**1. Introduction**

142 Computational Simulations and Applications

Tokuhiro, A. T. & Lykoudis, P. S. (1994). Natural Convection Heat Transfer from A Vertical

Tryggvason, G.; Scardovelli, R. & Zaleski, S. (2011). *Direct Numerical Simulations of Gas-Liquid* 

*Transfer*, Vol.37, No.6, pp.997-1003

UK

Plate-I. Enhancement with Gas Injection. *International Journal of Heat and Mass* 

*Multiphase Flows*, Cambridge University Press, ISBN 978-0-521-78240-1, Cambridge,

Fluidized beds are the units designed to provide fluid-solid contacting by the fluid flow through a bed of particles (Andrews and Arthur 2007). A number of thermal processes in technology take advantage of the importance of gas-solid interaction in fluidized beds to carry out gas-solid reactions, heterogeneous catalysis and particle drying. The gas-solid fluidization process in circulating fluidized beds is widely applied in many industrial branches. Characterization of the gas-solid particle flow in a circulating fluidized bed (CFB) riser is important for the process optimization. The particle size distribution has significant influence on the dynamics of gas-solid flow (He et al., 2008) along with another important property of the giving system, such as difference in the physical densities of the used materials. The gas-fluidized beds consist of fine granular materials that are subject to the gas flow from below giving the transport velocity that is large enough to overcome the gravity by the viscous drag force and thus the particles can suspend and be fluidized. When in the fluidized state, the moving particles work effectively as a mixer resulting in a uniform temperature distribution and high mass transfer rate, which are beneficial for the efficiency of many physical and chemical processes (Wang et al., 2005). For this reason the gasfluidized beds are widely applied in different industries: thermal, energy, chemical, petrochemical, metallurgical, and environmental industries in large-scale operations involving adhesion optimized coating, granulation, drying, and synthesis of fuels and base chemicals (Kunii & Levenspiel, 1991). In general, the lack of understanding of fundamentals of the dense gas–particle flows has led to severe difficulties in design and scale-up of these industrially important gas-solid contactors (van Swaaij, 1985). In most cases, the design and scale-up of fluidized bed reactors is a fully empirical process based on preliminary tests on pilot-scale model reactors, which is a very time consuming and thus expensive activity. Clearly, computer simulations can be a very useful tool to aid this design and scale-up process.

In the CFB furnaces the ash solids and inert materials like sand particles are mainly used as a solid heat carrier – separated in a hot cyclone and cooled after that in a heat exchanger

Mathematical Modelling of the Motion of

**Pipe data**

Table 1. The initial data for calculations.

which the traced particles might collide.

Dust-Laden Gases in the Freeboard of CFB Using the Two-Fluid Approach 145

be increased by the factor ten and decrease by factor three, respectively, versus their

*Generic name Dimension Minimal Average Maximal*

**Ash concentration** kg/nm3 10 20 20 **Ash density** kg/m3 2000 2000 2000 **Ash particle size** m 0.005 0.0075 0.01 **Sand density** kg/m3 2600 2600 2600 **Sand particle size** m 0.005 0.0075 0.01

For these media we have chosen the initial data very close to the medium in the Estonian oil-shale CFB furnace. The problems of two-phase flows in the CFB risers have been analyzed in certain publications (Hussain et al., 2005, Moscow Energija, 1973), but these studies do not consider dependence of the amount of sensible heat carried by solid particles on the mass flow loading magnitude. The numerical parametric study deals with the influence of the parameters of various riser exits on the hydrodynamics of gas-solid two-

The freeboard CFB used in the given research represents a cylindrical symmetric pipe flow domain occupied by the giving mixture of gas and two types of solid particles. Since the two-fluid model or Euler/Euler approach is applied for the description of the behaviour of solid particles as continuous co-existing phases, the numerical performance is carried out with the finite (control) volume method (Perić & Scheuerer, 1989 and Fertziger & Perić, 1996) written in numerical codes. Another mathematical modelling method, which also operates with the Euler/Euler (or coexisting) approximation deals with the high density or packed particulate flows and solution is obtained with applying the theory of granular flows, for example, that by book of Multiphase Flow and Fluidization: *Continuum and Kinetic Theory Descriptions* (Gidaspov, 1994). In such particulate flows, the particle-turbulence interaction phenomenon is less significant in comparison with the particle-particle collision phenomenon. On the contrary to the Euler/Euler approach, another well-known approximation, which is frequently applied for modelling, the dispersed phase is the Lagrangian Particle Tracking method. The Lagrangian method deals usually with huge numbers of tracking particles (up to several millions of tracking particles depending on the mass flow loading) to obtain a converged solution and also to take into account the particles feedback in the primary fluid (gas-phase). One mathematical technique that can be used for the calculation of flow parameters, including the coupling effect, is given by the particle-cell source method (Crowe et al., 1977). Helland et al., (2000) using the Lagrangian Particle-Tracking approach to calculate the two-dimensional gas-solid particles flow in a CFB riser with the 3% total volume concentration of solids. To take into account the effect of the interparticle collisions within the Lagrangian approach, (Sommerfeld, 2001) developed a stochastic inter-particle collision model with the introduction of a fictitious particle with

phase flow in the CFB riser (Hussain et al., 2005, Moscow Energija, 1973).

**Diameter** m 0.0305 0.0305 0.0305 **Height** m 1.525 1.525 1.525

magnitudes obtained for the normal flow conditions at the temperature 293K.

The following practical initial data were used in calculations:

while the ash particles come back into the furnace. The temperature level in the furnace can be held in the given range by circulating the ash/sand masses. While the heat capacity of ash is quite low, the circulating ash mass must be huge. One way of optimization is to keep up higher heat capacity by adding inertial sand particles. The high ash concentration in furnace gases can be attained with i) high velocity of gas in bed when the fuel particles carried out of bed are burned and their ash fills the whole volume of furnace and ii) ash circulation. The CFB combustion technology enables to bind the sulphur components with the carbonate components added to the fuel or existing within the mineral part of the fuel. A disadvantage of CFB is that some fuel ash particles become too fine during the circulation and therefore the size of ash particles contained in the fuel gas exiting the hot cyclone is too small. As a result of disintegration, the mass of fine ash particles, which are not separated from flue gases or captured in the connective flue ducts and multicyclone increases. The high concentration of particles in the fire gases of CFB furnace chamber contributes to the formation of particle clusters with the solid phase concentration within 0.1 – 0.2 m3/m3. At the exit of CFB boiler furnace the density of solid phase is within 5 – 20 kg/m3.

The given paper is an advanced research of two previous references: "Numerical simulation of uprising gas-solid particle flow in circulating fluidized bed" (Kartushinsky et al., 2009) and "Numerical simulation of uprising turbulent flow by 2D RANS for fluidized bed conditions" (Krupenski et al., 2010) where the mathematical modelling of CFB has been performed. The first one is related to the numerical simulations of CFB where the two-phase turbulent boundary layer approach (TBL) was included. The latter concerns modelling of CFB processes by the RANS approach, which has been developed for both, the gaseous and solid phases, implementing the Euler/Euler approximations or a two-fluid model. Both papers have their advantages and disadvantages. For instance, in the TBL approach the diffusive source terms were retained only in one direction, namely, in the transverse direction, and the magnitude of average transverse velocity components in the gas- and dispersed phases were much less than that of longitudinal components of the corresponding velocities in the gas- and dispersed phases. Such an approach is fully valid and used in the pipe channel flows as well as in the turbulent round jets and flows past the rigid shapes (Hussainov et al., 1995, 1996, Frishman et al., 1997). Nevertheless, a more rigorous and accurate solution was obtained with the help of RANS approach where there are no artificial predictions attached to the TBL approach. However, in both papers only one component of solid admixture, namely, the ash particles are used to simulate the motion of particulate solid phase as a whole.

The current mathematical performance assesses the effect of the presence of two coexisting solid substances, such as ash (light) and sand (heavy) particles with the particle size distribution for each component of solids. This system represents one step further for the mathematical approach to capture real physical processes in CFB. Besides, by making calculations in the real CFB conditions (high temperature of the process) we take into account the amount of heat that must be separated from the combustor by the sensible heat of ash and solid sand particles. The approach enables to optimize particle mass concentration of ash and sand solid particles in fire gases.

The problem is solved by using elaborated mathematical modelling with the help of 2D RANS approach that applies to two coexisting phases. The numerical simulations are performed in the vertical freeboard CFB flow conditions when the temperature of carrier gas-phase fluid is 1123K. Therefore the corresponding magnitudes of parameters of the gaseous phase such as kinematic viscosity coefficient and density of the gaseous phase must


be increased by the factor ten and decrease by factor three, respectively, versus their magnitudes obtained for the normal flow conditions at the temperature 293K. The following practical initial data were used in calculations:

Table 1. The initial data for calculations.

144 Computational Simulations and Applications

while the ash particles come back into the furnace. The temperature level in the furnace can be held in the given range by circulating the ash/sand masses. While the heat capacity of ash is quite low, the circulating ash mass must be huge. One way of optimization is to keep up higher heat capacity by adding inertial sand particles. The high ash concentration in furnace gases can be attained with i) high velocity of gas in bed when the fuel particles carried out of bed are burned and their ash fills the whole volume of furnace and ii) ash circulation. The CFB combustion technology enables to bind the sulphur components with the carbonate components added to the fuel or existing within the mineral part of the fuel. A disadvantage of CFB is that some fuel ash particles become too fine during the circulation and therefore the size of ash particles contained in the fuel gas exiting the hot cyclone is too small. As a result of disintegration, the mass of fine ash particles, which are not separated from flue gases or captured in the connective flue ducts and multicyclone increases. The high concentration of particles in the fire gases of CFB furnace chamber contributes to the formation of particle clusters with the solid phase concentration within 0.1 – 0.2 m3/m3. At

the exit of CFB boiler furnace the density of solid phase is within 5 – 20 kg/m3.

solid phase as a whole.

The given paper is an advanced research of two previous references: "Numerical simulation of uprising gas-solid particle flow in circulating fluidized bed" (Kartushinsky et al., 2009) and "Numerical simulation of uprising turbulent flow by 2D RANS for fluidized bed conditions" (Krupenski et al., 2010) where the mathematical modelling of CFB has been performed. The first one is related to the numerical simulations of CFB where the two-phase turbulent boundary layer approach (TBL) was included. The latter concerns modelling of CFB processes by the RANS approach, which has been developed for both, the gaseous and solid phases, implementing the Euler/Euler approximations or a two-fluid model. Both papers have their advantages and disadvantages. For instance, in the TBL approach the diffusive source terms were retained only in one direction, namely, in the transverse direction, and the magnitude of average transverse velocity components in the gas- and dispersed phases were much less than that of longitudinal components of the corresponding velocities in the gas- and dispersed phases. Such an approach is fully valid and used in the pipe channel flows as well as in the turbulent round jets and flows past the rigid shapes (Hussainov et al., 1995, 1996, Frishman et al., 1997). Nevertheless, a more rigorous and accurate solution was obtained with the help of RANS approach where there are no artificial predictions attached to the TBL approach. However, in both papers only one component of solid admixture, namely, the ash particles are used to simulate the motion of particulate

The current mathematical performance assesses the effect of the presence of two coexisting solid substances, such as ash (light) and sand (heavy) particles with the particle size distribution for each component of solids. This system represents one step further for the mathematical approach to capture real physical processes in CFB. Besides, by making calculations in the real CFB conditions (high temperature of the process) we take into account the amount of heat that must be separated from the combustor by the sensible heat of ash and solid sand particles. The approach enables to optimize particle mass

The problem is solved by using elaborated mathematical modelling with the help of 2D RANS approach that applies to two coexisting phases. The numerical simulations are performed in the vertical freeboard CFB flow conditions when the temperature of carrier gas-phase fluid is 1123K. Therefore the corresponding magnitudes of parameters of the gaseous phase such as kinematic viscosity coefficient and density of the gaseous phase must

concentration of ash and sand solid particles in fire gases.

For these media we have chosen the initial data very close to the medium in the Estonian oil-shale CFB furnace. The problems of two-phase flows in the CFB risers have been analyzed in certain publications (Hussain et al., 2005, Moscow Energija, 1973), but these studies do not consider dependence of the amount of sensible heat carried by solid particles on the mass flow loading magnitude. The numerical parametric study deals with the influence of the parameters of various riser exits on the hydrodynamics of gas-solid twophase flow in the CFB riser (Hussain et al., 2005, Moscow Energija, 1973).

The freeboard CFB used in the given research represents a cylindrical symmetric pipe flow domain occupied by the giving mixture of gas and two types of solid particles. Since the two-fluid model or Euler/Euler approach is applied for the description of the behaviour of solid particles as continuous co-existing phases, the numerical performance is carried out with the finite (control) volume method (Perić & Scheuerer, 1989 and Fertziger & Perić, 1996) written in numerical codes. Another mathematical modelling method, which also operates with the Euler/Euler (or coexisting) approximation deals with the high density or packed particulate flows and solution is obtained with applying the theory of granular flows, for example, that by book of Multiphase Flow and Fluidization: *Continuum and Kinetic Theory Descriptions* (Gidaspov, 1994). In such particulate flows, the particle-turbulence interaction phenomenon is less significant in comparison with the particle-particle collision phenomenon. On the contrary to the Euler/Euler approach, another well-known approximation, which is frequently applied for modelling, the dispersed phase is the Lagrangian Particle Tracking method. The Lagrangian method deals usually with huge numbers of tracking particles (up to several millions of tracking particles depending on the mass flow loading) to obtain a converged solution and also to take into account the particles feedback in the primary fluid (gas-phase). One mathematical technique that can be used for the calculation of flow parameters, including the coupling effect, is given by the particle-cell source method (Crowe et al., 1977). Helland et al., (2000) using the Lagrangian Particle-Tracking approach to calculate the two-dimensional gas-solid particles flow in a CFB riser with the 3% total volume concentration of solids. To take into account the effect of the interparticle collisions within the Lagrangian approach, (Sommerfeld, 2001) developed a stochastic inter-particle collision model with the introduction of a fictitious particle with which the traced particles might collide.

Mathematical Modelling of the Motion of

Dust-Laden Gases in the Freeboard of CFB Using the Two-Fluid Approach 147

*ri i Mi i ri*

<sup>2</sup> ( ) *<sup>t</sup> ri*

*v v C Fu*

*t t k k uk vk x x rr r* 

*u v v uv x r r rx* 

*h ri ri si si*

 *i i i si u rv D i si si si rD x rr x x rr r* 

*i si si uu r uv i si si*

 <sup>2</sup> <sup>1</sup> *ri i si i si si i Mi i ri*

*<sup>u</sup> u r uv C v g x rr* 

> <sup>2</sup> *i si si i si vu r v*

 <sup>2</sup> *ri i si si i si i Mi i si ri*

 

 

*i*

*i v uv r v C Fu*

2 22 2

22 2 2

(5)

(4)

0.5( ) *<sup>i</sup>*

*uv u v*

*<sup>u</sup> C v*

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

*i Mi i si ri*

(2)

(3)

(6)

(7)

1,3

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

2

1,3

1,3

*i i*

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

*x rr* 

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

*x rr* 

*x rr*

*i i*

2

4. Turbulence kinetic energy equation for the gaseous phase:

2 *<sup>t</sup>*

5. Mass conservation equation for the solid phase:

*r*

*i i*

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 and which is under numerical investigation.
