Effects of initial static bed height on fractional conversion and bed pressure drop in tapered-in and tapered-out fluidized bed reactors

Accepted Manuscript

Effects of initial static bed height on fractional conversion and bed pressure drop in tapered-in and tapered-out fluidized bed reactors Hossein Askaripour , Asghar Molaei Dehkordi PII: DOI: Reference:

S0301-9322(15)00178-0 10.1016/j.ijmultiphaseflow.2015.08.006 IJMF 2264

To appear in:

International Journal of Multiphase Flow

Received date: Revised date: Accepted date:

29 March 2015 6 June 2015 11 August 2015

Please cite this article as: Hossein Askaripour , Asghar Molaei Dehkordi , Effects of initial static bed height on fractional conversion and bed pressure drop in tapered-in and tapered-out fluidized bed reactors, International Journal of Multiphase Flow (2015), doi: 10.1016/j.ijmultiphaseflow.2015.08.006

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Highlights Chemical reaction behavior is investigated in tapered-in and -out fluidized beds.



Effects of static bed height are explored on the fractional conversion and pressure drop.



Two types of chemical reaction with gas volume reduction and increase are studied.



An appropriate static bed height exists in tapered-in beds from reaction point of view.



Changes of static bed height in tapered-out bed slightly affect the fractional conversion.

AC

CE

PT

ED

M

AN US

CR IP T



ACCEPTED MANUSCRIPT

Effects of Initial Static Bed Height on Fractional Conversion and Bed Pressure Drop in Tapered-in and Tapered-out Fluidized Bed Reactors Hossein Askaripour and Asghar Molaei Dehkordi* Department of Chemical and Petroleum Engineering, Sharif University of Technology, P.O. Box

CR IP T

11155–9465, Tehran, Iran

ABSTRACT

In this article, a standard 2D Two-Fluid Model (TFM) closed by the kinetic theory of granular flow (KTGF) has been applied to simulate the behavior of tapered-in and tapered-out fluidized bed

AN US

reactors. In this regard, two types of chemical reactions with gas volume reduction and increase were considered to investigate the effects of initial static bed height on the fractional conversion and bed pressure drop. To validate the CFD model predictions, the results of hydrodynamic simulations

M

concerning bed pressure drop and bed expansion ratio were compared against experimental data reported in the literature and excellent agreement was observed. The obtained simulation results

ED

clearly indicate that there is an appropriate static bed height in a tapered-in reactor in which the

PT

fractional conversion becomes maximum at this height; whereas variations of static bed height in a tapered-out reactor have insignificant influences on the fractional conversion. Moreover, it was

CE

found that the residence time, temperature, and intensity of turbulence of the gas phase are three important factors affecting the fractional conversion in tapered fluidized bed reactors. In addition, it

AC

was observed that increasing the static bed height increases the bed pressure drop for both the tapered-in and tapered-out fluidized bed reactors.

Keywords:

Initial static bed height; Bed pressure drop; Tapered fluidized beds; Gas volume

reduction and increase.

*

Corresponding author. Tel.: +98-21-66165401; fax +98-21-66022853. E-mail address: [email protected]

ACCEPTED MANUSCRIPT

1. Introduction Gas-solid fluidized beds as one of the most important contacting devices have been widely used in chemical and petroleum industries owing to their excellent gas-solid contact and favorable heat- and mass-transfer characteristics. Although most of the gas-solid fluidization behavior studies have been reported for cylindrical or columnar fluidized beds, a considerable portion of the

CR IP T

fluidized beds that have been proposed in the recent decades have inclined walls. The existence of a velocity gradient in the axial direction of the tapered fluidized beds leads to unique dynamic characteristics. Thus, tapered fluidized beds have found wide applications in various industrial processes such as wastewater treatment, incineration of waste materials, crystallization, coal

AN US

gasification, liquefaction, and food processing (Sau and Biswal, 2011; Peng and Fan, 1997). Two common approaches applied for modeling gas-solid fluidized beds are termed Eulerian-Lagrangian and Eulerian-Eulerian models. In the first approach, the gas and solid phases

M

are, respectively, treated as continuous and discrete phases, and the Newton’s second law of motion is solved for each individual particle taking into account the effects of particle collisions and forces

ED

acting on the particles by the gas phase. The Eulerian-Lagrangian models are normally limited to

PT

fluidized beds with a relatively small number of particles due to computational limitations (Taghipour et al., 2005). The Eulerian-Eulerian models (i.e., two-fluid models) are the most

CE

common approach in the simulation of gas-solid fluidized beds. With this approach, gas and solid phases are treated as interpenetrating continua and the integral balances of continuity, momentum,

AC

and thermal energy equations are applied for both the phases with appropriate boundary conditions and jump conditions for the interface. Because the resultant continuum approximation for the solid phase lacks variables such as viscosity and normal stress, certain averaging techniques and assumptions are required to derive a momentum balance for the solid phase (Taghipour et al., 2005; Pain et al., 2001).

ACCEPTED MANUSCRIPT Numerous improvements have been developed to simulate the gas-solid fluidized beds using two-fluid model (TFM). In recent years, mass conservation and momentum balance for gas and solid phases have been applied to simulate the hydrodynamics of bubbling gas-solid fluidized beds. By analogy with the use of kinetic theory of gases, the kinetic theory of granular flow (KTGF) was introduced into TFM to improve the description of particles collision (Chapman and Cowling,

CR IP T

1970). Hence, KTGF has been widely used by investigators who participated in the modeling and simulation of gas-solid flows in various types of fluidized beds. Moreover, it has been found that KTGF has a certain advantage in the perfect prediction of flow phenomena in fluidized beds and risers (Arastoopour, 2001; Bi et al., 2000; Gidaspow, 1994). Johansson et al. (2006) employed two

AN US

different approaches for solid phase stresses to simulate the behavior of columnar fluidized beds. The first approach uses a constant particle viscosity (CPV) and predicts the particle pressure, whereas the second approach uses the particle turbulent model (PT), which is based on KTGF

agreement with experimental data.

M

(Enwald and Almstedt,1999). They eventually reported that the results of KTGF model are in better

ED

Due to less pressure fluctuations in tapered fluidized beds, this kind of fluidized beds can be operated smoothly without any instability (Shi et al., 1984). For the fluidization of materials with

PT

a wide particle size distribution and various specifications, as well as for exothermic reactions (Kim

CE

et al., 2000 and solids mixing (Schaafsma et al., 2006), the tapered fluidized beds would be very beneficial and useful. Despite their widespread applications, much of the developments and designs

AC

of tapered fluidized bed reactors have been performed empirically and this is due to the complex nature of the behavior of gas-solid flow in this type of fluidized beds. In addition, numerical solution of the governing complex non-linear equations with moving phase boundaries is one of the other challenges in the simulation of tapered fluidized beds. However, with increasing computational capabilities and development of computational fluid dynamics (CFD) tools in recent years, some investigators have been involved in the hydrodynamic studying of tapered fluidized

ACCEPTED MANUSCRIPT beds, which would be beneficial in the design, optimization, and scale-up processes of tapered fluidized beds. Shi et al. (1984) and Peng and Fan (1997) experimentally investigated the hydrodynamic aspects of fluidization in liquid-solid tapered fluidized beds. Investigation of the flow pattern of the fluid and solid phases and measurement of the bed pressure drop at different superficial liquid

CR IP T

velocities led to the identification of five flow regimes in tapered fluidized beds. In addition, other hydrodynamic aspects of tapered fluidized beds, including maximum pressure drop, minimum velocity of partial fluidization, and minimum velocity of full fluidization were experimentally determined. Moreover, some models based on the balance between the forces, including effective

AN US

weight of particles, fluid frictional force, and drag force were developed.

Biswal et al. (1982) studied the fluctuation ratio for regular particles in gas-solid conical fluidized beds. On the basis of dimensional analysis, they derived a correlation for fluctuation ratio,

M

which is a function of static bed height, mean diameter of the bed, particle diameter, and mass

ED

velocity of fluid. Schaafsma et al. (2006) investigated the particles flow pattern and segregation phenomenon in a tapered fluidized bed granulator using segregation experiments and positron

PT

emission particle tracking experiments (PEPT) and concluded that the flow and segregation

CE

phenomena in tapered fluidized beds are totally different from those in columnar fluidized beds. Huilin et al. (2006) applied a two-dimensional TFM model based on KTGF to simulate the

AC

hydrodynamic behavior of gas-solid flow in both columnar and tapered risers. Comparison of the simulation results of tapered risers with the columnar risers demonstrated that the core-annular flow structure in the columnar risers disappeared in tapered risers and a better solids mixing can be obtained in tapered risers. In addition, as the tapered angle increases both the gas and particle velocities and solids holdup tend to be more uniformly distributed in the radial direction. Sau et al. (2008a,b) have developed several empirical models to predict the minimum fluidization velocity and maximum bed pressure drop for heterogeneous and homogeneous binary

ACCEPTED MANUSCRIPT mixtures of irregular particles in a gas-solid tapered fluidized bed. These models have been developed using dimensional analysis and estimating the adjustable parameters by nonlinear regression analysis of the experimental data. Eventually, similar correlations to what reported by Thonglimp et al. (1984) and Olazer et al. (1993) have been developed for minimum fluidization velocity and maximum bed pressure drop, respectively, that are functions of tapered angle, bed inlet

CR IP T

diameter, particle size, particle density, and initial static bed height. Sau et al. (2010) investigated the dynamics of tapered fluidized beds. They developed models for the prediction of bed expansion ratio for both spherical and non-spherical particles. Models were developed based on the dimensional analysis as a function of the geometry of tapered

AN US

bed, static bed height, particle diameter, density of solid and gas phases, and superficial gas velocity. In addition, they experimentally examined the effects of static bed height, particle diameter, tapered angle, and superficial gas velocity on the bed expansion ratio. Gan et al. (2014)

M

studied the hydrodynamic characteristics of the tapered fluidized beds. They experimentally examined the effects of tapered angle, static bed height, particle size, and bed inlet width on the

ED

minimum fluidization velocity and minimum velocity of full fluidization. They found that increasing the tapered angle, particle size, and static bed height increases both the minimum

PT

fluidization velocity and minimum velocity of full fluidization, while increasing the bed inlet width

CE

would decrease these two velocities.

AC

In the present work, TFM by applying KTGF approach was employed to simulate behavior of tapered fluidized beds in the presence of chemical reactions with gas volume reduction and increase. The main objective of the present work was to investigate the effects of initial static bed height on the fractional conversion and bed pressure drop in both tapered-in and tapered-out fluidized bed reactors. We believe that such a parametric study can be useful to explore the performance of a tapered fluidized bed reactor. Moreover, the results of hydrodynamic simulations concerning bed pressure drop and bed expansion ratio are compared with experimental data reported in the literature (Sau and Biswal, 2011).

ACCEPTED MANUSCRIPT

2. Two-Fluid Model Equations The description of gas-solid fluidized beds as two interpenetrating continua is the principle of two-fluid models (van Wachem and Almstedt, 2003; Acosta-Iborra et al., 2012). One continuum refers to the gas phase in the bed and the other to the particle phase (also known as the particulate or solid phase). This allows the Eulerian description of both phases without resorting the individual

CR IP T

Lagrangian tracking of each particle in the bed. Therefore, in applying the two-fluid approach for a fluidized bed reactor, the governing equations of continuity, momentum, and thermal energy balance for the gas and particle phases as well as the conservation equation of species for the gas phase can be used in the mathematical description of the phenomena. In this regard, in the

AN US

momentum equations, the particle phase is treated as a fluid with effective transport properties (Gidaspow, 1994). In general, to simulate the behavior of fluidized bed reactors, the conservation equations of mass, momentum, species, and thermal energy balance are solved together with a differential equation for the transport of granular temperature, which is based on KTGF and

ED

M

provides the level of random fluctuations of particle velocity due to collisions.

2.1. Continuity Equations

PT

The overall conservation of mass for the gas and solid phases in the absence of interphase

CE

mass transfer can be expressed as

AC

 ( g  g )    ( g  g v g )  0 t  ( s  s )    ( s  s v s )  0 t

(1)

(2)

where  g ,  s ,  g ,  s , v g and v s are, respectively, volume fraction of gas phase, volume fraction of the solid phase, gas density, solid density, local velocity of gas phase, and the local velocity of the solid phase. Since the bed spaces are occupied by solid or gas phases, the sum of the gas and solid volume fractions must be equal to one.

ACCEPTED MANUSCRIPT g  s  1

(3)

2.2. Momentum Equations The momentum equations for the gas and solid phases can be expressed as (Wang, 2006)

  g  g v g     g  g v g v g   g pg    g   g  g g  F d ,g  F l , g  F vm, g t

(4)

  s s v s     s s v s v s   spg    s  ps   s  s g  F d ,s  F l ,s  F vm,s  Ss t

(5)















CR IP T



where pg is the gas phase pressure, ps is the solid phase pressure,  g is the gas phase stress

AN US

tensor,  s is the solid phase stress tensor, g is the gravitational acceleration, and S s is the solid phase source term, which is exclusively introduced in the momentum equation of the solid phase in the tapered-in fluidized beds. In addition, F d , g and F d ,s are the drag forces, F l , g and F l ,s are the

M

lift forces, and F vm, g and F vm,s are the virtual mass forces for the gas and solid phases, respectively.

ED

Applying the Newtonian stress-strain for each phase, the stress tensor terms for the gas and solid phases can be evaluated as follows (Wang, 2006):



T

  

CE



   

2    g   g    v g I 3  

PT

 g   g  g  v g   v g

g

T 2    s   s s v s  v s    s  s  s    v s I





3



(6)

(7)

AC

where  g and  s are the shear viscosities, g and s are the bulk viscosities for the gas and solid phases, respectively, and I is the unit stress tensor. The solid shear viscosity is composed of kinetic and collisional terms arising from the momentum exchange of solid particles due to translation and collision, respectively. A frictional viscosity term is also included to account for the transition that occurs when the solids volume

ACCEPTED MANUSCRIPT fraction exceeds a critical value. The solid shear viscosity can be evaluated as follows (Chalermsinsuwan et al., 2011):

s  s ,col  s ,kin  s , fr

(8)

 s    

12

s , fr 

I2D 

 4  1  5 g , ss s 1  ess  

2

ps sin( ) 2 I2D

AN US

s ,kin

10  s d s s  96 s 1  ess  g , ss

(9)

CR IP T

4 5

s ,col   s  s d s g ,ss 1  ess  

2 2 1 2 D  D  D  D  D  D  Dxy2  Dyz2  Dzx2       xx yy yy zz zz xx     6

1 Dij  (v s  (v s )T ) 2

(10)

(11)

(12)

(13)

M

where d s , ess , g ,ss ,  s ,  , I 2D , and Dij are, respectively, the particle diameter, particle-particle

ED

restitution coefficient, solid radial distribution function, granular temperature, angle of internal friction, second invariant of the deviatory stress tensor, and the rate of strain tensor for solid phase.

PT

In addition, the subscripts col, kin, and fr denote collisional, kinetic, and frictional viscosities,

CE

respectively. The gas phase bulk viscosity is assumed to be a constant value of zero and the solid phase bulk viscosity accounting for the resistance of solid particles to compression and expansion,

AC

can be estimated as follows (Chalermsinsuwan et al., 2011):

4   s   s  s d s g ,ss 1  ess   s  3  

12

(14)

The solid pressure representing the normal solid phase forces due to particle-particle interactions consists of two terms, i.e., a kinetic term and a collisional term that can be determined by (Lun et al. 1984)

ACCEPTED MANUSCRIPT ps   s s s  2s 1  ess   s2 g ,ss s

(15)

The solid radial distribution function is a correction factor that modifies the probability of collision between the particles and can be evaluated as follows (Chalermsinsuwan et al., 2011): 1 3 1

   

  

(16)

where  s ,max is the solid volume fraction of maximum packing.

CR IP T

g , ss

    1   s    s ,max 

Because sustained contacts occur between the particles at high solid volume fractions, the

AN US

resulting frictional stresses should be taken into account for the description of solid phase stress. The majority of the available frictional models are based on the critical state theory of soil mechanics. On the basis of principles of soil mechanics, Johnson and Jackson (Johnson and Jackson, 1987) proposed semi-empirical equations for the frictional viscosity and pressure. This

M

model is used when the solid volume fraction exceeds a critical solid volume fraction. The frictional

ED

viscosity and pressure of the Johnson and Jackson model can be evaluated by (Johnson and Jackson, 1987) n

PT

  s ,cr 

s ,max   s 

p

,

Fr  0.05 , n  2 , p  5

p fr sin( ) 2 I2D

(17)

(18)

AC

 fr 

s

CE

p fr

  Fr 

where  s ,cr is the critical solid volume fraction, p fr is the frictional pressure,  fr is the frictional viscosity, and Fr , n , and p are empirical constants.

The momentum exchange between the gas and solid phases is carried out through drag, lift and virtual mass forces. The general form of the drag, lift, and virtual mass forces acting on gas and solid phases can be expressed as (ANSYS Inc., 2011)

ACCEPTED MANUSCRIPT





,

F d ,s   F d ,g



F l , g  0.5 g  s v s  v g   v g



 Dv s Dv g  F vm, g  0.5 s  g    Dt   Dt

,

,

(19)

F l ,s   F l , g

(20)

F vm,s   F vm, g

(21)

where  sg is the gas-solid momentum exchange coefficient, and time derivatives for solid and gas velocities, respectively.

Dv s Dv g and are the substantial Dt Dt

CR IP T



F d , g   sg v s  v g

Various drag models for gas-solid momentum exchange coefficient were reported in the

AN US

literature. In the present study, the drag model proposed by Gidaspow (Gidaspow, 1994) was used. This drag model is a combination of Wen and Yu model (1966) for dilute flows and Ergun equation (Ergun, 1952) for dense flows. In this regard, for  g  0.8 , Wen and Yu drag model was used,

M

whereas for  g  0.8 Ergun’s equation was applied.

24  g Re s

 g  0.8

1  0.15  Re 0.687  g s  

g ds vs  v g g

AC

Re s 

(22)

CE

CD 

 g  0.8

PT

ED

    v s  v g 2.65  3 CD s g g g  4 ds  gs   g s vs  v g  s 1   g   g   1.75 150  g d s2 ds 

(23)

(24)

where CD and Re s are, respectively, the drag coefficient and solid Reynolds number. In tapered-in fluidized beds, there exist three distinct regions, i.e., a dilute core, a dense annular region between the core and the wall named annulus, and a dilute hump region above the bed surface. From the simulation point of view, the structure of tapered-in fluidized beds should be divided into at least two regions that are a dilute fluidized region (including both the core and the

ACCEPTED MANUSCRIPT hump) and a dense defluidized region (annulus) (Sau and Biswal, 2011). To include the additional stresses exerted by the tapered side wall on the gas-solid flow, two solid phase source terms are defined for the annulus and core regions as follows (Wang, 2006):

Ptapered

(25)

Pcolumnar



kc  f  g , s , d s ,  g ,  g , v g



(26)

CR IP T

ka 

where k a is the pressure drop in a tapered-in fluidized bed-to-a columnar fluidized bed ratio. Although kc is a function of different parameters, for simplification of the problem, kc is assumed

AN US

to be a constant value of 1.

On the basis of what proposed in Ref. (Wang, 2006), the following simple expressions can

Ss   k  1  s s g  g  0.8 and y  H  g  0.8

(27)

(28)

ED

ka k   kc

M

be used to describe the solid phase source terms for both the annulus and core regions:

PT

where y is the vertical distance from the gas distributor and H 0 is the static bed height.

CE

2.3. Granular Temperature Equation

AC

The granular temperature of solid phase,  s , is defined as one-third of the mean square of velocity fluctuations of particles. The transport equation for the solid phase granular temperature can be evaluated by (Hamzehei and Rahimzad, 2010)

 



3  s s s      s s v s s    ps I   s : v s     s s   s  gs 2  t 







(29)

ACCEPTED MANUSCRIPT 

s

150  s d s s  384 1  ess  g , ss

s  6  2 1  5  s g ,ss 1  ess    2  s s d s 1  ess  g , ss  2

(30)





where  ps I   s : v s is the generation of granular energy by the solid stress tensor,  s s is

the diffusion flux of granular energy,  s is the diffusion coefficient of granular energy,  s is the collisional dissipation of energy, and gs is the energy exchange between gas and solid phases. The

CR IP T

collisional dissipation of energy,  s , represents the rate of energy dissipation within the solid phase due to inelastic collisions of solid particles and can be evaluated by (Hamzehei and

  s

12 1  ess2  g , ss ds 

AN US

Rahimzad, 2010)

 s s23s 2

(31)

The energy exchange between gas and solid phases, gs , is due to random fluctuations in

M

particle velocity and can be expressed as (Hamzehei and Rahimzad, 2010)

(32)

ED

gs  3 sg s

PT

The main characteristic of a turbulent system is the production of additional stresses due to the fluctuations of solid velocity in x, y, and z directions. The laminar or classical granular

CE

temperature,  s , is calculated using Eq. (29) to determine the random oscillations of individual

AC

solid particles, whereas the turbulent granular temperature, t , which is due to motion of clusters of particles or bubbles, is defined as the average of the solid normal Reynolds stresses and can be evaluated by (Chalermsinsuwan et al., 2009 and 2011)

1 1 1 t  vx vx  vy vy  vz vz 3 3 3

(33)

where vx , vy and vz are, respectively, the velocity fluctuations in x , y , and z directions and are defined as follows:

ACCEPTED MANUSCRIPT vivi 

vi 

1 k   vi (t )  vi  vi (t )  vi  k 1

(34)

1 k  vi (t ) k 1

(35)

where subscript i and k are the direction and total number of time steps, respectively. The sum of

solid particles is termed the total granular temperature.

2.4. Thermal Energy Equations

CR IP T

the granular temperatures due to both the individual particle oscillations and motion of clusters of

AN US

The thermal energy balances for both the gas and solid phases can be expressed as (ANSYS Inc, 2011)

p   g  g hg      g  g v g hg   g g   g : v g     g g Tg   Qsg  H rxn  t t

(36)

p   s s hs      s s v s hs   s s   s : v s     s sTs   Qsg t t

(37)





M





ED

where hg and hs are the enthalpies,  g and  s are the thermal conductivities, Tg and Ts are the

PT

temperatures for gas and solid phases, respectively, Qsg is the heat transfer between gas and solid

CE

phases, and H rxn is the heat of reaction.

AC

The heat transfer between gas and solid phases, Qsg , is a function of temperature difference between gas and solid phases and can be determines by (ANSYS Inc., 2011)

Qsg  hsg Ts  Tg 

hsg 

6 g  s g Nus d s2

(38)

(39)

ACCEPTED MANUSCRIPT where hsg and Nus are, respectively, the heat transfer coefficient between gas and solid phases and Nusselt number for the solid phase. Nusselt number is a function of the solid Reynolds number and Prandtl number and can be evaluated by (ANSYS Inc., 2011) 13 Nus   7  10 g  5 g2 1  0.7 Re0.2   1.33  2.4 g  1.2 g2  Re0.7s Pr1 3 s Pr

C p,g g

(41)

g

CR IP T

Pr 

(40)

where Pr is the Prandtl number and C p , g is the specific heat at constant pressure for the gas phase.

AN US

The definition of solid Reynolds number, Re s , is the same as Eq. (24).

2.5. Turbulence Equations

In Reynolds-averaged approach for turbulent flow modeling, flow variables of NavierStokes equations are decomposed into the mean and fluctuating components. Substituting flow

M

variables in terms of mean and fluctuating components into Navier-Stokes equations yields

ED

additional terms representing the effects of turbulence. To account for the effects of turbulent fluctuations, various types of turbulence models were presented in the literature. One of the most

PT

well-known models is the RNG k   two-equation model. The RNG k   model is derived using a statistical technique called renormalization group (ANSYS Inc., 2011; Orszag et al., 1993). The

CE

wide application of RNG k   model in industrial flow and heat transfer simulations is due to its

AC

robustness and reasonable accuracy. The RNG k   model is based on the transport equations for turbulent kinetic energy, its dissipation rate, and can be expressed as follows (ANSYS Inc., 2011):

  m k     m vm k     k t k   Gk ,m  m t

(42)

   m     m vm     t     C1 Gk ,m  C2 m   R t k

(43)









ACCEPTED MANUSCRIPT where  k and   are the inverse turbulent Prandtl number for k and  , respectively,  m is the mixture density, v m is the mixture velocity, Gk ,m is the generation of turbulence kinetic energy due to velocity gradient, t is the turbulent viscosity, and C1 and C2 are the model constants. More information about these parameters can be found in (ANSYS Inc., 2011).

CR IP T

2.6. Species Continuity Equation The species continuity equation for the gas phase can be expressed as follows (ANSYS Inc., 2011):

  g  g yg ,i      g g v g yg ,i     g J g ,i    g Rg ,i t



(44)

AN US



where yg ,i , J g ,i , and Rg ,i are, respectively, the mass fraction, diffusion flux and the net rate of generation/consumption of species i . Eqn. (44) could be solved for N  1 gas phase species, where

M

N is the total number of gas phase species that present in the system. Since the mass fraction of

N

y

g ,i

 1.0

(45)

PT

i 1

ED

gas phase species should sum to unity, the N th mass fraction can be determined as follows

CE

The diffusion flux of species i is due to the gradients of concentration and temperature and can be determined as follows (ANSYS Inc., 2011):

AC

  J g ,i     g Dgi ,m  t Sct 

Tg  i  yg ,i  DT Tg 

(46)

i ,m where Dg is the mass diffusion coefficient of species i , DTi is the thermal diffusion coefficient of

species i , and Sct is the turbulent Schmidt number. More information about the mass and thermal diffusion coefficients can be found in Ref. (ANSYS Inc., 2011).

2.7. Chemical Reactions

ACCEPTED MANUSCRIPT The required information for the modeling of chemical reactions in a fluidized bed reactor is the chemical reaction rate expressions. Due to the uncertainty about the chemical reaction rate expressions of fluidized bed reactors, and since the aim of this study was not to develop the details of any given chemical reaction, two simplified gas phase chemical reactions in Arrhenius forms are considered. The considered chemical reactions and their rate expressions can be expressed as

Reaction (1) (i.e., gas volume increase): k1 CH 4  0.5O2   CO  2H 2 ,  Hrxn  36.0 kJ / mol

Reaction (2) (i.e., gas volume reduction):

AN US

0.5 rCH4  k1CCH CO0.62 4

k2 CO2  3H 2   CH3OH  H 2O ,  Hrxn  48.97 kJ / mol

(47)

(48)

M

0.85 1.1 rCO2  k2CCO CH2 2

CR IP T

follows:

ED

where CCH 4 , CO2 , CCO2 , and CH 2 are, respectively, concentrations of CH4, O2, CO2 and H2, and k1 and k 2 are the reaction rate constants for reactions (1) and (2), respectively. The reactions rate

Ea ) RT

k2  k  exp(

Ea ) RT

AC

CE

k1  k exp(

PT

constants, k1 and k 2 , can be expressed as follows: (49)

(50)

where k and k  are the pre-exponential factors, and Ea and Ea are the activation energies of reactions (1) and (2), respectively, T is the gas phase temperature, and R is the universal gas constant.

2.8. Initial and Boundary Conditions

ACCEPTED MANUSCRIPT The initial conditions used for the 2D simulations of tapered-in and tapered-out fluidized bed reactors are shown in Fig. 1. As can be observed from this figure, at initial conditions the freeboard was free of any solid particle, whereas the bottom section of the bed was filled with solid particles with a volume fraction of  . Because the bed was initially considered at stagnant conditions; the x and y components of velocity field for both the gas and solid phases were set to

CR IP T

zero. Moreover, the mass fraction of gas phase species and the temperature of gas and solid phases were set to initial values.

In this model, the imposed boundary conditions are as follows: at the top boundary, constant atmospheric pressure was assumed so that particles are free to leave the system and for the inlet

AN US

boundary, a uniform distribution of the velocity and temperature of gas flow with specified mass fractions for gas phase species was considered with no particles entering the domain. For the lateral sidewalls, a simple no-slip wall boundary condition was set for the gas phase, while the Johnson

M

and Jackson (1987) boundary conditions were used for the particulate phase. Moreover, the walls were assumed isolated and the diffusion flux of gas phase species into the walls was considered to

ED

be zero.

PT

The Johnson and Jackson boundary conditions incorporate the specifications of the specularity coefficient and particle-wall restitution coefficient. It is well-known that the specularity

CE

coefficient has a significant influence on the global flow field. Unfortunately, the values assigned to both these variables vary widely throughout the literature. In this regard, literature sources could be

AC

classified into two general groups regarding the specification of the specularity coefficient: the group assuming near perfect slip (   0.001 ) (Benyahia et al., 2007; Almuttahar and Taghipour, 2008) and the group assuming partial slip ( 0.5    0.6 ) (Chalermsinsuwan et al. 2009; Lu et al.,2008). The partial slip boundary condition (   0.5 ) was chosen in the present work. This is because it seems to be more physically feasible than the perfect slip between particle and the wall. Furthermore, the particle-wall restitution coefficient ( esw ) was set to a value of 0.9. Finally, these

ACCEPTED MANUSCRIPT constants were implemented to evaluate the shear stress and granular temperature at the wall as follows:

nˆ  q 

 3 6

 3 6





s

 s ,max

s

 s ,max

 s g ,ss 0.5 s v sl ,|| 2

 s g ,ss 0.5 s v sl ,|| 

v sl ,||  v s  v wall

,

 3 s 2  s g ,ss 1  esw 1.5  s 4  s ,max

(51)

,

q   s s

(52)

CR IP T

nˆ  s  

where nˆ and v sl ,|| are the unit normal vector from the boundary into the particle assembly and the slip velocity of the particle assembly at the wall, respectively.

AN US

3. Simulation Conditions

A code of standard TFM closed by the KTGF has been developed based on the governing equations of mass, momentum, thermal energy, and turbulence for both the gas and solid phases, the granular temperature equation for solid phase and the conservation equation of species for gas

M

phase to carefully describe the behavior of the tapered-in and tapered-out fluidized beds in the

ED

presence of chemical reactions with gas volume reduction and increase. Due to the high computational time of the three-dimensional simulations, a two-dimensional CFD modeling was

PT

used to simulate the tapered fluidized bed reactors.

CE

3.1. Numerical Method

AC

In this study, a TFM computer code was developed to solve the governing partial differential equations. To solve the governing equations of mass, momentum, thermal energy, and turbulence for solid and gas phases, the conservation equation of species for gas phase constituents and the granular temperature for solid phase, we used the well-known finite volume method. In addition, the phase-coupled SIMPLE method was used for pressure-velocity coupling, and the first order implicit method and the second order upwind method were used to discretize the equations of mass, momentum, thermal energy, turbulence, species continuity, and granular temperature. These

ACCEPTED MANUSCRIPT methods provide satisfactory results for the prediction of bed behavior in the presence of chemical reactions. Moreover, the volumetric and laminar finite rate method was considered for the reaction mechanism. The simulations were performed with a uniform distribution of the velocity and temperature of gas flow and specified mass fractions for gas phase species with no solid particles imposed on

CR IP T

the gas distributor and the initial state of the bed was assumed at stagnant conditions. The time step of the simulations was set to a value of 610–4s and the number of iterations per each time step was set to 30. Examining the simulation results demonstrated that using this number of iteration guarantees that the relative error of all the variables would be less than 510–4 for most of the time

AN US

steps in the simulation runs. The simulation runs were performed for 15s, the first 5s of each run was ignored to eliminate the transient conditions and the rest of 10s was considered to obtain the

4. Results and Discussions

M

time-averaged variables reported in this work.

ED

In the simulation of the behavior of fluidized bed reactors, the governing equations of mass, momentum, energy, and turbulence for both the gas and solid phases, the species continuity

PT

equation for the gas phase and the granular temperature for solid phase would be concurrently

CE

solved. In this study, two kinds of exothermic chemical reactions were considered to study the behavior of tapered-in and tapered-out fluidized beds in the presence of chemical reactions. As

AC

described in section 2.7, the gas volume increases in the course of the first reaction, while the gas volume reduces for the second reaction. The values of the simulation parameters, gas and solid properties, fluidized bed dimensions, and reaction rate constants are summarized in Table 1. In the following subsections, simulation results are presented and discussed in three sections: the first section presents the comparison of bed expansion ratio and bed pressure drop between the hydrodynamic simulations of a tapered-in fluidized bed and the experimental data reported by Sau and Biswal (2011); the second section presents the study of grid independence for

ACCEPTED MANUSCRIPT both the tapered-in and tapered-out fluidized bed reactors; and the third section is devoted to the investigation of the influences of initial static bed height on the fractional conversion and bed pressure drop in both the tapered-in and tapered-out fluidized beds for the reactions with gas volume reduction and increase.

4.1. Validation of CFD Model with Experimental Data

CR IP T

To verify the CFD simulation results, careful validation against experimental data is always required. In this study, due to lack of experimental data for tapered-in or tapered-out fluidized beds in the presence of chemical reactions, the results obtained from the hydrodynamic simulations of a

AN US

tapered-in fluidized bed are compared against the experimental data reported in Ref. (Sau and Biswal , 2011). Sau and Biswal (2011) conducted an experimental work in a tapered-in fluidized bed of 520 mm height and 4.6 tapered angle to measure the bed expansion ratio and bed pressure drop. The tapered fluidized bed was made of Perspex sheet to permit the visual observation using a

M

high speed digital video camera and a manometer with two pressure taps, one just above the

ED

distributor and the other at the top of the fluidized bed, was used to record the pressure drop. The fluidized bed was filled with the spherical glass bead particles of 2.0 mm in diameter and was

PT

fluidized with air at a temperature of 301 K and atmospheric pressure.

CE

In hydrodynamic simulations, the governing equations of continuity and momentum for both the solid and gas phases, and the granular temperature equation for solid phase were solved

AC

together. The simulation parameters, solid specifications, and tapered-in fluidized bed dimensions were considered the same as those summarized in Table 1, except in the hydrodynamic simulations, air flow at velocities of 1.42, 2.83 and 3.54 m/s, and a temperature of 301 K (ρair =1.17 kg/m3 and μair = 1.82×10–5 Pa s) was used as the fluidizing agent. Figures 2(a) and (b) show the bed pressure drop and bed expansion ratio versus superficial gas velocity for both the experimental data and numerical simulations in a tapered-in fluidized bed, respectively. The initial static bed height in Figs. 2(a) and (b) is considered to be 6.5 cm. As can be

ACCEPTED MANUSCRIPT observed from Figs. 2(a) and (b), there is an excellent agreement between the experimental data and numerical simulations for both the bed pressure drop and bed expansion ratio. It can be also observed that there is a slight difference between the experimental data and numerical simulations, and this slight discrepancy may be due to the simplification of the bed geometry from 3D

4.2. Study of Grid Independence

CR IP T

configuration to 2D configuration.

The numerical simulations are incomplete without the study of grid independence. In this study, to investigate that if the simulation results are independent of grid size, the number of grid

AN US

points was incrementally increased in both the radial and axial directions. In this regard, four kinds of grid number of 32×125, 36×140, 40×145, and 44×150, and three kinds of grid number of 32×125, 36×140 and 40×145 were examined for reaction (1) in the tapered-in and tapered-out fluidized bed reactors, respectively. Figures 3(a) and (b) demonstrate the variations of CO and CH4

M

mole fractions versus column height for reaction (1) in a tapered-in fluidized bed reactor,

ED

respectively, and Fig. 4 shows the variations of CO mole fraction versus column height for reaction (1) in a tapered-out fluidized bed reactor. Moreover, it should be mentioned that all the mole

PT

fractions presented in Figs. 3 and 4 are time- and area-averaged and the initial static bed height, H 0 ,

CE

in Figs. 3 and 4 were considered to be 10.0 and 1.9 cm, respectively. As can be seen in Figs. 3(a) and (b), there is an appreciable discrepancy between the mole

AC

fraction profiles for the grid numbers of 32×125 and 36×140 and this discrepancy is reduced for the grid numbers of 36×140 and 40×145. It is also observed that the incrementally increasing the grid number from 40×145 to 44×150 would slightly affect the mole fraction profiles of CO and CH4. Because the grid size should be sufficiently fine, so, further refinement does not change the simulation results appreciably, the grid number of 40×145 was chosen for reaction (1) in the tapered-in fluidized bed reactors. Fig. 4 shows that the incrementally increasing the grid numbers, 32×125, 36×140, and 40×145, would insignificantly affect the mole fraction profile of CO.

ACCEPTED MANUSCRIPT Therefore, the mesh number of 36×140 was chosen for reaction (1) in the tapered-out fluidized bed reactors. Similar works were conducted to study the grid independence regarding reaction (2) in the tapered-in and tapered-out fluidized bed reactors. The obtained results clearly indicate that the grid sizes of 40×150 and 36×145 are appropriate for the tapered-in and tapered-out fluidized bed reactors, respectively. As a result of insignificant effects of incrementally increasing the grid

exclusively demonstrated on one of the species of the gas phase.

4.3. Effects of Initial Static Bed Height

CR IP T

numbers in Fig. 4 for tapered-out fluidized bed reactors, the effects of different grid numbers are

AN US

In this section, the effects of initial static bed height on the fractional conversion and bed pressure drop in both the tapered-in and tapered-out fluidized bed reactors are investigated. In this regard, five initial static bed heights, i.e., 6.0, 8.0, 10.0, 12.5, and 15.0 cm, and four initial static bed heights, i.e., 3.0, 4.0, 5.0, and 6.0 cm, were considered for the tapered-in and tapered-out fluidized

M

bed reactors, respectively. Figures 5(a) and 6(a) show the outlet mole percent of CO and CH3OH

ED

versus initial static bed height for reactions (1) and (2) in the tapered-in fluidized bed reactors, respectively. Figures 7(a) and 8(a) show the outlet mole percent of CO and CH3OH versus initial

PT

static bed height for reactions (1) and (2) in the tapered-out fluidized bed reactors, respectively. In addition, Figs. 5(b) and 6(b) show the bed pressure drop versus initial static bed height for reactions

CE

(1) and (2) in the tapered-in fluidized bed reactors, respectively. Figures 7(b) and 8(b) also show the

AC

bed pressure drop versus initial static bed height for reactions (1) and (2) in the tapered-out fluidized bed reactors, respectively. Moreover, it should be noted that all the simulation results in Figs. 5 to 8 are time- and area-averaged.

4.3.1. Fractional Conversion in Tapered-in Reactors It can be observed from Fig. 5(a) that for small static bed heights, the outlet mole percent of CO increases with increasing initial static bed height. Because of the solid particles and their

ACCEPTED MANUSCRIPT continuous movement in the fluidized bed reactor, the contacts between solid and gas phases increase that result in increasing the gas phase turbulence. As a result, the intensity of mass transfer in the gas phase and the degree of fractional conversion would increase. Figures 9(a), (b) and (c) respectively shows the variations of time- and area-averaged laminar, turbulent and total granular temperatures versus column height, which are closely related to those shown in Fig. 5(a). The

CR IP T

laminar granular temperature demonstrates the random oscillations of individual solid particles, whereas the turbulent granular temperature demonstrates the oscillations of clusters of solid particles or bubbles. The sum of the laminar and turbulent granular temperatures is called total granular temperature; therefore, total granular temperature shows the overall system oscillations.

AN US

Overall comparison of Fig. 9(c) at different static bed heights shows that increasing static bed height boosts the total granular temperature; hence, it verifies that with increasing static bed height in tapered-in reactors, the intensity of gas phase turbulence would be increased. Therefore, with increasing initial static bed height within a reasonable range, the degree of fractional conversion

M

would be increased.

ED

It can be also observed from Fig. 5(a) that the outlet mole percent of CO would decrease after it reaches a maximum value at a specified static bed height. A similar trend to what explained

PT

for the outlet mole percent of CO in Fig. 5(a) can be also observed for the outlet mole percent of

CE

CH3OH in Fig. 6(a). Moreover, the comparison of Figs. 5(a) and 6(a) clearly shows that the appropriate initial static bed height required to attain the maximum fractional conversion would

AC

vary with reaction type. Figure 10 shows the variations of gas phase temperature versus column height as a function of initial static bed heights, which are closely related to those shown in Fig. 5(a). Comparison of the gas phase temperature profiles of Fig. 10 for various H0 , i.e., 8.0, 10.0, and 12.5 cm shows that the increasing initial static bed height reduces the distribution of gas phase temperature along the bed height. Existence of solid particles in the bottom section of tapered-in fluidized bed reactors creates a low temperature region. Increasing static bed height results in more expanding of low-temperature region, and this is due to more expansion of solid particles. The

ACCEPTED MANUSCRIPT expansion of low-temperature region in the tapered-in fluidized beds has negative effects on the fractional conversion. However, if the initial static bed height exceeds a specific value, the negative effects of low-temperature region dominate the positive effects of solid particles, which result in increasing the turbulence and intensity of mass transfer of gas phase, consequently, the outlet mole percent of CO would be reduced.

CR IP T

It can be observed from Fig. 3(a) for grid number of 40×145 that the slope of CO mole fraction profile in the upper section of tapered-in fluidized beds is more than that in the lower section, and this phenomenon represents that the main portion of fractional conversion occurred in the upper section of fluidized bed. It is also observed from Fig. 10 that the slope of gas phase

AN US

temperature profiles in the upper section of tapered-in fluidized bed reactors is more than that in the lower section and this phenomenon agrees well with what explained for CO mole fraction profile shown in Fig. 3(a). Because the gas phase velocity in the upper section of tapered-in fluidized beds

M

is less than that in lower section, the gas phase reactants have more time to react and convert to the products. Therefore, increasing the residence time of the gas phase in the upper section of tapered-

ED

in fluidized bed reactors would result in increasing the degree of fractional conversion. Similar explanations to what mentioned for reaction (1) in Figs. 3(a), 5(a), 9(a-c) and 10 are also validate

CE

PT

for reaction (2) in the tapered-in fluidized bed reactors.

4.3.2. Fractional Conversion in Tapered-out Reactors

AC

Figure 7(a) shows that the increasing initial static bed height has slight influence on the outlet mole percent of CO in the tapered-out fluidized beds. A similar trend can be also observed for the outlet mole percent of CH3OH in Fig. 8(a). Figure 11 shows the variations of gas phase temperature versus column height for three initial static bed heights, which are related to those shown in Fig. 7(a). The comparison of the gas phase temperature profiles presented in Fig. 11 for the cases of H0=3.0, 4.0, and 5.0 cm shows that the increasing initial static bed height has no appreciable effect on the gas phase temperature profiles. As a result of insignificant changes of gas

ACCEPTED MANUSCRIPT phase temperature profiles at different initial static bed heights, the degree of fractional conversion in tapered-out fluidized bed reactors remains approximately constant. Similar explanations to what mentioned for reaction (1) in Figs. 7(a) and 11 can be given for reaction (2) in the tapered-out fluidized bed reactors.

4.3.3. Bed Pressure Drop in Tapered-in and Tapered-out Reactors

CR IP T

It can be observed from part (b) of Figs. 5 to 8 that increasing initial static bed height increases the bed pressure drop in both tapered-in and tapered-out fluidized bed reactors. The measured pressure drop in fluidized bed reactors is classified into two different parts. The first part

AN US

is the unrecoverable or frictional pressure drop and the second one is the static pressure drop. Thus the total pressure drop in fluidized beds is sum of frictional and static pressure drops. The frictional pressure drop due to frictional forces between the gas and solid phases is an unrecoverable energy loss, whereas the static pressure drop is due to changes in the static head of the gas phase. Because

M

of low density of the gas phase in gas-solid fluidized beds, the static pressure drop can be negligible

ED

in comparison with the frictional pressure drop and the main portion of total pressure drop is due to frictional pressure drop. Therefore, with an increase in the initial static bed height the contact

PT

between solid and gas phases increases, hence, the frictional pressure drop and total pressure drop

CE

of tapered fluidized bed reactors increase.

5. Conclusions

AC

A multitude of studies has been conducted on the tapered fluidized beds over the past few

decades. However, most of the studies were concerned with the experimental investigation of hydrodynamic behavior and little attentions have been paid to the behavior of tapered fluidized beds in the presence of chemical reactions. In addition, most of the hydrodynamic studies focused on the tapered-in fluidized beds and tapered-out fluidized beds have received little attentions. Therefore, in the present study, 2D simulations of chemical reactions with gas volume reduction and increase were carried out in both the tapered-in and tapered-out fluidized bed reactors to investigate the

ACCEPTED MANUSCRIPT effects of initial static bed height on the fractional conversion and bed pressure drop. Moreover, to validate the model predictions, the results of the hydrodynamic simulations regarding bed pressure drop and bed expansion ratio were compared against experimental data reported in the literature. On the basis of the present simulation results, the following conclusions can be drawn: a) For small static bed heights in tapered-in fluidized bed reactors, the fractional conversion

CR IP T

increases with increasing initial static bed height. With further increase in the static bed height, the fractional conversion reaches a maximum value and then decreases.

b) The appropriate initial static bed height to reach the maximum fractional conversion depends on the reaction type.

AN US

c) Expansion of the low-temperature region in the tapered-in fluidized-bed reactors has negative effects on the fractional conversion.

d) Increasing the gas phase residence time in the upper section of tapered-in fluidized bed reactors would increase the degree of fractional conversion.

ED

degree of fractional conversion.

M

e) Increasing the initial static bed height in tapered-out reactors has slight influences on the

f) With an increase in the initial static bed height of both the tapered-in and tapered-out

PT

reactors, the bed pressure drop increases. g) The residence time, temperature, and the intensity of turbulence of the gas phase are three

CE

important factors affecting the fractional conversion in tapered fluidized bed reactors.

AC

Acknowledgment

The present authors would like to thank Sharif University of Technology (Tehran, Iran).

ACCEPTED MANUSCRIPT

References

Acosta-Iborra, A., Hernández-Jiménez, F., de Vega, M., Briongos, J.V., 2012. A novel methodology for simulating vibrated fluidized beds using two-fluid models, Chem. Eng. J. 198-199, 261-274. Almuttahar, A., Taghipour, F., 2008. Computational fluid dynamics of high density circulating fluidized bed riser: study of modeling parameters, Powder Technol. 185, 11-23. ANSYS Inc., ANSYS Fluent Theory Guide (2011).

CR IP T

Arastoopour, H., 2001. Numerical simulation and experimental analysis of gas–solid flow systems: 1999 Fluor-Daniel Plenary lecture, Powder Technol. 119, 59–67. Benyahia, S., Syamlal, M., O'Brien, T.J., 2007. Study of the ability of multiphase continuum models to predict core-annulus flow, AIChE J. 53, 2549-2568.

AN US

Bi, H.T., Ellis, N., Abba, I.A., Grace, J.R., 2000. A state-of-the-art review of gas–solid turbulent fluidization, Chem. Eng. Sci. 55, 4789–4825. Biswal, K.C., Sahu, S., Roy, G.K., 1982. Prediction of the fluctuation ratio for gas-solid fluidization of regular particles in a conical vessel, Chem. Eng. J. 23, 97-100. Chapman, S., Cowling, T.G., 1970. The mathematical theory of non-uniform gases, Cambridge University Press.

M

Chalermsinsuwan, B., Piumsomboon, P., Gidaspow, D., 2009. Kinetic theory based computation of PSRI riser: part I- Estimate of mass transfer coefficient, Chem. Eng. Sci. 64. 1195-1211.

ED

Chalermsinsuwan, B., Gidaspow, D., Piumsomboon, P., 2011. Two- and three-dimensional CFD modeling of Geldart A particles in a thin bubbling fluidized bed: Comparison of turbulence and dispersion coefficients, Chem. Eng. J. 171, 301-313.

PT

Cloete, S., Zaabout, A., Johansen, S.T., Annaland, M.V.S., Gallucci, F., Amini, S., 2013. The generality of the standard 2D TFM approach in predicting bubbling fluidized bed hydrodynamics, Powder Technol. 235, 735-746.

CE

Enwald, H., Almstedt, A.E., 1999. Fluid dynamics of a pressurized fluidized bed: comparison between numerical solutions from two-fluid models and experimental results, Chem. Eng. Sci. 54, 329-342.

AC

Ergun, S., 1952. Fluid flow through packed columns, Chem. Eng. Prog. 48, 89-94. Gan, L., Lu, X., Wang, Q., 2014. Experimental and theoretical study on hydrodynamic characteristics of tapered fluidized beds, Adv. Powder Technol. 25, 824-831. Gidaspow, D., 1994. Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. Academic Press, San Diego. Hamzehei, M., Rahimzadeh, H., 2010. Numerical and experimental investigation of a fluidized bed chamber hydrodynamics with heat transfer, Korean J. Chem. Eng. 27, 355-363.

ACCEPTED MANUSCRIPT Huilin, L., Yunhua, Z., Zhiheng, S., Ding, J., Jiying, J., 2006. Numerical simulations of gas-solid flow in tapered risers, Powder Technol. 169, 89-98. Johnson, P.C., Jackson, R., 1987. Frictional-collisional constitutive relations for granular materials, with application to plane shearing, J. Fluid Mech. 176, 67-93. Johansson, K., van Wachem, B.G.M., Almstedt, A.E., 2006. Experimental validation of CFD models for fluidized beds: Influence of particle stress models, gas phase compressibility and air inflow models, Chem. Eng. Sci. 61, 1705-1717.

CR IP T

Kim, H.G., Lee, I.O., Chung, U.C., Kim, Y.H., 2000. Fluidization characteristics of iron ore fines of wide size distribution in a cold tapered gas-solid fluidized bed, ISIJ Int. 40, 16–22. Lu, H., Wang, S., He, Y., Ding, J., Liu, G., Hao, Z., 2008. Numerical simulation of flow behaviour of particles and clusters in riser using two granular temperatures, Powder Technol. 182, 282-293. Lun, C.K.K., Savage, S.B., Jeffrey, D.J., Chepurniy, N., 1984. Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field, J. Fluid Mech. 140, 223-256.

AN US

Orszag, S.A., Yakhot, V., Flannery, W.S., Boysan, F., Choudhury, D., Maruzewski, J., Patel, B., 1993. Renormalization Group Modeling and Turbulence Simulations, International Conference on Near-Wall Turbulent Flows, Tempe, Arizona. Olazer, M., San Jose, M.J., Aguayo, A.T., Arandes, J.M., Bilbao, J., 1993. Pressure drop in conical spouted beds. Chem. Eng. J. 51, 53–60.

M

Peng, Y., Fan, L.T., 1997. Hydrodynamic characteristics of fluidization in liquid-solid tapered beds. Chem. Eng. Sci. 52, 2277-2290.

ED

Pain, C.C., Mansoorzadeh, S., de Oliveira, C.R.E, 2001. A study of bubbling and slugging fluidised beds using the two-fluid granular temperature model, Int. J. Multiphase Flow 27, 527–551.

PT

Sau, D.C., Mohanty, S., Biswal, K.C., 2010. Experimental studies and empirical models for the prediction of bed expansion in gas-solid tapered fluidized beds, Chem. Eng. Process. 49, 418-424.

CE

Sau, D.C., Mohanty, S., Biswal, K.C., 2008a. Correlations for critical fluidization velocity and maximum bed pressure drop for heterogeneous binary mixture of irregular particles in gas-solid tapered fluidized beds, Chem. Eng. Process. 47, 2386-2390.

AC

Sau, D.C., Mohanty, S., Biswal, K.C., 2008b. Critical fluidization velocities and maximum bed pressure drops of homogeneous binary mixture of irregular particles in gas-solid tapered fluidized beds, Powder Technol. 186, 241-246. Shi, Y.F., Yu, Y.S., Fan, L.T., 1984. Incipient fluidization condition for a tapered fluidized bed, Ind. Eng. Chem. Fund. 23, 484-489. Schaafsma, S.H., Marx, T., Hoffmann, A.C., 2006. Investigation of the particle flow pattern and segregation in tapered fluidized bed granulators, Chem. Eng. Sci. 61, 4467–4475. Taghipour, F., Ellis, N., Wong, C., 2005. Experimental and computational study of gas-solid fluidized bed hydrodynamics, Chem. Eng. Sci. 60, 6857-6867. Sau, D.C.; Biswal, K.C., 2011. Computational fluid dynamics and experimental study of the hydrodynamics of a gas-solid tapered fluidized bed, Appl. Math. Model. 35, 2265-2278.

ACCEPTED MANUSCRIPT Thonglimp, V., Hiquily, N., Laguerie, C., 1984. Vitesse minimale de fluidization et expansion des couches de mélanges de particules solids fluidisees par ungaz, Powder Technol. 39, 223–239. van Wachem, B.G.M., Almstedt, A.E., 2003. Methods for multiphase computational fluid dynamics, Chem. Eng. J. 96, 81-98. Wang, Z., 2006. Experimental studies and CFD simulations of conical spouted bed hydrodynamics, Ph.D. Thesis, The University of British Columbia.

AC

CE

PT

ED

M

AN US

CR IP T

Wen, C.Y., Yu, Y.H., 1966. Mechanics of fluidization, Chemical Engineering Progress Symposium Series 62, 100-111.

ACCEPTED MANUSCRIPT

Table Captions

Table 1. Simulation parameters and chemical reaction rate constants used in the CFD simulation of tapered-

AC

CE

PT

ED

M

AN US

CR IP T

in and tapered-out fluidized bed reactors.

ACCEPTED MANUSCRIPT

M

AN US

CR IP T

Figures Captions

ED

Fig. 1. Schematic diagram of the problem and initial conditions used.

AC

CE

PT

(a) tapered-in; and (b) tapered-out fluidized bed reactors.

ACCEPTED MANUSCRIPT Fig. 2. Comparison of simulation results against experimental data.

AN US

CR IP T

(a) Bed pressure drop; and (b) Bed expansion ratio in a tapered-in fluidized bed; H0=6.5 cm.

AC

CE

PT

ED

in fluidized bed reactor, H0=10 cm.

M

Fig. 3. Variations of mole fractions of (a) CO; and (b) CH4 vs. column height for reaction (1) in the tapered-

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

ED

Fig. 4. Variations of mole fraction of CO vs. column height for reaction (1) in the tapered-out fluidized bed

AC

CE

PT

reactor, H0=1.9 cm.

ACCEPTED MANUSCRIPT Fig. 5. (a) Outlet mole percent of CO; and (b) bed pressure drop vs. initial static bed height for reaction (1)

AN US

CR IP T

in the tapered-in fluidized bed reactors.

AC

CE

PT

ED

(2) in the tapered-in fluidized bed reactors.

M

Fig. 6. (a) Outlet mole percent of CH3OH; and (b) bed pressure drop vs. initial static bed height for reaction

CR IP T

ACCEPTED MANUSCRIPT

Fig. 7. (a) Outlet mole percent of CO; and (b) bed pressure drop vs. initial static bed height for reaction (1)

AC

CE

PT

ED

M

AN US

in the tapered-out fluidized bed reactors.

Fig. 8. (a) Outlet mole percent of CH3OH; and (b) bed pressure drop vs. initial static bed height for reaction (2) in the tapered-out fluidized bed reactors.

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 9. Variations of (a) laminar granular temperature; (b) turbulent granular temperature; and (c) total

AC

CE

granular temperature vs. column height for reaction (1) in the tapered-in fluidized bed reactors.

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

ED

Fig. 10. Variations of gas phase temperature vs. column height for reaction (1) in the tapered-in fluidized

AC

CE

PT

bed reactors.

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 11. Variations of gas phase temperature vs. column height for reaction (1) in the tapered-out fluidized

AC

CE

PT

ED

bed reactors.

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

Graphical Abstract

ACCEPTED MANUSCRIPT Table 1 Simulation parameters and chemical reaction rate constants used in the CFD simulation of tapered-in and tapered-out fluidized bed reactors. Value

Particle density, kg/m3 Particle size, mm

2600 2

Initial solid volume fraction (ε0) Particle-wall restitution coefficient Particle-particle restitution coefficient Specularity coefficient Bed height, m Inlet diameter of tapered-in reactor, m Inlet diameter of tapered-out reactor, m Tapered angle,  Superficial gas velocity, m/s Inlet gas temperatures (reactions (1) and (2)), K Maximum packing limit (εs,max) Angle of internal friction,  Critical solid volume fraction (Cloete, 2013) Turbulent Schmidt number (ANSYS Inc. ,2011) Activation energies ( Ea and Ea ), J/kmol Pre-exponential factors ( k and k  ) Inlet mole fractions of CH4, O2 and N2 (reaction (1)) Inlet mole fractions of CO2 and H2 (reaction (2))

0.48 0.9 0.9 0.5 0.52 0.05 0.135 4.6 4.0 600, 550 0.63 30 0.55 0.7 1.345108, 1.39108 2.1191011, 9.51013 0.5, 0.35, 0.15 0.3, 0.7

AC

CE

PT

ED

M

AN US

CR IP T

Parameter