**2. Theoretical terms of the model**

#### **2.1 Governing equations for the two dimensional RANS model**

In the area of multiphase flows there has been developed a lot of models for particulate flows in several papers e.g.(Pfeffer et al., 1966, Michaelides, 1984). A "two-fluid model" is being used in the modelling of dispersed two-phase systems where the gas and particles are considered as two coexisting phases that reach the entire flow domain. To describe the flow of the particulate phase, one of the possibilities is using the Reynolds-Averaged Navier-Stokes (RANS) method. The general equations of this method were examined by plenty of experiments, which showed that with using this method it is possible to discover, for example, the boundary conditions using the wall-functions approach and it is quite easy to implement it numerically. In this work the RANS method is used for both coexisting phases, namely the gas- and solid phases with the closure equations. Two basic predictions were used for closure of the governing equations of gaseous and dispersed phases:

i) the four-way coupling model (Crowe, 2000) by that captures one capture the particleturbulence interaction phenomena and ii) the inter-particle collision closure model (Kartushinsky & Michaelides 2004) to assess an the particles interaction. The both models are used for receiving an output of necessary data which are the axial and radial velocities, turbulent energy and the particle mass concentration. The information on these parameters will be much useful for evaluation of the relevant processes occurred in particulate flows like CFB processes.

This model is based on the complete averaged Navier-Stokes equations applied for the axisymmetrical upward gas-solid particle turbulent flow in the freeboard CFB processes. The governing equations present the carrier fluid (gas-phase) and solid (polydispersed) phase which is considered a co-existing flow and consists of a continuity equation for the gas-phase and mass conservation equation in the dispersed phase together with the momentum equations for both phases in the longitudinal and radial directions. In addition, the moment of momentum equation is included for the solid phase because of Magnus lift force and plausible particle rotation stemmed from the wall interaction. The solid phase is considered a polydispersed phase, which consists of two particle fractions – light (ash) particles and heavy (sand) particles. The present governing equations along with the corresponding boundary conditions are given by Kartushinsky and Michaelides (2004, 2006, 2009) and are the following:

1. Continuity for the gaseous phase:

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial (rv)}{r \partial r} = 0 \tag{1}$$

2. Linear momentum equation in the longitudinal direction for the gaseous phase:

$$\frac{\partial}{\partial \mathbf{x}} \left( \mathbf{u}^2 - \mathbf{v}\_t \frac{\partial \mathbf{u}}{\partial \mathbf{x}} \right) + \frac{\partial}{r \partial r} r \left( \mu \mathbf{v} - \mathbf{v}\_t \frac{\partial \mathbf{u}}{\partial r} \right) = -\frac{\partial p}{\rho \partial \mathbf{x}} + \frac{\partial}{\partial \mathbf{x}} \mathbf{v}\_t \frac{\partial \mathbf{u}}{\partial \mathbf{x}} + \frac{\partial}{r \partial r} r \mathbf{v}\_t \frac{\partial \mathbf{v}}{\partial \mathbf{x}}$$

To sum up with introduction one can underline an importance of sought problem, that is the key study of given process is taken place in freeboard of furnace of CFB steam-generator

In the area of multiphase flows there has been developed a lot of models for particulate flows in several papers e.g.(Pfeffer et al., 1966, Michaelides, 1984). A "two-fluid model" is being used in the modelling of dispersed two-phase systems where the gas and particles are considered as two coexisting phases that reach the entire flow domain. To describe the flow of the particulate phase, one of the possibilities is using the Reynolds-Averaged Navier-Stokes (RANS) method. The general equations of this method were examined by plenty of experiments, which showed that with using this method it is possible to discover, for example, the boundary conditions using the wall-functions approach and it is quite easy to implement it numerically. In this work the RANS method is used for both coexisting phases, namely the gas- and solid phases with the closure equations. Two basic predictions were

i) the four-way coupling model (Crowe, 2000) by that captures one capture the particleturbulence interaction phenomena and ii) the inter-particle collision closure model (Kartushinsky & Michaelides 2004) to assess an the particles interaction. The both models are used for receiving an output of necessary data which are the axial and radial velocities, turbulent energy and the particle mass concentration. The information on these parameters will be much useful for evaluation of the relevant processes occurred in particulate flows

This model is based on the complete averaged Navier-Stokes equations applied for the axisymmetrical upward gas-solid particle turbulent flow in the freeboard CFB processes. The governing equations present the carrier fluid (gas-phase) and solid (polydispersed) phase which is considered a co-existing flow and consists of a continuity equation for the gas-phase and mass conservation equation in the dispersed phase together with the momentum equations for both phases in the longitudinal and radial directions. In addition, the moment of momentum equation is included for the solid phase because of Magnus lift force and plausible particle rotation stemmed from the wall interaction. The solid phase is considered a polydispersed phase, which consists of two particle fractions – light (ash) particles and heavy (sand) particles. The present governing equations along with the corresponding boundary conditions are given by Kartushinsky and Michaelides (2004, 2006,

> ( ) <sup>0</sup> *u rv x rr*

*t t tt u u uv <sup>p</sup> <sup>u</sup> r uv <sup>r</sup> x x rr r x x x rr x* 

2. Linear momentum equation in the longitudinal direction for the gaseous phase:

(1)

and which is under numerical investigation.

**2.1 Governing equations for the two dimensional RANS model** 

used for closure of the governing equations of gaseous and dispersed phases:

**2. Theoretical terms of the model** 

like CFB processes.

2009) and are the following:

1. Continuity for the gaseous phase:

2

$$-\sum\_{i=1,3} \alpha\_i \left(\frac{\mu\_{ri}}{\sigma\_i'} + \mathbb{C}\_{Mi} \boldsymbol{\Omega}\_i \boldsymbol{\upsilon}\_{ri}\right) \tag{2}$$

3. Linear momentum equation in the radial direction for the gaseous phase:

$$\frac{\partial}{\partial \mathbf{x}} \left( \mu \upsilon - \mathbf{v}\_t \frac{\partial \upsilon}{\partial \mathbf{x}} \right) + \frac{\partial}{r \partial r} r \left( \upsilon^2 - \mathbf{v}\_t \frac{\partial \upsilon}{\partial r} \right) = -\frac{\partial p}{\rho \hat{c} r} + \frac{\partial}{\partial \mathbf{x}} \mathbf{v}\_t \frac{\partial \upsilon}{\partial r} + \frac{\partial}{r \partial r} r \mathbf{v}\_t \frac{\partial \upsilon}{\partial r}$$

$$-\frac{2 \mathbf{v}\_t \upsilon}{r^2} - \sum\_{i=1,3} \alpha\_i \left( \frac{\upsilon\_{ri}}{\mathbf{r}\_i'} - (\mathbf{C}\_{Mi} \mathbf{Q}\_i + \mathbf{F}\_{si}) \mu\_{ri} \right) \tag{3}$$

4. Turbulence kinetic energy equation for the gaseous phase:

$$
\frac{\partial}{\partial \mathbf{x}} \left( \nu \mathbf{k} - \mathbf{v}\_t \frac{\partial \mathbf{k}}{\partial \mathbf{x}} \right) + \frac{\partial}{r \partial r} \left( \nu \mathbf{k} - \mathbf{v}\_t \frac{\partial \mathbf{k}}{\partial r} \right) =$$

$$
\nabla\_t \left\{ 2 \left[ \left( \frac{\partial u}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial v}{\partial r} \right)^2 + \left( \frac{v}{r} \right)^2 \right] + \left( \frac{\partial u}{\partial r} + \frac{\partial v}{\partial \mathbf{x}} \right)^2 \right\}$$

$$

5. Mass conservation equation for the solid phase:

$$\frac{\partial}{\partial \mathbf{x}} (\alpha\_i \mu\_{si}) + \frac{\partial}{r \partial r} (r \alpha\_i v\_{si}) = -\frac{\partial}{\partial \mathbf{x}} \left( D\_{si} \frac{\partial \alpha\_i}{\partial \mathbf{x}} \right) - \frac{\partial}{r \partial r} \left( r D\_{si} \frac{\partial \alpha\_i}{\partial r} \right) \tag{5}$$

6. Momentum equation in the longitudinal direction for the solid phase:

$$
\frac{\partial}{\partial \mathbf{x}} \left( \alpha\_i \mu\_{si} \mu\_{si} \right) + \frac{\partial}{r \partial r} \left( r \alpha\_i \mu\_{si} \upsilon\_{si} \right) = 
$$

$$
$$

7. Momentum equation in the radial/transverse direction for the solid phases:

$$
\frac{\partial}{\partial \mathbf{x}} \left( \alpha\_i \boldsymbol{\upsilon}\_{si} \boldsymbol{\mu}\_{si} \right) + \frac{\partial}{r \partial r} \left( r \boldsymbol{\alpha}\_i \boldsymbol{\upsilon}^2\_{si} \right) = 
$$

$$
$$

Mathematical Modelling of the Motion of

2 *v u rotV*

model of Kartushinsky & Michaelides (2004).

**3. Results and discussions** 

y+= *y v yc k*

value at (r=0).

2 *us* , <sup>2</sup>

*x r* 

coefficient is calculated as <sup>0</sup>

2D motion, as <sup>1</sup>

Dust-Laden Gases in the Freeboard of CFB Using the Two-Fluid Approach 149

. The average values, *us* , *<sup>s</sup> v* , <sup>2</sup> *us*

and *s s v* are the particles stress tensors originated from their turbulent fluctuation along with their inter-collisions calculated from (Kartushinsky & Michaelides, 2004). Due to the particle turbulent diffusion and particle collision the diffusion coefficient has two components: *DD D s sturb col* . The first term in expression for the particle diffusion

<sup>2</sup> <sup>T</sup> D k <sup>τ</sup> T 1 exp <sup>3</sup> <sup>τ</sup>

Zaichik & Alipchenkov (2005) and the second term is taken from the particle's collision

*The numerical method:* In the given RANS computations the *control volume* (cv) method was used. The governing equations (1-9) were solved using a strong implicit procedure with the lower and upper matrix decomposition and up-wind scheme for convective fluxes (Perić & Scheuerer, 1989 and Fertziger & Perić, 1996). For the considered computations, 145,000 uniformly distributed control volumes were utilized for running the numerical codes. The wall functions were incorporated at a dimensionless distance from the wall as follows,

All computations were extended from the pipe entrance to a short distance up to *x/D=50* (D is the pipe diameter) similar to the height of the freeboard of CFB. For the particulate phase, when the size of particles is often larger than the size of the viscous boundary sub-layer, the volume domain occupied by the dispersed phase has slightly shrunk, which gives always positive values for the solids' velocities in the wall vicinity. This method follows the

All results are presented in the dimensionless way: the velocities of both phases are related to the gas-phase velocity at the centre of the flow (r=0), the turbulent energy is normalized to a square of the gas-phase velocity, and particle mass concentration is normalized to its

*The numerical results.* The effect of inter-particle collisions is very important for the particulate flows when the ratio of / 1 *<sup>c</sup>* (where *<sup>c</sup>* is the time of inter-particle collision and is the particle response time). In the considered freeboard CFB, for the particulate flows with a high mass flow ratio about or above 10kg dust/kg air the given ratio of *<sup>c</sup>* / is less unit resulted in accounting of the collision process in CFB by utilizing "collision terms" in equations (5-8). These terms are responsible for inter-particle collisions. These are terms for the production of longitudinal and radial components of linear velocity correlations and deriving linear and angular velocity correlations of the solid phase, such as

*si v* , *u vs s* , *us s* , *s s v* . These velocity correlations are due to the particle collision between various fractions and they are computed from the difference in average velocities

=10, where *y* and *c* are the control volume size and the empirical

, <sup>2</sup>

from the PDF model of

is defined for the

*si v* , *u vs s* , *us s* ,

the differential mathematical operator "rot" over the gas velocity vector *V*

sturb 0

constant that equals to 0.09 *c* and *k* is the turbulent energy, respectively.

numerical approach by Hussainov et al., (1996) has been employed here.

8. Angular momentum equation for the solid phases:

$$\frac{\partial}{\partial \mathbf{x}} \left( \alpha\_i \mathbf{o}\_{si} \boldsymbol{\mu}\_{si} \right) + \frac{\partial}{r \partial r} \left( r \mathbf{a}\_i \mathbf{o}\_{si} \boldsymbol{v}\_{si} \right) = $$
 
$$ -\frac{\partial}{\partial \mathbf{x}} \left( \alpha\_i \overline{\boldsymbol{u}\_{si}^{\prime} \boldsymbol{\alpha}\_{si}^{\prime}} \right) - \frac{\partial}{r \partial r} \left( r \mathbf{a}\_i \overline{\boldsymbol{v}\_{si}^{\prime} \boldsymbol{\alpha}\_{si}^{\prime}} \right) - \alpha\_i \mathbf{C}\_{oi} \frac{\Omega\_i}{\mathsf{r}\_i} \tag{8} $$

Here *p* is the pressure, *u*, *us* , *<sup>s</sup> v* , *s* are the longitudinal, radial, angular velocity components of gas- and solid phases (subscript s), respectively, and the particle mass concentration. The subscript "i" corresponds to the number of particle fraction and varies in the range ( 1 3 *i* ), which composes the polydispersed phase. The particle void fraction is linked with the particle mass concentration as / *<sup>p</sup>* ( is solids void fraction). The closure equations of gas-phase are performed by using *<sup>h</sup> k L* four-way coupling model of Crowe (2000) where *k* is the turbulent energy of carrier fluid and *Lh* is the hybrid length scale. This parameter is computed as a harmonic average of the integral turbulence length scale of single phase pipe flow, *L*0 ( 3/2 *L k* 00 0 / ) and inter-particle spacing, , defined as <sup>3</sup> / 1 *<sup>p</sup>* . Thus, the hybrid length scale or scale of dissipation rate of turbulent energy in particulate flows is determined as *LL L <sup>h</sup>* 2 / 0 0 [Crowe 2000]. The values, *p* and are the densities of the particle materials and gas-phase, is the particle size. The coefficient of turbulent viscosity is calculated as 0 *kL* , by the turbulent energy of particulate flow and turbulence length scale related to the single phase flow. Thus, the parameters of the single phase flow (subscript 0), the average velocity components, turbulent energy, 0 *k* , its dissipation rate, 0 , together with L0 (while *T k* 0 00 / is integral turbulence time scale) have to be calculated in advance (in preliminary calculations) for completion modelling in the pipe gas-solid turbulent flow system. An advantage of the fourway coupling model of (Crowe, 2000) (with the inclusion of particle collision) is that it includes the turbulence enhancement by the presence of particles, expressed via the term 2 2 *u v r r* ( *u uu r s* and *r s v vv* is the slip velocity between the gas- and solid phases along the streamwise and radial directions) and turbulence attenuation via the increase of its dissipation rate by particles, 3/2 *h h k L* / (in the right-hand side terms of Eq. 4). 2 18 *<sup>p</sup>* is the particle response time for the Stokes regime ( is the kinematic viscosity coefficient) and /*CD* for the non-Stokes regime expressed via the particle Reynolds number, 2 2 Re*s rr u v* / and 0.687 *CD s* 1 0.15Re . By determining of the coefficients of *CMi* and *Fsi* one can correct the values of the lift Magnus and Saffman forces and *Ci* for the particles rotation are taken from (Crowe et al., 1998) for relevant range of change of the particle Reynolds number, Res. *<sup>s</sup> rotV* is the angular velocity slip of particles while the differential mathematical operator "rot" over the gas velocity vector *V* is defined for the 2D motion, as <sup>1</sup> 2 *v u rotV x r* . The average values, *us* , *<sup>s</sup> v* , <sup>2</sup> *us* , <sup>2</sup> *si v* , *u vs s* , *us s* , and *s s v* are the particles stress tensors originated from their turbulent fluctuation along with their inter-collisions calculated from (Kartushinsky & Michaelides, 2004). Due to the particle turbulent diffusion and particle collision the diffusion coefficient has two components: *DD D s sturb col* . The first term in expression for the particle diffusion coefficient is calculated as <sup>0</sup> sturb 0 <sup>2</sup> <sup>T</sup> D k <sup>τ</sup> T 1 exp <sup>3</sup> <sup>τ</sup> from the PDF model of

Zaichik & Alipchenkov (2005) and the second term is taken from the particle's collision model of Kartushinsky & Michaelides (2004).
