.25} k^{0.5} y} \tag{10}$$

**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>

RANS Numerical Simulation of Turbulent Particulate Pipe Flow for Fixed Reynolds Number

**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

.

http://dx.doi.org/10.5772/57216

27

.

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

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