Fluid dynamic modelling and simulation of circulating fluidized bed reactors: importance of the interface turbulence transfer

Abstract

The main objective of this work is to analyze the importance of the gas-solid interface transfer of the kinetic energy of the turbulent motion on the accuracy of prediction of the fluid dynamic of Circulating Fluidized Bed (CFB) reactors. CFB reactors are used in a variety of industrial applications related to combustion, incineration and catalytic cracking. In this work a two-dimensional fluid dynamic model for gas-particle flow has been used to compute the porosity, the pressure, and the velocity fields of both phases in 2-D axisymmetrical cylindrical co-ordinates. The fluid dynamic model is based on the two fluid model approach in which both phases are considered to be continuous and fully interpenetrating. CFB processes are essentially turbulent. The model of effective stress on each phase is that of a Newtonian fluid, where the effective gas viscosity was calculated from the standard k-epsilon turbulence model and the transport coefficients of the particulate phase were calculated from the kinetic theory of granular flow (KTGF). This work shows that the turbulence transfer between the phases is very important for a better representation of the fluid dynamics of CFB reactors, especially for systems with internal recirculation and high gradients of particle concentration. Two systems with different characteristics were analyzed. The results were compared with experimental data available in the literature. The results were obtained by using a computer code developed by the authors. The finite volume method with collocated grid, the hybrid interpolation scheme, the false time step strategy and SIMPLEC (Semi-Implicit Method for Pressure Linked Equations - Consistent) algorithm were used to obtain the numerical solution.

Circulating fluidized bed; kinetic theory of granular flows; mathematical modeling; simulation; finite volume method

Fluid Dynamic Modelling and Simulation of Circulating Fluidized Bed Reactors: Importance of the Interface Turbulence Transfer

José Jailson Nicácio Alves

Departamento de Engenharia Química. Centro de Ciências e Tecnologias

Universidade Federal da Paraíba. Caixa Postal 10057

58109-000. Campina Grande, Pb. Brazil

[e-mail]

Waldir Pedro Martignoni

Petrobrás. Superintendência da Industrialização do Xisto

GETEC. Caixa Postal 28

83900-000 São Mateus do Sul, PR. Brazil

martig@petrobrás.com.br

Milton Mori

Departamento de Processos Químicos. Faculdade de Engenharia Química (FEQ)

Universidade Estadual de Campinas. Caixa Postal 6066

13081-970. Campinas, SP. Brazil

[e-mail]

The main objective of this work is to analyze the importance of the gas-solid interface transfer of the kinetic energy of the turbulent motion on the accuracy of prediction of the fluid dynamic of Circulating Fluidized Bed (CFB) reactors. CFB reactors are used in a variety of industrial applications related to combustion, incineration and catalytic cracking. In this work a two-dimensional fluid dynamic model for gas-particle flow has been used to compute the porosity, the pressure, and the velocity fields of both phases in 2-D axisymmetrical cylindrical co-ordinates. The fluid dynamic model is based on the two fluid model approach in which both phases are considered to be continuous and fully interpenetrating. CFB processes are essentially turbulent. The model of effective stress on each phase is that of a Newtonian fluid, where the effective gas viscosity was calculated from the standard k-e turbulence model and the transport coefficients of the particulate phase were calculated from the kinetic theory of granular flow (KTGF). This work shows that the turbulence transfer between the phases is very important for a better representation of the fluid dynamics of CFB reactors, especially for systems with internal recirculation and high gradients of particle concentration. Two systems with different characteristics were analyzed. The results were compared with experimental data available in the literature. The results were obtained by using a computer code developed by the authors. The finite volume method with collocated grid, the hybrid interpolation scheme, the false time step strategy and SIMPLEC (Semi-Implicit Method for Pressure Linked Equations - Consistent) algorithm were used to obtain the numerical solution.

Keywords: Circulating fluidized bed; kinetic theory of granular flows; mathematical modeling; simulation; finite volume method

Introduction

CFB reactors are used in a variety of industrial applications related to coal combustion, residue incineration and catalytic cracking of oils. A simplified CFB system is shown in Figure 1.a. Figure 1.b shows the idealized symmetrical riser with the inlet and outlet boundaries. Many authors have demonstrated experimentally that there is a variation in the particle concentration and in the velocity fields of both gas and particle phases in the radial and axial directions. A non-uniform particle distribution influences the particle and the gas flow, the residence time and consequently, the reactor performance. High particle concentration near the wall and internal circulation are characteristics of CFB reactors. Currently, efforts have been expended to develop mathematical models capable of predicting the CFB fluid dynamics satisfactorily. To estimate the turbulence in the gas phase, the k-e turbulence model, modified to consider the presence of the particle phase, was used to calculate the effective viscosity of the gas phase. For the particle phase, the effective stress on particle phase was obtained from the kinetic theory of granular flows (KTGF) model. The KTGF model have been used by many authors to predict the particle-particle interactions and consequently the particulate phase stresses (Sinclair and Jackson, 1989; Ding and Gidaspow, 1990; Louge et al., 1991; Pita and Sundaresan, 1993; SEU-Kim and Arastoopour, 1995; Samuelsberg and Hjertager, 1996; Nieuwlande et al., 1996). In a rigorous form, the mathematical model for the granular temperature contains a term of interface turbulence transfer between the gas and particulate phase, which is a function of the correlation between the velocity fluctuations of the gas and those of the particles. When applying the KTGF model to predict gas solid flow, SEU-Kim and Arastoopour (1995), Samuelsberg and Hjertager (1996), and Nieuwland et al. (1996) have neglected the term of turbulence transfer between the phases; Pita and Sundaresan (1993), and Ding and Gidaspow (1990) have neglected the correlation between the velocity fluctuations of the gas and those of the particles. Bolio et al. (1995) has included the turbulence transfer between the phases and applied the model to dilute gas solid flow. This work shows that the turbulence transfer between the phases is important for a better representation of the fluid dynamics of CFB reactors, establishing a way to estimate the correlation between the velocity fluctuations of the gas and those of the particles assuming that the global energy of the fluctuating movement is conserved (this is only a preliminary hypothesis to be investigated).

Nomenclature

a_i = matrix of the coefficients

A_f = plane area, m2

b = independent term in the linear system

d_p = particle diameter, m

e = coefficient of particle-particle restitution

e_w = coefficient of particle-wall restitution

F_k = mass flow across plane "k," kg/s

g_r, g_z = gravity acceleration, m/s2

g₀ = radial distribution function

G = rate of irreversible conversion to internal energy, kg/(m.s³)

H = tube height, m

k = turbulent kinetic energy of the gas, m²/s2

n = vector unitary

P = pressure, Pa, or point in the grid

r = radial co-ordinate, m, r*=r/R

R = tube radius, m

S = source term, Equation 1

T = granular temperature, m²/s2

T₀ = granular temperature at inlet, m²/s²

v = vector velocity (v_i,_z, v_i,_r), m/s

v_g,₀ = axial gas phase velocity at inlet, m/s

v_s0 = axial particulate phase velocity at inlet, m/s

= correlation between the gas and particle

velocities fluctuation, m²/s2

z = axial co-ordinate, m, z^*=z/H

Greek letters

b = coefficient of transfer momentum in the gas-particle interface, kg/m³.s

e = dissipation rate of turbulent kinetic energy of the gas, m²/s3

e_i = volumetric fraction of phase "i"

f_i = generic variable

F = particle-wall factor of friction

g₁ = rate of transfer of the kinetic energy of the fluctuating motion to gas phase from the particulate phase, kg/(m.s³)

g₂ = dissipation rate of pseudo-thermal energy due to inelastic collisions between particles, kg/(m.s³)

g₃ = rate of interface transfer of kinetic energy of the fluctuating motion to particulate phase from the gas phase, kg/(m.s³)

g_w = dissipation rate of pseudo-thermal energy due to inelastic collisions between the particles and wall, kg/s3

h = generic co-ordinate (r, z), m

j = particle sphericity

m_i = viscosity of phase "i," kg/m.s

r_i = density of phase "i," kg/m3

J_P = discrete volume of numerical mesh Mathematical Model

The fluid dynamics model is based on the two-fluid model that treats the fluid phase and the particulate phase as a two-fluid continuum, which is interpenetrating (Whitaker, 1973). The average conservation equations in a two-dimensional cylindrical co-ordinate can be written in a general form as (Alves and Mori, 1998):

Where: f_i represents a generic transport property (table 1); velocity components, volumetric fraction, granular temperature, kinetic turbulent energy, etc.; v_i,j is the velocity component j; r_i is the density and e_i is the volumetric fraction of phase i; and G_f is the transport coefficient of the variable f_i in the phase i.

The turbulent viscosity of the gas phase is calculated from the Komogorov-Prandtl relation (Dasgupta et al., 1994):

The effective particulate phase viscosity is dependent on the granular temperature (T=1/3á v’_{s,h ñ} ², v’_s,h =v’_s,z=v’_s,r) for which a transport equation is derived from the kinetic theory of granular flows (Gidaspow, 1994). In the KTGF (Samuelsberg and Hjertager, 1996),

is the dissipation rate of pseudo-thermal energy - kinetic energy of the fluctuating motion of the particulate phase - due to particle-particle collisions, where "e" is the coefficient of restitution particle-particle, that have its value between zero, for perfectly inelastic collisions, and one, for perfectly elastic collisions.

is the effective pressure on the particle phase due to particle-particle contact,

is the radial distribution function, were e_s,max is the maximum particle concentration (fixed bed),

is the bulk viscosity of the particle phase (or second coefficient of viscosity),

is the effective viscosity of the particulate phase, and

is the diffusion coefficient of pseudo-thermal energy due to gradients in the granular temperature profile.

The coefficient of momentum transfer between the gas and particulate phases, b , is given by (Nieuwland et al., 1996):

where js is the particle sphericity.

The analysis which we are working on is related to terms g₁ and g₃, the interface transfer of the kinetic energy of the fluctuating motion, given by (Louge et al, 1991; Bolio et al., 1995):

By a simple inspection of equations 10 and 11 it can be seen that the correlation should have its value between 2k and 3T. If it is assumed that the global energy of the fluctuating motion is conserved -neglecting the conversion to other energy forms -, one obtains:

This work analysis the results obtained with and without the term given by equations 10 and 11, that is, this work analysis the importance of turbulence transfer between the gas and particle phases in the fluid dynamics of Circulating Fluidized Bed Reactors. Two systems with different characteristics were used in the investigations.

Boundary Conditions

Except for pressure that is obtained by extrapolation from the flow domain, the following boundary conditions were assumed: 1) at the inlet all variables have their values specified; 2) at the tube center the symmetry condition (zero radial velocity component and zero radial gradient of all other variables); 3) at the exit the continuity condition (diffusion in the flow direction is neglected); and 4) at the wall the non-slip condition was specified for the two velocity components of the gas phase. The turbulent kinetic energy and its dissipation rate are calculated from the well-known wall functions (Martinuzzi and Pollard, 1989). For the particulate phase the impermeable wall condition, v_s,r=0, is applied for the radial component, and the axial velocity component is obtained from a balance of momentum transfer between the particles and the wall, just as given by Sinclair and Jackson (1989).

where F_w is the friction factor particle-wall, that have its value between 0 and 1. The granular temperature at wall is obtained from the flow of the pseudo-thermal energy to the wall, q_PT, - balance at wall - (Sinclair and Jackson, 1989),

where,

is the dissipation rate of pseudo-thermal energy due to inelasticity of the particle-wall collisions.

The particle concentration at the wall is obtained from a radial momentum balance at the wall, just as given by Dasgupta et al (1994) for the fully developed region:

The right hand side term in the Eq. 16 limits the particle concentration to the maximum value physically possible, where g = g (e_s,T) is obtained by applying the limit of e_sêw® e_s,max when T ê_w® 0.

Numerical Method

We have one system of non-linear, differential partial equations that can be solved numerically. In this work the finite volume method (FVM) with collocated grid was used for the discretization of the system of equations (Peric et al., 1988). Patankar (1980) and Maliska (1995) show the FVM in details.

The solution procedure used is an extension of the SIMPLEC procedure to two-phase flows. Meier et al. (1999) show details of the algorithm used. The momentum conservation equation for each phase is used to calculate the velocity components of each phase. The continuity equation of the dispersed phase is used to calculate phase concentrations. The continuity equation of continuous phase, coupled with the gas velocity fields, is used to calculate the gas pressure field by the SIMPLEC algorithm (Van Doormall and Raythby, 1984). In the finite volume method the calculation domain is divided into non-overlapping finite volumes (Fig 2.a) in which the conservation equations are integrated. Thus, integration of Equation 1 into the generic control volume, J_p, shown in Fig 2.b, results in the discrete conservation equation for variable f_i (Alves and Mori, 1998):

where n is the normal unit vector which is external to plane "k"; A_f is the area of this plane and j is the base vector; and h represents co-ordinate directions, r and z. By linearization of Eq. 17 and interpolating f_i between the main points of the mesh (Fig. 2.b), we obtain the general discretization equation for a generic variable f_i on point P:

where "nb" represents the neighbors of point P (N, S, E and W). The hybrid interpolation scheme was used to calculate the coefficients a_nb and b (Patankar, 1980).

Results and Discussion

The capability of prediction of internal recirculations and high particle concentration is the objective in the developing mathematical models of the fluid dynamic of Circulating Fluidized Bed Reactors. Currently, there is no consensus about the mathematical models for the Circulating Fluidized Bed Reactors in the literature. The systems analyzed in this work consist of vertical tubes (Figure 1.b) - CFB’s risers -, whose characteristics are given in Table 2 (Tsuo and Gidaspow, 1990; Pita and Sundaresan, 1993). The granular temperature at the tube entrance and the parameters of the kinetic theory of granular flows in Table 2 were obtained from the literature (Alves and Mori, 1997).

The flow is considered to be axisymmetrical, steady state and isothermal. The gas phase is assumed to be Newtonian and incompressible. The particulate phase is composed of same size, non-coalescent particles. The results were obtained using a personal computer with a computer code developed by the authors in FORTRAN language. A grid with 80 x 20 (zxr) internal volumes - a mesh with 82 x 22 (r x z) points including the boundaries - was used for the two systems. The convergence criterion employed required that the Euclidean norm of the mass source with relation to the inlet mass flow should be less than 10^-8. For the two systems the false time step strategy was necessary in order to guarantee the convergence. A increment of 10^-3 seconds was used for the two systems.

Figures 3 and 4 show the particle concentration on an axial position for the two systems given in table 2. The numerical results of the particle concentration are compared to experimental data available in the literature (Tsuo and Gidaspow, 1990; Nieuwland et al., 1996). A good agreement between predicted and experimental data was obtained for system 1, while for system 2 only the qualitative behavior of the experimental data was predicted. It can be seen from the Figures 3 and 4 that the turbulence transfer between the phases, as modeled in this work, affect the predicted flow patterns. It can be seen also that for system 1, the predicted results are in better agreement with experimental data of particle concentration, when the turbulence transfer between the phases was considered. The results obtained for system 2, considering or not the term of turbulence transfer between the phases, are identical.

Figures 5 and 6 compare the results obtained with and without the term of turbulence transfer between the phases. This figures show the streamlines of the particulate phases for the two systems. For system 1 there are differences between the solutions obtained if the interface turbulence transfer is or not considered in the model. These differences show the importance of considering fluctuating motion transfer between the phases in multi phase flows.

To emphasize the flow patterns predicted in this work, the granular temperature and the particle concentration are presented. The results are shown in the form contour maps in Figures 7 to 10 and this results were obtained considering the term of turbulence transfer between the phases in the mathematical model. For the gas-solid system investigated in this work, it can be seen that there is a tendency of the particles to concentrate in the regions of lower granular temperature. This behavior can be seen comparing Figures 7 and 8, for system 1, and comparing Figures 9 and 10 for system 2. Near the wall is a region of low granular temperature and high particle concentration.

Conclusions

We have analyzed in this work the importance of the interface turbulence transfer between the phases in a CFB riser reactor and concluded that this term was important for one system of the two systems analyzed. Due to this it should not be neglected in the prediction of the fluid dynamics of CFB reactors. We have analyzed the importance of turbulence transfer between the phases modeling the term given by equation 12, by applying a condition of global conservation of the fluctuating motion. This condition is not right because there is loss of fluctuating motion due to conversion of the kinetic energy of the fluctuating motion to internal energy. However, considering the hypothesis of global conservation of kinetic energy of the fluctuating motion, the results are in better agreement with the experimental data than the results obtained when the interface turbulence transfer is neglected.

We have seen that the interface turbulence term affect the predicted particle concentration and stream lines of particulate phase for the system 1. For system 2, the turbulence transfer between the phases, in the way that it was modeled, did not affect considerably the flow patterns. Internal recirculation and high particle concentration closing to the wall characterize the system 1. Then, the interface turbulence transfer, frequently neglected in the literature, is very important for systems with high particle concentration and internal recirculation.

The analyses presented here for two systems should be extended for other systems and the final conclusions should contribute to consolidate the mathematical models for Circulating Fluidized Bed Reactors based on phenomenological models. At present, we can only conclude that the qualitative behavior of gas solid flow in risers of CFB reactors can be predicted, and that the quantitative behavior is more realistic when the turbulence transfer between the phases is considered.

Manuscript received: March 1999. Technical Editor: Angela Ourívio Nieckele.

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Publication Dates

• Publication in this collection
  28 Sept 2001
• Date of issue
  2001

History

• Received
  Mar 1999