**3. Results and discussions**

148 Computational Simulations and Applications

*i si si u rv i si si*

 

 *<sup>i</sup> i si si i si si i i*

*u rv C x rr* 

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

( *u uu r s* and *r s v vv* is the slip velocity between the gas- and solid phases

1 0.15Re . By determining of the coefficients of *CMi* and

2

18 *<sup>p</sup>* 

along the streamwise and radial directions) and turbulence attenuation via the increase of its

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,

*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

dissipation rate by particles, 3/2 *h h k L* / (in the right-hand side terms of Eq. 4).

2 2 Re*s rr u v* / and 0.687 *CD s*

*i*

(8)

*x rr* 

8. Angular momentum equation for the solid phases:

 2 2 *u v r r* 

*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,

y+= *y v yc k* =10, where *y* and *c* are the control volume size and the empirical

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

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

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 value at (r=0).

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

Mathematical Modelling of the Motion of

**0.5**

20kg/kg.

**0.55**

**mixture1:ash mixture1:sand mixture3:ash mixture3:sand**

**Axial velocity of solids components**

**0.6**

**0.65**

Dust-Laden Gases in the Freeboard of CFB Using the Two-Fluid Approach 151

the velocity profile to its shape of "fully" turbulent regime. At the same time the average magnitude of the longitudinal velocity component of solids slightly increases with the growth of the mass flow ratio (cf. straight dashed and bold solid lines in Fig. 1). Such tendency in the two-phase turbulent jet has been experimentally observed by Laats & Mulgi (1979). Fig. 1 gives also the distribution of longitudinal velocity components of different solid particles. As the modelling shows, the velocity distribution of solid phase is less sensible to the variation of particle sizes than to the change of mass flow loading (cf.

The following Figs. 2 and 3 show the detailed distribution of longitudinal velocity components for each particle fraction of solids presented separately. The cases of mixture1 and mixture3 show similar particle sizes of 500µm, but different material densities (light and heavy particles) and also different mass flow ratios: 10 and 20kg/kg (Fig. 2). The other cases are the mixture1 and the mixture2 with the particle sizes and material densities distributions obtaining for the same total mass flow loading 10kg/kg (Fig. 3). As one can notice, the ash particles have higher velocity than heavy sand particles (cf. dashed and solid dashed lines, Fig.2) that could be observed for both mass flow loadings: 10kg/kg (mixture1) and 20kg/kg (mixture3). Mixture2 is a more complicated case of particle composition. Considering the above, we can see that the larger ash particles have a lower velocity value than the smaller ones (cf. light and dark diffused lines for 500 and 1000µm particles, Fig.3). However, at the same time the heavier sand particles of 500µm have larger velocity magnitude than the lighter ash particles of 500 and 1000µm, which show smaller velocity magnitude (Fig.3). This trend is probably caused by the higher rate of particle collision between the light ash particles of different sizes than that between the light and heavy particles of the same size. It is difficult to predict such a tendency, but, it can be observed in numerical simulations.

> **0 0.25 0.5 0.75 1 Radial distance, r/R**

Fig. 2. Axial velocity profiles of ash and sand particles for various flow conditions: mixture1:

ash and sand particles of 0.5mm for 10kg/kg and mixture3: ash and sand 0.5mm for

longitudinal velocity profiles for mixture1 and mixture3, straight lines in Fig. 1).

of various sizes of particles and from that in particle material densities (light and heavy particles). An analytical expression for given velocity correlations of the dispersed phase along with the closure approach of the governing equations of polydispersed phase, Eqs. (5-8), are given in model of (Kartushinsky & Michaelides, 2004).

The results of numerical simulation are shown in the following Figures 1-7. The axial velocity distribution of dispersed phase is calculated as an average velocity of the mixture of ash particles of different sizes or as a mixture of ash and sand particles with applying the formulae, *s i si i* / *i i u u* where *usi* , and *i* are the axial linear velocity and particle mass concentration of ash or ash and sand particles, respectively. The Fig.1 shows longitudinal distribution of the gas- and solid phases for three examined cases: mixture1 is the ash ( *<sup>p</sup>* 2000kg/m3) and sand particles ( *<sup>p</sup>* 2600kg/m3) of the same size, 500µm with the total mass flow loading of 10kg/kg equally distributed between the ash and sand particle fractions; mixture2 is the composition of ash particles of two sizes, 500 and 1000µm and sand particles of 500µm with the total mass flow loading of 10kg/kg, which are equally distributed between these three particle fractions, and finally, mixture3 consists of ash and sand particles of the same size 500µm with the higher total mass flow ratio of 20kg/kg where the mass fractions are equally distributed between the ash and sand particles. The calculations were performed for the conditions of CFB, namely, when the density of the gaseous carrier fluid was ρ=0.3kg/m3 and kinematic viscosity of the carrier fluid <sup>4</sup> 1.5 10 m2/s. This corresponds to the flow parameters of hot gases at the temperature of T=1123K.

Fig. 1. Axial velocity distribution of gas- and dispersed phases for their average axial velocity by different flow conditions for mixtures 1, 2 and 3.

As one can notice, the gas velocity profile is similar to the typical turbulent velocity profile, however in a dense flow with high mass loading, e.g., 20kg/kg loaded by coarse particles, the velocity profile of the carrier gas-phase becomes flatter (diffusive line in Fig.1). It comes from the effect of turbulence enhancement by the motions of coarse particles, which modify

of various sizes of particles and from that in particle material densities (light and heavy particles). An analytical expression for given velocity correlations of the dispersed phase along with the closure approach of the governing equations of polydispersed phase, Eqs.

The results of numerical simulation are shown in the following Figures 1-7. The axial velocity distribution of dispersed phase is calculated as an average velocity of the mixture of ash particles of different sizes or as a mixture of ash and sand particles with applying the

mass concentration of ash or ash and sand particles, respectively. The Fig.1 shows longitudinal distribution of the gas- and solid phases for three examined cases: mixture1 is the ash ( *<sup>p</sup>* 2000kg/m3) and sand particles ( *<sup>p</sup>* 2600kg/m3) of the same size, 500µm with the total mass flow loading of 10kg/kg equally distributed between the ash and sand particle fractions; mixture2 is the composition of ash particles of two sizes, 500 and 1000µm and sand particles of 500µm with the total mass flow loading of 10kg/kg, which are equally distributed between these three particle fractions, and finally, mixture3 consists of ash and sand particles of the same size 500µm with the higher total mass flow ratio of 20kg/kg where the mass fractions are equally distributed between the ash and sand particles. The calculations were performed for the conditions of CFB, namely, when the density of the gaseous carrier fluid was ρ=0.3kg/m3 and kinematic viscosity of the carrier fluid <sup>4</sup> 1.5 10 m2/s. This corresponds to the flow parameters of hot gases at the temperature

> **0 0.25 0.5 0.75 1 Radial distance, r/R**

Fig. 1. Axial velocity distribution of gas- and dispersed phases for their average axial

As one can notice, the gas velocity profile is similar to the typical turbulent velocity profile, however in a dense flow with high mass loading, e.g., 20kg/kg loaded by coarse particles, the velocity profile of the carrier gas-phase becomes flatter (diffusive line in Fig.1). It comes from the effect of turbulence enhancement by the motions of coarse particles, which modify

*u u* where *usi* , and *i* are the axial linear velocity and particle

(5-8), are given in model of (Kartushinsky & Michaelides, 2004).

formulae, *s i si i* /

of T=1123K.

*i i*

**0**

velocity by different flow conditions for mixtures 1, 2 and 3.

**0.25**

**0.5**

**gas:mixture1 gas:mixture3 solids:mixture1 solids:mixture2 solids:mixture3**

**Axial velocity of gas and solids**

**0.75**

**1**

the velocity profile to its shape of "fully" turbulent regime. At the same time the average magnitude of the longitudinal velocity component of solids slightly increases with the growth of the mass flow ratio (cf. straight dashed and bold solid lines in Fig. 1). Such tendency in the two-phase turbulent jet has been experimentally observed by Laats & Mulgi (1979). Fig. 1 gives also the distribution of longitudinal velocity components of different solid particles. As the modelling shows, the velocity distribution of solid phase is less sensible to the variation of particle sizes than to the change of mass flow loading (cf. longitudinal velocity profiles for mixture1 and mixture3, straight lines in Fig. 1).

The following Figs. 2 and 3 show the detailed distribution of longitudinal velocity components for each particle fraction of solids presented separately. The cases of mixture1 and mixture3 show similar particle sizes of 500µm, but different material densities (light and heavy particles) and also different mass flow ratios: 10 and 20kg/kg (Fig. 2). The other cases are the mixture1 and the mixture2 with the particle sizes and material densities distributions obtaining for the same total mass flow loading 10kg/kg (Fig. 3). As one can notice, the ash particles have higher velocity than heavy sand particles (cf. dashed and solid dashed lines, Fig.2) that could be observed for both mass flow loadings: 10kg/kg (mixture1) and 20kg/kg (mixture3). Mixture2 is a more complicated case of particle composition. Considering the above, we can see that the larger ash particles have a lower velocity value than the smaller ones (cf. light and dark diffused lines for 500 and 1000µm particles, Fig.3). However, at the same time the heavier sand particles of 500µm have larger velocity magnitude than the lighter ash particles of 500 and 1000µm, which show smaller velocity magnitude (Fig.3). This trend is probably caused by the higher rate of particle collision between the light ash particles of different sizes than that between the light and heavy particles of the same size. It is difficult to predict such a tendency, but, it can be observed in numerical simulations.

Fig. 2. Axial velocity profiles of ash and sand particles for various flow conditions: mixture1: ash and sand particles of 0.5mm for 10kg/kg and mixture3: ash and sand 0.5mm for 20kg/kg.

Mathematical Modelling of the Motion of

the flow.

the flow.

*s i si i* / *i i*

> **0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8**

**Angular velocity of solids**

Dust-Laden Gases in the Freeboard of CFB Using the Two-Fluid Approach 153

profile (cf. dashed and solid lines, Fig. 4). Different behaviour in the distribution of radial velocity components of solid and gas-phase can also be observed. Namely, the profiles of solid phase have concave shape while the profile of gas-phase is convex. This means that the increase of mass flow loading results in the increase of radial velocity component of both fractions whatever particles are considered – the ash or sand particles (cf. solid and diffused lines for mixture3 versus the dashed bold and diffused bold lines for mixture1, Fig. 4). This comes from the effect of attenuation of the gas-phase by a larger amount of solid particles in

The effect of lift forces is given in Fig. 5 in terms of the distribution of angular velocity of particles. Fig. 5 shows the distribution of angular velocity of ash and sand particles separately and also their average angular velocity calculated analogously to the abovementioned calculation of average linear velocity of solids in the given form:

where *si* is the angular velocity of composed particle fractions. As the

figure shows, the angular velocity of particles is gradually increasing towards the wall and light ash particles have higher rotation in the vicinity of the wall than heavy sand particles. The increase of particle rotation is obviously stemmed from effect of diminishing of the particles inertia. The particle rotation results indirectly from the intensification of the mixing process, because of the growth of Magnus lift force that causes the particle migration across

> **mixture1,ash mixture1:sand mixture1:average mixture3:ash mixture3:sand mixture3:average**

**0 0.25 0.5 0.75 1 Radial distance, r/R**

Fig. 5. Angular velocity profiles of ash and sand particles and average profile of the mixture

Fig. 6 shows the distribution of particle mass concentration across the two regimes of the flow, which depends on the mass flow ratio: mixture1 with 10kg/kg and mixture3 with 20kg/kg. As the figure shows the lower mass flow ratio results in the slower decrease of mass concentration towards the wall versus the increase of mass concentration of particles for higher mass loading of the flow. As numerical results show, the profile of particle mass concentration is close to the flat shape, which can be observed in the flow loaded by coarse

of particles *s* for the same flow conditions in Fig. 4 for mixtures 1 and 3.

Fig. 3. Axial velocity profiles of ash and sand particles for various flow conditions: mixture1: ash and sand particles of 0.5mm and mixture2: ash 0.5 and 1mm and sand 0.5mm for 10kg/kg.

Fig. 4. Radial velocity profiles of gas phase and two components of solids for various flow conditions: mixture1 and mixture3 for ash and sand solid phases with different mass loadings, 10 and 20 kg/kg, respectively.

Fig. 4 shows the distribution of radial velocity components of gas- and dispersed phases. The profiles of radial velocities of ash and sand particles have been plotted separately for different mass loadings: 10 (mixture1) and 20kg/kg (mixture3). As one can notice, increase of mass flow loading results in change of shape and magnitude of the radial gas velocity

**mixture1:ash mixture1:sand mixture2:ash-0.5mm mixture2:ash-1mm mixture2:sand0.5mm**

**0 0.25 0.5 0.75 1 Radial distance, r/R**

**0 0.25 0.5 0.75 1**

**Radial distance, r/R**

Fig. 4. Radial velocity profiles of gas phase and two components of solids for various flow conditions: mixture1 and mixture3 for ash and sand solid phases with different mass

Fig. 4 shows the distribution of radial velocity components of gas- and dispersed phases. The profiles of radial velocities of ash and sand particles have been plotted separately for different mass loadings: 10 (mixture1) and 20kg/kg (mixture3). As one can notice, increase of mass flow loading results in change of shape and magnitude of the radial gas velocity

Fig. 3. Axial velocity profiles of ash and sand particles for various flow conditions: mixture1:

**gas,c=10 gas,c=20 mixture1,ash mixture1,sand mixture3,ash mixture3,sand**

ash and sand particles of 0.5mm and mixture2: ash 0.5 and 1mm and sand 0.5mm for

**0.5**

**-0.0003**

**-0.00015**

**0.00015**

**Radial velocity of gas & solids** 

**components**

loadings, 10 and 20 kg/kg, respectively.

**0.0003**

**0.00045**

**0**

**0.55**

**Axial velocity of solids components**

10kg/kg.

**0.6**

**0.65**

profile (cf. dashed and solid lines, Fig. 4). Different behaviour in the distribution of radial velocity components of solid and gas-phase can also be observed. Namely, the profiles of solid phase have concave shape while the profile of gas-phase is convex. This means that the increase of mass flow loading results in the increase of radial velocity component of both fractions whatever particles are considered – the ash or sand particles (cf. solid and diffused lines for mixture3 versus the dashed bold and diffused bold lines for mixture1, Fig. 4). This comes from the effect of attenuation of the gas-phase by a larger amount of solid particles in the flow.

The effect of lift forces is given in Fig. 5 in terms of the distribution of angular velocity of particles. Fig. 5 shows the distribution of angular velocity of ash and sand particles separately and also their average angular velocity calculated analogously to the abovementioned calculation of average linear velocity of solids in the given form: *s i si i* / *i i* where *si* is the angular velocity of composed particle fractions. As the

figure shows, the angular velocity of particles is gradually increasing towards the wall and light ash particles have higher rotation in the vicinity of the wall than heavy sand particles. The increase of particle rotation is obviously stemmed from effect of diminishing of the particles inertia. The particle rotation results indirectly from the intensification of the mixing process, because of the growth of Magnus lift force that causes the particle migration across the flow.

Fig. 5. Angular velocity profiles of ash and sand particles and average profile of the mixture of particles *s* for the same flow conditions in Fig. 4 for mixtures 1 and 3.

Fig. 6 shows the distribution of particle mass concentration across the two regimes of the flow, which depends on the mass flow ratio: mixture1 with 10kg/kg and mixture3 with 20kg/kg. As the figure shows the lower mass flow ratio results in the slower decrease of mass concentration towards the wall versus the increase of mass concentration of particles for higher mass loading of the flow. As numerical results show, the profile of particle mass concentration is close to the flat shape, which can be observed in the flow loaded by coarse

Mathematical Modelling of the Motion of

in CFB units can be obtained.

**0**

**0.016**

**0.032**

**Turbulent energy** 

**4. Comparison of the results** 

and 20 kg/kg).

perceptible fact:

**5. Conclusions** 

**0.048**

Dust-Laden Gases in the Freeboard of CFB Using the Two-Fluid Approach 155

Feeding of particles into the flow field may create some reason for additional turbulence generation and it is much appreciated because the intensification of mixing process in CFB can be substantially improved, and as a result, higher efficiency of the combustion process

> **single phase gas phase,mixture1 gas phase,mixture2 gas phase,mixture3**

**0 0.25 0.5 0.75 1 Radial distance**

Fig. 7. Turbulent energy profiles of single and gas phases for mixtures 1, 2 and 3 with different particle sizes of ash: 0.5 and 1 mm and sand 0.5 mm and for the mass loadings (10

Comparing the results with our previous research (Kartushinsky et al., 2009 and Krupenski et al., 2010) in which the theoretical initial data were used we can notice the following

Inclusion of second (heavier) particle fraction modifies the turbulence of carrier fluid

The numerical study of particulate turbulent flow modelled by 2D RANS (Euler/Euler) approach showed importance of addition of second solid fraction, characterized by heavy (sand) particles along with existence of first solid fraction of lighter (ash) particles in the mixing process taken place in freeboard CFB process. The main contribution to the flow formation stems from the inclusion of inter-particle collisions and four-way coupling turbulence modulation due to the presence of polydispersed solid particles with various physical properties. Other forces exerted on the motion of solids are: the gravitation, viscous

a. variation of solids material properties results in the enhancement of flow turbulence in comparison with the turbulence level of the flow loaded by one particle fraction;

resulting in the intensification of mixing process in the freeboard area of CFB.

drag and lift forces. On the basis of the performed calculations one can conclude:

particles. In fact, the numerically obtained profiles of particle mass concentration are highly appreciated because of the efficient operation of CFB units. On the contrary, in the flow domain the gradient profiles of mass concentration can cause retard of enhancement of the combustion process. Thus, an additional sand mass fraction brought to the flow domain may contribute to the improvement of the combustion process in CFB cycles.

Fig. 6. Distribution of particle mass concentration for ash and sand solid phases in different flow conditions shown in previous Figs. 4 and 5: for mixtures 1 and 3.

Finally, the Fig. 7 shows distribution of turbulent the energy across the flow. All considered results for given three different regimes: mixture1, mixture2 and mixture3 are matched between each other and versus also turbulent energy of single phase flow. As a whole, the trend shows that the particles in all the observed regimes generate the turbulence, which stems from the vortex shedding phenomenon behind the particles which is input to the level of turbulence generated by the flow itself. This effect of turbulence modulation, namely, the turbulence enhancement due to the presence of coarse particles is explained and computed using the four-way coupling model by Crowe (2000). This amount of an additional turbulent energy is proportional to the square of velocity slip between the gaseous and the solid phases following the model by Crowe (2000) and it is substantial because of large velocity slip between the phases owing to high inertia of large particle size. Following to the model of Crowe (2000), this generation term is balanced by the introduced dissipation rate of turbulent energy and calculated via the hybrid turbulence length scale (last term in the right-hand side of Eq. 5). The given four-way coupling model by Crowe (2000) is based on the criteria of turbulence modulation by particles considering the ratio of particle size to the integral turbulence length scale. In accordance with this criterion for the considered cases of two-phase turbulent flow loaded by 500 and 1000µm particles, this scale ratio is far above 0.1 and therefore the particles enhanced the turbulence of the carrier gas-phase flow. In addition, the effect of increase of polydispersity grade, i.e. particle size variation from 500 up to 1000µm occurred for the mixture2 (only with ash particles) is less pronounced than that with increase of mass flow ratio up to 20kg/kg occurred in the case of the mixture3 (cf. bold dashed line in Fig. 7), on forming the shape and magnitude level of turbulent energy. Feeding of particles into the flow field may create some reason for additional turbulence generation and it is much appreciated because the intensification of mixing process in CFB can be substantially improved, and as a result, higher efficiency of the combustion process in CFB units can be obtained.

Fig. 7. Turbulent energy profiles of single and gas phases for mixtures 1, 2 and 3 with different particle sizes of ash: 0.5 and 1 mm and sand 0.5 mm and for the mass loadings (10 and 20 kg/kg).
