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CFD Study of hydrodynamic and heat transfer in a 2D spouted bed: Assessment of radial distribution function
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Journal of the Taiwan Institute of Chemical Engineers 000 (2015) 1–10
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CFD Study of hydrodynamic and heat transfer in a 2D spouted bed: Assessment of radial distribution function Seyyed Hossein Hosseini a,∗, Mohsen Fattahi b, Goodarz Ahmadi c a
Department of Chemical Engineering, Ilam University, Ilam 69315–516, Iran Department of Chemical Engineering, Faculty of Engineering, University of Kurdistan, Sanandaj 66177, Iran c Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699-5725, USA b
a r t i c l e
i n f o
Article history: Received 4 March 2015 Revised 9 June 2015 Accepted 21 June 2015 Available online xxx Keywords: CFD Spouted bed Transient heat transfer Hydrodynamic Radial distribution function
a b s t r a c t The hydrodynamic and transient gas to particle heat transfer for spouted regime was studied using an Eulerian–Eulerian two-fluid model (TFM) including the kinetic theory of granular flows (KTGF). The influence of different expressions for the radial distribution function (RDF) on the CFD results of the spouted beds was evaluated. Detailed analyses of the CFD results showed that the RDF affects the CFD results especially in dense regions on both particle velocity and temperature distributions. Moreover, the best agreement with the experimental data was obtained by the RDF of Lun and Savage for the region near the wall and by RDF of Ma and Ahmadi, as well as, Arastoopour in the fountain region. © 2015 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
1. Introduction The spouted bed was first developed by Gishler [1] as an alternative drying method of moist wheat particles instead of a slugging fluidized bed. A spouted bed typically displays three distinct regions that are: spout, annulus, and fountain. The spouted beds are widely used in various industrial applications such as the drying of grains, spray drying, coating, heterogeneous catalysis, and gasification of biomass and coal [2–7]. Rapid increase of computational hardware capacity, recent advances in numerical algorithms, and deeper understanding of physics of multiphase flows, computational fluid dynamics (CFD) has emerged as an effective tool for predicting the behavior of gas–solid flows in the recent years. The main advantage of CFD modeling is that a wide range of the gas and solid flow properties may be studied rather economically [8]. Including the heat transfer process in the CFD simulations, the thermal behavior in addition to the hydrodynamics of two-phase systems can be analyzed. In many applications of spouted beds such as drying technology, chemical reactors and biological processes, understanding of thermal behavior is important, and some cases, the temperature distribution in the bed need to be controlled. Therefore, for effective usage of the
∗
Corresponding author: Tel: +98 913 7944470. E-mail address: [email protected], [email protected] (S.H. Hosseini).
spouted beds, knowledge of the temperature distribution in the bed is essential. Despite the importance of knowledge of particle temperature distribution in the spouted beds, there are rather few CFD studies on the thermal effects were reported in the literature. This issue is exasperated due to the lack of controlled experimental data on thermal behavior of multiphase flows. Several researchers have studied the wall to bed heat transfer in gas–solid bubbling fluidized beds using the two-phase flow modeling (TFM) approach [9,10]. Other researchers have investigated the heat transfer between a bubbling fluidized bed and an immersed body by the TFM [11,12] and discrete element method (DEM) [13,14]. A few authors studied the particles temperature distribution in gas–solid systems using the CFD without quantitative comparison with the experimental data, in part due to the lack of experimental measurements [15–18]. While there have been numerous studies of the effect of drag function, particle-particle restitution coefficient, particle-wall restitution coefficient and specularity coefficient (ϕ ) on dynamics of two-phase flows, rather few works have been published on the effect of RDF on the CFD results of gas–solid systems. Only van Wachem et al. [19] and Ahuja and Patwardhan [20] showed that the simulation results of gas–solid systems are not sensitive to the use of different solid stress models and RDFs in dense flow regimes. Clearly to arrive at an accurate heat transfer model for simulation of gas–solid flows, using an accurate hydrodynamics model is essential. In the past numerous investigations were performed and the influence of the hydrodynamics parameters such as drag function
http://dx.doi.org/10.1016/j.jtice.2015.06.027 1876-1070/© 2015 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Please cite this article as: S.H. Hosseini et al., CFD Study of hydrodynamic and heat transfer in a 2D spouted bed: Assessment of radial distribution function, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.06.027
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Nomenclature CD ds Dp H W es g g0 H0 Í I2D ks
β βErgun βWen−Yu
P Ps t U
vi Umf T K Cp h Nu Re Pr
drag coefficient [-] particle diameter [m] Depth of the bed [m] Vessel height [m] width of the bed [m] particle–particle restitution coefficient [-] acceleration due to gravity [m/s2 ] radial distribution coefficient [-] Static bed depth [mm] stress tensor [-] second invariant of the deviatoric stress tensor [-] diffusion coefficient for granular energy [kg/m s] gas/solid momentum exchange coefficient [kg/m3 s] gas/solid momentum exchange coefficient by Ergun equation [kg/m3 s] gas/solid momentum exchange coefficient calculated by Wen-Yu equation[kg/m3 s] pressure [N/m2 ] solids pressure [N/m2 ] time [s] superficial gas velocity [m/s] velocity [m/s] minimum fluidization velocity [m/s] Temperature [K] thermal conductivity [w/m.k] specific heat capacity [J/Kg.K] Enthalpy [J/Kg] Nusselt number [-] Reynolds number [-] Prandtl number [-]
Greek letters αi volume fraction [-] γs the collisional dissipation of energy [kg/s3 m] s granular temperature [m2 /s2 ] λs solid bulk viscosity [kg/m s] μi shear viscosity [kg/m s] ρi density [kg/m3 ] τi stress tensor [N/m2 ] φ angle of internal friction [deg] ϕ specularity coefficient [-] Subscripts col collision fr friction kin kinetic g gas p particle q phase type (solid or gas) s solids T stress tensor
[21–25], particle viscosity model [24,25], particle–particle restitution coefficient [24–26], particle-wall restitution coefficient, and specularity coefficient [27–29] on the CFD results of the spouted beds were investigated. In the present study, the optimized values of hydrodynamics parameters that were obtained in our earlier work [30] were used and the effect of the RDF on the CFD results for the 2D spouted beds was evaluated. Note that the Gidaspow drag function, which is widely used in gas–solid systems, was used in the current CFD model. The simulation results for the hydrodynamic and heat transfer in the spouted bed were compared with the experimental data of Brown
and Lattimer [31]. Comments regarding the accuracy of different radial distribution model were also provided. 2. CFD modeling In this work, an Eulerian–Eulerian two-fluid model is utilized for predicting the transient heat transfer in the pseudo two-dimensional spouted bed and for the particle flow and temperature distributions. A commercially available ANSYS-FLUENT code is used for these simulations. The heat transfer model and Lun and Savage’s RDF [32] are incorporated in the code using the User’s Defined Function (UDF). 2.1. Governing equations The governing equations of balance and the associated constitutive models of the Eulerian–Eulerian TFM that are used in the simulation of spouted beds are summarized in this section. It is assumed that the solid-phase is made of spherical granular particles with a fixed diameter. The kinetic theory of granular flow was used in the model. The virtual mass and lift effects, which are expected to be small, are neglected. The gas–solid interphase exchange coefficient, β , was modeled using the Gidaspow and coauthors [33] drag model. The solid shear viscosity is composed of collisional, kinetic, and frictional components. For dense flows when the solid volume fraction approaches the packing limit, the frictional viscosity becomes quite important, as the friction between particles generates large stresses. Here the expression suggested by Johnson and Jackson [34] is used for the frictional viscosity in dense cases. The bulk viscosity and the solids pressure are calculated using the expressions suggested by [35]. The continuity equation for the qth phase without any mass transfer between the phases is given by:
∂ (α ρ ) + ∇ .(αq ρq υ q ) = 0 ∂t q q
(1)
where αq , ρq and vq , respectively, are the volume fraction, density and velocity of the qth phase. The conservation of momentum for the gas and solid phases are given as: Gas phase:
∂ (α ρ υ ) + ∇ .(αg ρg υ g υ g ) ∂t g g g g ) + αg ρg g s − υ = − αg ∇ p + ∇ .τ g + β(υ
(2)
Solid phase:
∂ (α ρ υ ) + ∇ .(αs ρs υ s υ s ) ∂t s s s g − υ s ) + αs ρs g = − αs ∇ p − ∇ ps + ∇ .τ s + β(υ
(3)
where αs = 1 − αg . The conservation of energy for the gas and solid phases are given as:Gas phase:
∂ (α ρ H ) + ∇ .(αg ρg υ g Hg ) = ∇ .αg Kg,e f f ∇ .Tg − hgs (Ts − Tq ) ∂t g g g
(4)
Solid phase:
∂ (α ρ H ) + ∇ .(αs ρs υ s Hs ) = ∇ .αs Ks,e f f ∇ .Ts + hsg (Ts − Tq ) ∂t s s s
(5)
The transport equation for granular temperature, s (fluctuation kinetic energy of particles), is given as:
3 2
∂ (ρs αs s ) + ∇ .(αs ρs υ s s ) ∂t s + ∇ .(ks ∇ s ) − γs − 3β s = −∇ ps I¯ + τ g :∇ υ
(6)
Please cite this article as: S.H. Hosseini et al., CFD Study of hydrodynamic and heat transfer in a 2D spouted bed: Assessment of radial distribution function, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.06.027
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2.2. The constitutive equations for hydrodynamic
Solid pressure [35]:
Ps = αs ρs s +2ρs (1 + es )αs2 g0 s
Solid and gas phase stress tensors:
2 τ s = αs μs ∇ υ s + (∇ υ s )T + (αs λs − αs μs )∇ .υs I¯
(7)
2 ∇ υ g + (∇ υ g )T − ∇ .υg I¯
(8)
3
τ g = αg μg
3
s+d p s
(9)
1 3αs αs + + (1 − αs ) 2(1 − αs )2 2(1 − αs )3 2
(10)
Note that in this model, g0 is independent of the maximum packing and when maximum packing is approached, g0 remains finite. Ogawa et al. [38] proposed a model for the RDF, which has been widely used by the researchers for different gas–solid systems. That is,
g0 = 1 −
1/3 −1
αs
(11)
αs,max
Additional expressions for the RDF that are examined in the present study are: Lun and Savage [32]
g0 = 1 −
g0 =
3 1− 5
(12)
αs αs,max
1/3 −1 (13)
1 + 2.5αs + 4.59α + 4.52α
αs 1 − αs,max
2 s
3 0.678
3 s
(14)
The Ma and Ahmadi expression for the RDF is based on the virial series for dense gases and has been widely used for CFD simulation of bubbling gas–solid fluidized beds [40–43]. Iddir and Arastoopour [44]:
g0 = 1 −
αs αs,max
−1
(15)
Collisional energy dissipation: The energy dissipation due to inter-particle collisions is given as
12(1 − es )g0 γs = ρs αs2 3/2 √ s ds π 2
(16)
12
2 π s 4 1+ αs g0,ss (1+ess ) 5 96(1+ess )g0,ss
(18)
Frictional viscosity of Johnson and Jackson [34]:
μs, f ric = F r.
(αs − αs,min )n . sin φ (αs, max − αs ) p
(19)
where Fr, n and p are empirical model parameters. The diffusivity of granular temperature [45]:
ks =
2 π s 6 1+ αs g0 (1+es ) 5 384(1+es )g0 12 s + 2αs ρs ds g0 (1+es ) π 150ρs ds
(20)
Solid bulk viscosity [35]:
4 λ = αs ρs ds g0 (1+es ) 3
s π
(21)
Gas–solid drag coefficient of Gidaspow and co-workers [33]:
β =(1−ϕgs )βErgun +ϕgs βWen−Yu
(22)
where
→ − αs ρg − υs−→ υ g αs2 μg βErgun = 150 +1.75 f or αg ≤ 0.8 ds αg ds2 − → − → 3 αs αg ρg υ s − υ g −2.65 βWen−Yu = CD αg f or αg > 0.8 4
CD =
ϕgs =
ds
24 αg Res 1 + 0.15(αg Res )
0.687
CD = 0.44, Arctan 150 × 1.75(0.2−αs )
π
,
Res < 1000 Res ≥ 1000
+0.5
Constitutive equations for the thermal energy balance Interphase heat transfer The heat transfer coefficient is related to the Nusselt number, Nus , using the following:
hsg =
Ma and Ahmadi [39]:
g0 =
+
10ds ρs
Gidaspow and Huilin:
4 s = αs ρs ds g0 (1+ess ) 5 π
−2.5αs,max
αs αs,max
μs
where s is the distance between grains. From Eq. (9) it follows that for a dilute solid phase s → ∞, and therefore g0 → 1. In the limit when the solid phase compacts, s → 0 and g0 → ∞. The RDF is closely connected to the factor χ of Chapman and Cowling’s [36] theory of nonuniform gases. χ is equal to 1 for a rarefied gases, and increases and tends to infinity when the molecules are too close so that their motions are not possible. There has been several suggested expression for the RDF in the literature. Carnahan and Stirling [37] obtained a correlation for the RDF given as:
g0 =
(17)
Solid shear viscosity as given by Gidaspow et al. [45]:
Radial distribution function models The RDF, g0 , is a correction that modifies the probability of collisions between grains when the solid granular phase becomes dense. This function may also be interpreted as the non-dimensional distance between grains:
g0 =
3
6kg αs αg Nus ds2
(23)
An empirical relation for the interphase heat transfer coefficient was proposed by Gunn [46] which relates the Nusselt number with the particle Reynolds and Prandtl numbers. That is,
Nus = 7 − 10αg +5α 2g
1
1 + 0.7(Re p )2 (Pr) 3
1
+ 1.33 − 2.4αg +1.2α 2g (Re p )0.2 (Pr) 3 where Pr
(24)
C μ = pqk q . q
Several successful simulation results of heat transfer for gas–solid fluidized beds using Gunn [46] model have been reported in the literature [9–12,15,18]. Effective conductivities of the gas and solids phases For description of the heat-transfer rate in the two-fluid continuum model, it the overall bulk thermal conductivity is decomposed
Please cite this article as: S.H. Hosseini et al., CFD Study of hydrodynamic and heat transfer in a 2D spouted bed: Assessment of radial distribution function, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.06.027
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S.H. Hosseini et al. / Journal of the Taiwan Institute of Chemical Engineers 000 (2015) 1–10 Table 1 Physical and numerical parameters.
Fig. 1. Transient inlet air temperature ramp for the spouted regime [30].
into the effective thermal conductivities of the gas and the solids phase [9]. For the gas phase, the effective thermal conductivity is given by [9]:
Kg,e f f =
√ 1− αs Kg
(25)
αg
The thermal conductivity of the particulate phase differs from the particle material and depends on the contacts of the particles in the fluidized bed. The effective thermal conductivity of the solids phase can be expressed as:
1 Ks,e f f = √ kg ωA + (1 − ω)
(26)
αs
where
=
A (B − 1) (B + 1) (A − 1) B ln B − B − 2 B B 2 A
2
1− A
1− A
1− A
and for spherical particles
Ks A= , Kg
αs B = 1.25 αg
10/9 ,
ω = 7.26 × 10−3
In Eq. (26) ω represents the ratio of the particle contact area to the total particle surface area. In experiments conducted by Brown and Lattimer [31], the inlet gas temperature is not constant and increases with time as shown in Fig. 1. This temperature variation of the inlet gas is implemented in the ANSYS-FLUENT code by means of the user-defined functions (UDF) written in C-format. 2.2. Simulation conditions The corresponding physical and numerical parameters selected for the present simulation are listed in Table 1. 2.3. Simulation procedure A three-dimensional modeling is used in the present study. Due to the symmetry behavior of the particles half of the bed is selected for simulations. As noted before, the ANSYS-FLUENT CFD code is utilized for simulating the hydrodynamics and heat transfer of the fluidized bed studied experimentally by Brown and Lattimer [31]. The set of governing equations described in Section 2.1 is solved by the finite control volume technique. The phase coupled SIMPLE algorithm,
Symbol
Description
Simulation
ds (μm) ρs (Kg/m3 ) ρg (Kg/m3 ) μg (Kg/m.s) Umf (m/s) Ug,s φs (-) H0 (mm) φ (-) es (-) ew (-) αs,max (−) ϕ (-) H (mm) W (mm) D (mm) Ks (W/m.k) Cp,s (KJ/Kg.k) Cp,g (KJ/Kg.k)
Particle diameter Particle density Gas density Gas viscosity Minimum fluidization velocity Superficial velocity in spouted regime Sphericity Static bed depth Internal friction angle of particles Geldart group Particle–particle restitution coefficient Wall to particle restitution coefficient Maximum volume fraction of particles Specularity coefficient Bed height Bed width Bed depth Thermal conductivity of solids Specific heat capacity of solids Specific heat capacity of gas
550 2500 1.22 1.789 × 10−5 0.24 3Umf 1 49.75 28.5 B 0.8 0.2 0.62 0.025 280 56.4 4.95 1.05 0.84 1.006
which is an extension of the SIMPLE algorithm for multiphase flow, is used for the pressure–velocity coupling and correction. The momentum and volume fraction equations are discretized by a first-order upwind scheme. The adaptive time step in the range of 0.00001– 0.0005 is used with about 100 iterations per time step. A convergence criterion of 10−3 for each scaled residual component is specified for the relative error between two successive iterations. In this study the configuration and the operating conditions of the spouted bed studied experimentally by Brown and Lattimer [31] are used in the simulations. The corresponding pseudo two-dimensional fluidized bed is shown schematically in Fig. 2. The base was made of a stainless steel distributor plate with a single slit jet of 1.6 mm wide and the full depth of the bed. The outside of the plate was coated with a low thermal conductivity silicon rubber insulation to reduce thermal energy loss. The back and side walls consisted of polymethyl– methacrylate coated with paint to ensure surface flatness. The top of the distributor plate was covered by a fine mesh screen and the bed was filled with glass particles. The Geldart B classified particle type with mean diameter of 550 μm was used. It is worth mentioning that Brown and Lattimer [31] used the infrared thermography technique to measure the heat transfer in the entire bed. In this method the instantaneous particle temperatures can be evaluated by measuring the infrared emitted thermal radiation. The main advantage of this approach is that it provides nonintrusively temperature data for gas–solid flows. This method is not limited to steady-state condition and the detailed localized and timedependent data can be measured. Such experimental data is invaluable for validation of CFD and heat transfer models. The grid structures for the system under study are illustrated in Fig. 3. The mesh size is about 1 mm near the lateral bed wall, 0.2 mm at the center of the spout section, and varied along the axial direction of the bed from 0.5 to 3 mm. 2.5. Initial and boundary conditions A Dirichlet boundary condition at the bottom of the bed is used to specify a uniform gas inlet velocity. The outflow condition with zero velocity gradients is specified at the top of the freeboard. A noslip boundary condition for all walls is assumed for the gas phase. The particle normal velocity is set to zero at the wall. The Johnson and Jackson [34] wall boundary condition is used for the tangential particle velocity, and granular temperature of the solid phase at the wall. As mentioned before, the inlet air temperature varies roughly linearly with time. The walls are assumed adiabatic. Initially, the
Please cite this article as: S.H. Hosseini et al., CFD Study of hydrodynamic and heat transfer in a 2D spouted bed: Assessment of radial distribution function, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.06.027
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Fig. 3. Grid structures for the computational domain.
Fig. 2. Schematic of the pseudo 2D flat bottom rectangular column setup with dimensions in mm [30]. Table 2 Initial conditions. Description
Initial value
Gas temperature (K) Particle temperature (K) Static bed depth (mm) Solid volume fraction (-) Gas and particles velocities (m/s) Gas volume fraction in freeboard (-)
296 296 49.75 0.619 0 1
particle concentration in the spouted bed is specified, and gas velocity inside the spouted bed is set to zero. The particle concentration in the freeboard region is also set to zero. The initial conditions selected for the present simulation are listed in Table 2. 3. Results and discussion 3.1. Quantitative analysis of the radial distribution function expressions Eq. (18) shows that the solid viscosity depends on the radial distribution function. For α s,max = 0.62, which is the experimental finding
Fig. 4. Comparison of various forms of radial distribution functions.
of Brown and Lattimer [31], variations of different expressions for the redial distribution function as a function of solid volume fraction are plotted in Fig. 4. It can be seen that at the close packing the correlation of Carnahan and Stirling [37] unlike other expressions does not tend to the correct limit. As noted before, at the maximum solid packing, particles are in contact and the RDF tends to infinity. As the solid volume fraction approaches zero, all expressions except for Gidaspow correlation reach to the correct limit of one. As the solids volume fraction increases to more than 0.55 and get closer to the close packing, some differences between different expressions appear. Fig. 4 also shows that the RDF of Ma and Ahmadi [39] and Arastoopour [44] are very close. When the particle volume fraction is higher than 0.55, the expression suggested by Lun and Savage [32] gives the largest value compared to the others correlations. The solids volume fraction in spouted beds can vary from almost zero to the maximum packing limit, so the proper choice of RDF markedly affects the CFD results for the spouted beds.
Please cite this article as: S.H. Hosseini et al., CFD Study of hydrodynamic and heat transfer in a 2D spouted bed: Assessment of radial distribution function, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.06.027
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Fig. 5. Solids shear viscosity from different radial distribution functions.
Fig. 5 indicates the influence of the various forms of RDF on the dimensionless solid viscosity of Gidaspow et al. [45] for α s,max = 0.62 and es = 0.8. Ma and Ahmadi [39] and Arastoopour [44] correlations give nearly identical solid viscosity results. The Lun and Savage [32] correlation, however, leads to highest value for the solid shear viscosity at solid volume fraction higher than 0.55 compared to the other correlations. It is noteworthy that the higher shear viscosity values are calculated for the annulus region, where the solid volume fraction is higher than 0.55. Furthermore, an increase in solid viscosity leads to an increase in the resistance of solid particles against their downward movement. Therefore, the Lun and Savage [32] expression is expected to lead to lower particles velocity in the annulus region. It should be noted that in the spouted beds, the model needs to predict the particle behavior through the spout (dilute region) and the annulus (dense region) for accurate description of the fountain zone. 3.2. Effect of the different radial distribution function correlations on the CFD results 3.2.1. Voidage distribution As noted before, different expressions for the RDF is used to compute the solid shear viscosity that is applied in the solid phase momentum equation. Fig. 5 shows the variation of solid shear viscosity as a function of solid volume fraction. It is clearly seen that the value of solid viscosity varies depending on the RDF model used. The differences are particularly large at high solid volume fractions.
For U = 3Umf , ϕ = 0.025, es = 0.8 and ew = 0.2, Fig. 6 shows the instantaneous gas volume fraction contours for different RDF correlations. All models are for t = 5 sec, which the same as the experimental data of Brown and Lattimer [31]. In this figure the model predictions for different RDF are compared with the experimental data of Brown and Lattimer [31] in terms of fountain height and particles distribution through the bed. This figure shows that the RDF of Carnahan and Stirling [37] does not predict the proper gas volume fraction in the bed. Ma and Ahmadi and Arastoopour models predict results similar to the experiment of Brown and Lattimer [31] in terms of the fountain height and particles concentration near the top of the fountain. However, the simulation results with the RDF of Ma and Ahmadi and Arastoopour predict slightly lower particle concentrations near the wall compared those observed in the experiments. It should be noted that all boundary conditions are the same for all RDF models used. Fig. 6 also shows that the simulation results for RDFs of Ogawa et al. [38] and Lun and Savage [32] are quite similar; however, the CFD results with the use of Lun and Savage [32] expression are in a better agreement with the experimental data in term of fountain height and incoherent spouting. The Models of MA and Ahmadi and Arastoopour, however, better predict the high level of particle concentration region near the top of the fountain compared to the models of Ogawa et al. [38] and Lun and Savage [32]. The experimental contours for gas volume fraction of [31] shows that incoherent spouting occurs and also the bed surface, upper zone of annulus, is not flat but concave. This trend is predicted by the Lun and Savage [32] correlation for the RDF reasonably well. Fig. 6 also shows that dead zones predicted by all RDF expressions, except for Carnahan and Stirling [37] correlation, are roughly the same. 3.2.2. Temperature distribution The results of particle temperature distribution predicted by different RDFs are compared with the experimental observation of Brown and Lattimer [31] in Fig. 7. It is seen that the temperature differences across the bed is about 2 °C. The experimental data for the particle temperature distribution shown in Fig. 7 indicates that the highest particle temperature is not at the gas inlet zone but at certain height slightly above the gas inlet zone. Also the temperature is decreasing from spout toward the fountain base, because of increasing particle concentration. In addition, the temperature is increasing from the fountain base to the fountain top. This is in part because of the decrease in the particle concentration. Experimental result also clearly shows the dead zones near the wall surrounding the jet inlet. While the temperature contours predicted by various correlations appears to be quite different, the actual temperature field vary only slightly. All correlations shows that the particles temperature varies from the maximum value in fountain region and decrease with distance from the bed center consistent with the experiment of Brown
Fig. 6. Contour plot of voidage distribution obtained by different radial distribution functions.
Please cite this article as: S.H. Hosseini et al., CFD Study of hydrodynamic and heat transfer in a 2D spouted bed: Assessment of radial distribution function, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.06.027
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Fig. 7. Contour plot of particle temperature distribution obtained by different radial distribution functions.
and Lattimer [31] shown in Fig. 7. The minimum temperature at lower elevation occurs between the fountain and the walls, but at higher elevation occurs at the wall. Similar to the CFD results for voidage distribution, only the temperature distribution obtained by Carnahan and Stirling [41] expression differs from that observed by the experiment. The results of Ma and Ahmadi [39] and Arastoopour [44] for temperature distribution are similar. The correlations of Ogawa et al. [38] Ma and Ahmadi [39] and Arastoopour [44] for the RDF lead to better agreement with the experimental data in the fountain and around the spout regions compared with that of Lun and Savage [32]. However, these correlations over-predict the temperature in annulus region near the wall at higher elevation. The Lun and Savage [32] correlation shows that the particles temperature varies from maximum value in spout to minimum in the annulus region. Figs. 6 and 7 suggests that the RDF of Lun and Savage [32] lead to slightly better results in predicting particles temperature distribution and particles behavior in the dead zone in the bed among the other forms of RDFs. 3.2.3. Particle velocity distribution Fig. 8 shows the predicted particles velocity in the bed for different forms of RDFs. As can be seen from this figure, except for Carnahan and Stirling [37], all RDFs models lead to similar particle velocity distributions. The expression of Lun and Savage [32], however, leads to the very low magnitude of particles velocity in region surrounding
the jet, as the particle temperature in that region does not change or varies slight with little gas to particle heat transfer. This figure also reveals that beyond the region surrounding the jet inlet, the results predicted by Ogawa et al. [38] and Ma and Ahmadi [39] correlations similar to Lun and Savage [32] model lead to the maximum velocity magnitude in spout region of about 1.15 m/s. As can be seen in Fig. 8, the particles velocity obtained by the Carnahan and Stirling [37] model in the zone surrounding the jet inlet is high which also lead to the over-prediction of the particles temperature in this region shown in Fig. 7. For different RDF models, the predicted vertical particle velocities at different levels of y = 0.02 and 0.03 m from the bottom of the bed are plotted, respectively, in Fig. 9 (a, b). For the sake of clarity, the CFD results obtained from the Carnahan and Stirling [37] correlation, which differs significantly from the other expressions, was not shown in Fig. 9. It is seen that all models show that the particles velocity reaches to their maximum at the center of spout and sharply decreases to zero at the intersection of spout and annular regions. The decreasing trend is remarkable in the spout region, while in the annulus region, the speed of particles moving downward is very small. The DEM results obtained by Deb and Tafti [47] for the same spouted bed are also reproduced in Fig. 9 (a, b) for comparison. It is seen that the CFD results using all RDF are consistent with the DEM of Deb and Tafti in the fountain region. The Lun and Savage model, however, leads to
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Fig. 8. Contour plot of particle velocity obtained by different radial distribution functions.
Fig. 10. particle temperature distribution at y = 0.02 m obtained by different radial distribution functions.
particle velocities that are more comparable with the DEM of Deb and Tafti in the annular region for both levels. That is Fig. 9 (a, b) shows that the speed of particles moving downward by Lun and Savage RDF is smaller than those observed by the other RDFs. In the top left corner of Fig. 9a the slight downward movement of the particles predicted by Lun and Savage model is shown. As mentioned before, Lun and Savage [32] expression leads to high solid shear viscosity at closest packing, thus leading to the lowest particles velocity in the annulus region. The vertical particle velocities predicted by the other correlations are larger due to lower solid viscosity in the annulus region. Fig. 10 represent the particles temperature at y = 0.02 m for different RDFs. As can be seen from this figure, all expressions predict the maximum solid temperature at the spout portion and Ma and Ahmadi [39] and Ogawa et al. [38] expressions lead to the highest temperature of particles. Note that the particles temperature distribution predicted by almost all expressions except Lun and Savage [32] is greatly differ from experimental observation (see Fig. 7). 3.3. Effect of angle of base of the bed Fig. 9. Vertical velocity of particles at y = 0.02 m obtained by different radial distribution functions.
For many applications of spouted beds such as drying and coating of particles, in order to have uniform products, behavior of particles
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4. By changing the angle of the spouted bed base, the dead zone highly decreased and the particles temperature distribution was more uniform in the bed. References
Fig. 11. Effect of changing the angle of base of the bed to 60◦ on CFD results, (a) Contour plot of voidage distribution, (b) Contour plot of particle temperature distribution.
should be stable and uniform. Hosseini et al. [25] investigated different angle of base of the conical spouted bed. They showed that conical angle in spouted beds is a critical parameter in appearance of unstable spouting. Liu et al. [48] also showed that unstable spouting would occur when the conical angle is less than 30° by CFD tool. In this study the conical angle of the bed is changed to 60◦ and behavior of particles at this condition is investigated. Fig. 11a shows the voidage distribution and temperature distribution of particles for a bed with angle base of 60◦ using Lun and Savage’s [32] RDF. As seen in Fig. 11 by changing the angle of base of the bed to 60◦ , incoherent spouting is not occurred and stable spouting is reached. In addition, the figure shows that with changing the angle of the base of the bed to 60◦ dead zone is highly reduced. 4. Conclusions In the present study, the hydrodynamics and the transient gas to particle heat transfer in a 2D rectangular spouted bed were investigated using an Eulerian–Eulerian two-fluid model (TFM) in conjunction with the kinetic theory of granular materials. The Gidaspow drag model was used to obtain the interphase interaction of gas and solid phases. The boundary condition of the Johnson–Jackson was applied to characterize the particle-wall collisions and sliding. The empirical relation for interphase heat transfer coefficient of Gunn [46] was also used. In addition, frictional viscosity of Johnson and Jackson [34] was applied in the model. The inlet gas temperature was not constant and increased with time. The influence of different forms of RDF on the CFD results of the spouted beds was evaluated. The main conclusions are summarized as follows: 1. A quantitative comparison between different forms of RDF was made. It was found that there are some differences in the model predictions for the closest packing region in the annulus near the wall. 2. While all model predictions for particle velocity in the bed, under the same conditions listed in Table 1, are reasonable, the RDF Lun and Savage’s [32] lead to slightly better results in the annular region near the wall. The incoherent spouting was predicted reasonably by Lun and Savage’s [32] expression. 3. For prediction of particles temperature in the bed, under the same conditions listed in Table 1, CFD model including Lun and Savage’s [32] RDF showed better agreement with the experimental data in the region near the wall. The models of Ma and Ahmadi [39] and Arastoopour [44], however, do better prediction in the fountain region.
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[25] Hosseini SH, Ahmadi G, Saeedi Razavi B, Zhong W. Computational Fluid Dynamic Simulation of Hydrodynamic Behavior in a Two-Dimensional Conical Spouted Bed. Energy Fuels 2010;24:6086–98. [26] Du W, Bao X, Xu J, Wei W. Computational fluid dynamics (CFD) modeling of spouted bed: Influence of frictional stress, maximum packing limit and coefficient of restitution of particles. Chem Eng Sci 2006;61:4558–70. [27] Lan X, Xu C, Gao J, Al-Dahhan M. Influence of solid-phase wall boundary condition on CFD simulation of spouted beds. Chem Eng Sci 2012;69:419–30. [28] Hosseini SH, Ahmadi G, Olazar M. CFD Study of Particle Velocity Profiles inside a Draft Tube in a Cylindrical Spouted Bed with Conical Base. J Taiwan Inst Chem Eng 2014;45:2140–9. [29] Hosseini S.H. Influences of Geometric Factors on CFD Results of a Draft Tube Cylindrical Spouted Bed. Prog Comput Fluid Dy 2015 (accepted for publication ). [30] Hosseini SH, Fattahi M, Ahmadi G. 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[32] Lun CKK, Savage SB. The effects of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials. Acta Mechanica 1986;63:15–44. [33] Huilin L, Gidaspow D, Bouillard J, Wentie L. Hydrodynamic Simulation of Gas– Solid Flow in a Riser Using Kinetic Theory of Granular Flow. Chem Eng J 2003;95:1–13. [34] Johnson PC, Jackson R. Frictional-Collisional Constitutive Relations for Granular Materials, with Application to Plane Shearing. J Fluid Mech 1987;176:67–93. [35] Lun CKK, Savage SB, Jeffrey DJ, Chepurniy N. Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J Fluid Mech 1984;140:223–56. [36] Chapman S, Cowling TG. The Mathematical Theory of Non-Uniform Gases. 3rd edition. Cambridge, England: Cambridge University Press; 1990. [37] Carnahan NF, Starling KE. Equations of State for Non-Attracting Rigid Spheres. J Chem Phys 1969;51:635–6. [38] Ogawa S, Umemura A, Oshima N. On the Equation of Fully Fluidized Granular Materials. J Appl Math Phys 1980;31:483–93. [39] Ma D, Ahmadi G. An Equation of State for Dense Rigid Sphere Gases. J Chem Phys 1986;84:3449–50.
[40] Wang J, van der Hoef MA, Kuipers JAM. CFD study of the minimum bubbling velocity of Geldart A particles in gas-fluidized beds. Chem Eng Sci 2010;65:3772–85. [41] Ye M, Wang J, van der Hoef MA, Kuipers JAM. Two-fluid modeling of Geldart A particles in gas-fluidized beds. Particuology 2008;6:540–8. [42] Wang J, van der Hoef MA, Kuipers JAM. Coarse grid simulation of bed expansion characteristics of industrial-scale gas–solid bubbling fluidized beds. Chem Eng Sci 2010;65:2125–31. [43] Hosseini SH, Zivdar M, Rahimi R, Samimi A. CFD simulation of gas–solid bubbling fluidized bed containing the FCC particles. Korean J Chem Eng 2009;26:1405–13. [44] Ibdir H, Arastoopour H. Modeling of multi-type particle flow using kinetic approach. AIChE J 2005;51:1620–32. [45] Gidaspow D, Bezburuah R, Ding J. Hydrodynamics of circulating fluidized beds. In: Proceedings of the Seventh Engineering Foundation Conference on Fluidization Kinetic Theory Approach, Fluidization VII; 1992. p. 75–82. [46] Gunn DJ. Transfer of Heat or Mass to Particles in Fixed and Fluidised Beds. Int J Heat Mass Transfer 1978;21:467–76. [47] Deb S, Tafti D. Investigation of flat bottomed spouted bed with multiple jets using DEM–CFD framework. Powder Technol 2014;254:387–402. [48] Liu X, Zhong W, Shao Y, Ren B, Jin B. Evaluation on the effect of conical geometry on flow behaviours in spouted beds. Can J Chem Eng 2014;92:768–74.
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Contents lists available at ScienceDirect
Journal of the Taiwan Institute of Chemical Engineers journal homepage: www.elsevier.com/locate/jtice
CFD Study of hydrodynamic and heat transfer in a 2D spouted bed: Assessment of radial distribution function Seyyed Hossein Hosseini a,∗, Mohsen Fattahi b, Goodarz Ahmadi c a
Department of Chemical Engineering, Ilam University, Ilam 69315–516, Iran Department of Chemical Engineering, Faculty of Engineering, University of Kurdistan, Sanandaj 66177, Iran c Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699-5725, USA b
a r t i c l e
i n f o
Article history: Received 4 March 2015 Revised 9 June 2015 Accepted 21 June 2015 Available online xxx Keywords: CFD Spouted bed Transient heat transfer Hydrodynamic Radial distribution function
a b s t r a c t The hydrodynamic and transient gas to particle heat transfer for spouted regime was studied using an Eulerian–Eulerian two-fluid model (TFM) including the kinetic theory of granular flows (KTGF). The influence of different expressions for the radial distribution function (RDF) on the CFD results of the spouted beds was evaluated. Detailed analyses of the CFD results showed that the RDF affects the CFD results especially in dense regions on both particle velocity and temperature distributions. Moreover, the best agreement with the experimental data was obtained by the RDF of Lun and Savage for the region near the wall and by RDF of Ma and Ahmadi, as well as, Arastoopour in the fountain region. © 2015 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
1. Introduction The spouted bed was first developed by Gishler [1] as an alternative drying method of moist wheat particles instead of a slugging fluidized bed. A spouted bed typically displays three distinct regions that are: spout, annulus, and fountain. The spouted beds are widely used in various industrial applications such as the drying of grains, spray drying, coating, heterogeneous catalysis, and gasification of biomass and coal [2–7]. Rapid increase of computational hardware capacity, recent advances in numerical algorithms, and deeper understanding of physics of multiphase flows, computational fluid dynamics (CFD) has emerged as an effective tool for predicting the behavior of gas–solid flows in the recent years. The main advantage of CFD modeling is that a wide range of the gas and solid flow properties may be studied rather economically [8]. Including the heat transfer process in the CFD simulations, the thermal behavior in addition to the hydrodynamics of two-phase systems can be analyzed. In many applications of spouted beds such as drying technology, chemical reactors and biological processes, understanding of thermal behavior is important, and some cases, the temperature distribution in the bed need to be controlled. Therefore, for effective usage of the
∗
Corresponding author: Tel: +98 913 7944470. E-mail address: [email protected], [email protected] (S.H. Hosseini).
spouted beds, knowledge of the temperature distribution in the bed is essential. Despite the importance of knowledge of particle temperature distribution in the spouted beds, there are rather few CFD studies on the thermal effects were reported in the literature. This issue is exasperated due to the lack of controlled experimental data on thermal behavior of multiphase flows. Several researchers have studied the wall to bed heat transfer in gas–solid bubbling fluidized beds using the two-phase flow modeling (TFM) approach [9,10]. Other researchers have investigated the heat transfer between a bubbling fluidized bed and an immersed body by the TFM [11,12] and discrete element method (DEM) [13,14]. A few authors studied the particles temperature distribution in gas–solid systems using the CFD without quantitative comparison with the experimental data, in part due to the lack of experimental measurements [15–18]. While there have been numerous studies of the effect of drag function, particle-particle restitution coefficient, particle-wall restitution coefficient and specularity coefficient (ϕ ) on dynamics of two-phase flows, rather few works have been published on the effect of RDF on the CFD results of gas–solid systems. Only van Wachem et al. [19] and Ahuja and Patwardhan [20] showed that the simulation results of gas–solid systems are not sensitive to the use of different solid stress models and RDFs in dense flow regimes. Clearly to arrive at an accurate heat transfer model for simulation of gas–solid flows, using an accurate hydrodynamics model is essential. In the past numerous investigations were performed and the influence of the hydrodynamics parameters such as drag function
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Nomenclature CD ds Dp H W es g g0 H0 Í I2D ks
β βErgun βWen−Yu
P Ps t U
vi Umf T K Cp h Nu Re Pr
drag coefficient [-] particle diameter [m] Depth of the bed [m] Vessel height [m] width of the bed [m] particle–particle restitution coefficient [-] acceleration due to gravity [m/s2 ] radial distribution coefficient [-] Static bed depth [mm] stress tensor [-] second invariant of the deviatoric stress tensor [-] diffusion coefficient for granular energy [kg/m s] gas/solid momentum exchange coefficient [kg/m3 s] gas/solid momentum exchange coefficient by Ergun equation [kg/m3 s] gas/solid momentum exchange coefficient calculated by Wen-Yu equation[kg/m3 s] pressure [N/m2 ] solids pressure [N/m2 ] time [s] superficial gas velocity [m/s] velocity [m/s] minimum fluidization velocity [m/s] Temperature [K] thermal conductivity [w/m.k] specific heat capacity [J/Kg.K] Enthalpy [J/Kg] Nusselt number [-] Reynolds number [-] Prandtl number [-]
Greek letters αi volume fraction [-] γs the collisional dissipation of energy [kg/s3 m] s granular temperature [m2 /s2 ] λs solid bulk viscosity [kg/m s] μi shear viscosity [kg/m s] ρi density [kg/m3 ] τi stress tensor [N/m2 ] φ angle of internal friction [deg] ϕ specularity coefficient [-] Subscripts col collision fr friction kin kinetic g gas p particle q phase type (solid or gas) s solids T stress tensor
[21–25], particle viscosity model [24,25], particle–particle restitution coefficient [24–26], particle-wall restitution coefficient, and specularity coefficient [27–29] on the CFD results of the spouted beds were investigated. In the present study, the optimized values of hydrodynamics parameters that were obtained in our earlier work [30] were used and the effect of the RDF on the CFD results for the 2D spouted beds was evaluated. Note that the Gidaspow drag function, which is widely used in gas–solid systems, was used in the current CFD model. The simulation results for the hydrodynamic and heat transfer in the spouted bed were compared with the experimental data of Brown
and Lattimer [31]. Comments regarding the accuracy of different radial distribution model were also provided. 2. CFD modeling In this work, an Eulerian–Eulerian two-fluid model is utilized for predicting the transient heat transfer in the pseudo two-dimensional spouted bed and for the particle flow and temperature distributions. A commercially available ANSYS-FLUENT code is used for these simulations. The heat transfer model and Lun and Savage’s RDF [32] are incorporated in the code using the User’s Defined Function (UDF). 2.1. Governing equations The governing equations of balance and the associated constitutive models of the Eulerian–Eulerian TFM that are used in the simulation of spouted beds are summarized in this section. It is assumed that the solid-phase is made of spherical granular particles with a fixed diameter. The kinetic theory of granular flow was used in the model. The virtual mass and lift effects, which are expected to be small, are neglected. The gas–solid interphase exchange coefficient, β , was modeled using the Gidaspow and coauthors [33] drag model. The solid shear viscosity is composed of collisional, kinetic, and frictional components. For dense flows when the solid volume fraction approaches the packing limit, the frictional viscosity becomes quite important, as the friction between particles generates large stresses. Here the expression suggested by Johnson and Jackson [34] is used for the frictional viscosity in dense cases. The bulk viscosity and the solids pressure are calculated using the expressions suggested by [35]. The continuity equation for the qth phase without any mass transfer between the phases is given by:
∂ (α ρ ) + ∇ .(αq ρq υ q ) = 0 ∂t q q
(1)
where αq , ρq and vq , respectively, are the volume fraction, density and velocity of the qth phase. The conservation of momentum for the gas and solid phases are given as: Gas phase:
∂ (α ρ υ ) + ∇ .(αg ρg υ g υ g ) ∂t g g g g ) + αg ρg g s − υ = − αg ∇ p + ∇ .τ g + β(υ
(2)
Solid phase:
∂ (α ρ υ ) + ∇ .(αs ρs υ s υ s ) ∂t s s s g − υ s ) + αs ρs g = − αs ∇ p − ∇ ps + ∇ .τ s + β(υ
(3)
where αs = 1 − αg . The conservation of energy for the gas and solid phases are given as:Gas phase:
∂ (α ρ H ) + ∇ .(αg ρg υ g Hg ) = ∇ .αg Kg,e f f ∇ .Tg − hgs (Ts − Tq ) ∂t g g g
(4)
Solid phase:
∂ (α ρ H ) + ∇ .(αs ρs υ s Hs ) = ∇ .αs Ks,e f f ∇ .Ts + hsg (Ts − Tq ) ∂t s s s
(5)
The transport equation for granular temperature, s (fluctuation kinetic energy of particles), is given as:
3 2
∂ (ρs αs s ) + ∇ .(αs ρs υ s s ) ∂t s + ∇ .(ks ∇ s ) − γs − 3β s = −∇ ps I¯ + τ g :∇ υ
(6)
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2.2. The constitutive equations for hydrodynamic
Solid pressure [35]:
Ps = αs ρs s +2ρs (1 + es )αs2 g0 s
Solid and gas phase stress tensors:
2 τ s = αs μs ∇ υ s + (∇ υ s )T + (αs λs − αs μs )∇ .υs I¯
(7)
2 ∇ υ g + (∇ υ g )T − ∇ .υg I¯
(8)
3
τ g = αg μg
3
s+d p s
(9)
1 3αs αs + + (1 − αs ) 2(1 − αs )2 2(1 − αs )3 2
(10)
Note that in this model, g0 is independent of the maximum packing and when maximum packing is approached, g0 remains finite. Ogawa et al. [38] proposed a model for the RDF, which has been widely used by the researchers for different gas–solid systems. That is,
g0 = 1 −
1/3 −1
αs
(11)
αs,max
Additional expressions for the RDF that are examined in the present study are: Lun and Savage [32]
g0 = 1 −
g0 =
3 1− 5
(12)
αs αs,max
1/3 −1 (13)
1 + 2.5αs + 4.59α + 4.52α
αs 1 − αs,max
2 s
3 0.678
3 s
(14)
The Ma and Ahmadi expression for the RDF is based on the virial series for dense gases and has been widely used for CFD simulation of bubbling gas–solid fluidized beds [40–43]. Iddir and Arastoopour [44]:
g0 = 1 −
αs αs,max
−1
(15)
Collisional energy dissipation: The energy dissipation due to inter-particle collisions is given as
12(1 − es )g0 γs = ρs αs2 3/2 √ s ds π 2
(16)
12
2 π s 4 1+ αs g0,ss (1+ess ) 5 96(1+ess )g0,ss
(18)
Frictional viscosity of Johnson and Jackson [34]:
μs, f ric = F r.
(αs − αs,min )n . sin φ (αs, max − αs ) p
(19)
where Fr, n and p are empirical model parameters. The diffusivity of granular temperature [45]:
ks =
2 π s 6 1+ αs g0 (1+es ) 5 384(1+es )g0 12 s + 2αs ρs ds g0 (1+es ) π 150ρs ds
(20)
Solid bulk viscosity [35]:
4 λ = αs ρs ds g0 (1+es ) 3
s π
(21)
Gas–solid drag coefficient of Gidaspow and co-workers [33]:
β =(1−ϕgs )βErgun +ϕgs βWen−Yu
(22)
where
→ − αs ρg − υs−→ υ g αs2 μg βErgun = 150 +1.75 f or αg ≤ 0.8 ds αg ds2 − → − → 3 αs αg ρg υ s − υ g −2.65 βWen−Yu = CD αg f or αg > 0.8 4
CD =
ϕgs =
ds
24 αg Res 1 + 0.15(αg Res )
0.687
CD = 0.44, Arctan 150 × 1.75(0.2−αs )
π
,
Res < 1000 Res ≥ 1000
+0.5
Constitutive equations for the thermal energy balance Interphase heat transfer The heat transfer coefficient is related to the Nusselt number, Nus , using the following:
hsg =
Ma and Ahmadi [39]:
g0 =
+
10ds ρs
Gidaspow and Huilin:
4 s = αs ρs ds g0 (1+ess ) 5 π
−2.5αs,max
αs αs,max
μs
where s is the distance between grains. From Eq. (9) it follows that for a dilute solid phase s → ∞, and therefore g0 → 1. In the limit when the solid phase compacts, s → 0 and g0 → ∞. The RDF is closely connected to the factor χ of Chapman and Cowling’s [36] theory of nonuniform gases. χ is equal to 1 for a rarefied gases, and increases and tends to infinity when the molecules are too close so that their motions are not possible. There has been several suggested expression for the RDF in the literature. Carnahan and Stirling [37] obtained a correlation for the RDF given as:
g0 =
(17)
Solid shear viscosity as given by Gidaspow et al. [45]:
Radial distribution function models The RDF, g0 , is a correction that modifies the probability of collisions between grains when the solid granular phase becomes dense. This function may also be interpreted as the non-dimensional distance between grains:
g0 =
3
6kg αs αg Nus ds2
(23)
An empirical relation for the interphase heat transfer coefficient was proposed by Gunn [46] which relates the Nusselt number with the particle Reynolds and Prandtl numbers. That is,
Nus = 7 − 10αg +5α 2g
1
1 + 0.7(Re p )2 (Pr) 3
1
+ 1.33 − 2.4αg +1.2α 2g (Re p )0.2 (Pr) 3 where Pr
(24)
C μ = pqk q . q
Several successful simulation results of heat transfer for gas–solid fluidized beds using Gunn [46] model have been reported in the literature [9–12,15,18]. Effective conductivities of the gas and solids phases For description of the heat-transfer rate in the two-fluid continuum model, it the overall bulk thermal conductivity is decomposed
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Fig. 1. Transient inlet air temperature ramp for the spouted regime [30].
into the effective thermal conductivities of the gas and the solids phase [9]. For the gas phase, the effective thermal conductivity is given by [9]:
Kg,e f f =
√ 1− αs Kg
(25)
αg
The thermal conductivity of the particulate phase differs from the particle material and depends on the contacts of the particles in the fluidized bed. The effective thermal conductivity of the solids phase can be expressed as:
1 Ks,e f f = √ kg ωA + (1 − ω)
(26)
αs
where
=
A (B − 1) (B + 1) (A − 1) B ln B − B − 2 B B 2 A
2
1− A
1− A
1− A
and for spherical particles
Ks A= , Kg
αs B = 1.25 αg
10/9 ,
ω = 7.26 × 10−3
In Eq. (26) ω represents the ratio of the particle contact area to the total particle surface area. In experiments conducted by Brown and Lattimer [31], the inlet gas temperature is not constant and increases with time as shown in Fig. 1. This temperature variation of the inlet gas is implemented in the ANSYS-FLUENT code by means of the user-defined functions (UDF) written in C-format. 2.2. Simulation conditions The corresponding physical and numerical parameters selected for the present simulation are listed in Table 1. 2.3. Simulation procedure A three-dimensional modeling is used in the present study. Due to the symmetry behavior of the particles half of the bed is selected for simulations. As noted before, the ANSYS-FLUENT CFD code is utilized for simulating the hydrodynamics and heat transfer of the fluidized bed studied experimentally by Brown and Lattimer [31]. The set of governing equations described in Section 2.1 is solved by the finite control volume technique. The phase coupled SIMPLE algorithm,
Symbol
Description
Simulation
ds (μm) ρs (Kg/m3 ) ρg (Kg/m3 ) μg (Kg/m.s) Umf (m/s) Ug,s φs (-) H0 (mm) φ (-) es (-) ew (-) αs,max (−) ϕ (-) H (mm) W (mm) D (mm) Ks (W/m.k) Cp,s (KJ/Kg.k) Cp,g (KJ/Kg.k)
Particle diameter Particle density Gas density Gas viscosity Minimum fluidization velocity Superficial velocity in spouted regime Sphericity Static bed depth Internal friction angle of particles Geldart group Particle–particle restitution coefficient Wall to particle restitution coefficient Maximum volume fraction of particles Specularity coefficient Bed height Bed width Bed depth Thermal conductivity of solids Specific heat capacity of solids Specific heat capacity of gas
550 2500 1.22 1.789 × 10−5 0.24 3Umf 1 49.75 28.5 B 0.8 0.2 0.62 0.025 280 56.4 4.95 1.05 0.84 1.006
which is an extension of the SIMPLE algorithm for multiphase flow, is used for the pressure–velocity coupling and correction. The momentum and volume fraction equations are discretized by a first-order upwind scheme. The adaptive time step in the range of 0.00001– 0.0005 is used with about 100 iterations per time step. A convergence criterion of 10−3 for each scaled residual component is specified for the relative error between two successive iterations. In this study the configuration and the operating conditions of the spouted bed studied experimentally by Brown and Lattimer [31] are used in the simulations. The corresponding pseudo two-dimensional fluidized bed is shown schematically in Fig. 2. The base was made of a stainless steel distributor plate with a single slit jet of 1.6 mm wide and the full depth of the bed. The outside of the plate was coated with a low thermal conductivity silicon rubber insulation to reduce thermal energy loss. The back and side walls consisted of polymethyl– methacrylate coated with paint to ensure surface flatness. The top of the distributor plate was covered by a fine mesh screen and the bed was filled with glass particles. The Geldart B classified particle type with mean diameter of 550 μm was used. It is worth mentioning that Brown and Lattimer [31] used the infrared thermography technique to measure the heat transfer in the entire bed. In this method the instantaneous particle temperatures can be evaluated by measuring the infrared emitted thermal radiation. The main advantage of this approach is that it provides nonintrusively temperature data for gas–solid flows. This method is not limited to steady-state condition and the detailed localized and timedependent data can be measured. Such experimental data is invaluable for validation of CFD and heat transfer models. The grid structures for the system under study are illustrated in Fig. 3. The mesh size is about 1 mm near the lateral bed wall, 0.2 mm at the center of the spout section, and varied along the axial direction of the bed from 0.5 to 3 mm. 2.5. Initial and boundary conditions A Dirichlet boundary condition at the bottom of the bed is used to specify a uniform gas inlet velocity. The outflow condition with zero velocity gradients is specified at the top of the freeboard. A noslip boundary condition for all walls is assumed for the gas phase. The particle normal velocity is set to zero at the wall. The Johnson and Jackson [34] wall boundary condition is used for the tangential particle velocity, and granular temperature of the solid phase at the wall. As mentioned before, the inlet air temperature varies roughly linearly with time. The walls are assumed adiabatic. Initially, the
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Fig. 3. Grid structures for the computational domain.
Fig. 2. Schematic of the pseudo 2D flat bottom rectangular column setup with dimensions in mm [30]. Table 2 Initial conditions. Description
Initial value
Gas temperature (K) Particle temperature (K) Static bed depth (mm) Solid volume fraction (-) Gas and particles velocities (m/s) Gas volume fraction in freeboard (-)
296 296 49.75 0.619 0 1
particle concentration in the spouted bed is specified, and gas velocity inside the spouted bed is set to zero. The particle concentration in the freeboard region is also set to zero. The initial conditions selected for the present simulation are listed in Table 2. 3. Results and discussion 3.1. Quantitative analysis of the radial distribution function expressions Eq. (18) shows that the solid viscosity depends on the radial distribution function. For α s,max = 0.62, which is the experimental finding
Fig. 4. Comparison of various forms of radial distribution functions.
of Brown and Lattimer [31], variations of different expressions for the redial distribution function as a function of solid volume fraction are plotted in Fig. 4. It can be seen that at the close packing the correlation of Carnahan and Stirling [37] unlike other expressions does not tend to the correct limit. As noted before, at the maximum solid packing, particles are in contact and the RDF tends to infinity. As the solid volume fraction approaches zero, all expressions except for Gidaspow correlation reach to the correct limit of one. As the solids volume fraction increases to more than 0.55 and get closer to the close packing, some differences between different expressions appear. Fig. 4 also shows that the RDF of Ma and Ahmadi [39] and Arastoopour [44] are very close. When the particle volume fraction is higher than 0.55, the expression suggested by Lun and Savage [32] gives the largest value compared to the others correlations. The solids volume fraction in spouted beds can vary from almost zero to the maximum packing limit, so the proper choice of RDF markedly affects the CFD results for the spouted beds.
Please cite this article as: S.H. Hosseini et al., CFD Study of hydrodynamic and heat transfer in a 2D spouted bed: Assessment of radial distribution function, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.06.027
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Fig. 5. Solids shear viscosity from different radial distribution functions.
Fig. 5 indicates the influence of the various forms of RDF on the dimensionless solid viscosity of Gidaspow et al. [45] for α s,max = 0.62 and es = 0.8. Ma and Ahmadi [39] and Arastoopour [44] correlations give nearly identical solid viscosity results. The Lun and Savage [32] correlation, however, leads to highest value for the solid shear viscosity at solid volume fraction higher than 0.55 compared to the other correlations. It is noteworthy that the higher shear viscosity values are calculated for the annulus region, where the solid volume fraction is higher than 0.55. Furthermore, an increase in solid viscosity leads to an increase in the resistance of solid particles against their downward movement. Therefore, the Lun and Savage [32] expression is expected to lead to lower particles velocity in the annulus region. It should be noted that in the spouted beds, the model needs to predict the particle behavior through the spout (dilute region) and the annulus (dense region) for accurate description of the fountain zone. 3.2. Effect of the different radial distribution function correlations on the CFD results 3.2.1. Voidage distribution As noted before, different expressions for the RDF is used to compute the solid shear viscosity that is applied in the solid phase momentum equation. Fig. 5 shows the variation of solid shear viscosity as a function of solid volume fraction. It is clearly seen that the value of solid viscosity varies depending on the RDF model used. The differences are particularly large at high solid volume fractions.
For U = 3Umf , ϕ = 0.025, es = 0.8 and ew = 0.2, Fig. 6 shows the instantaneous gas volume fraction contours for different RDF correlations. All models are for t = 5 sec, which the same as the experimental data of Brown and Lattimer [31]. In this figure the model predictions for different RDF are compared with the experimental data of Brown and Lattimer [31] in terms of fountain height and particles distribution through the bed. This figure shows that the RDF of Carnahan and Stirling [37] does not predict the proper gas volume fraction in the bed. Ma and Ahmadi and Arastoopour models predict results similar to the experiment of Brown and Lattimer [31] in terms of the fountain height and particles concentration near the top of the fountain. However, the simulation results with the RDF of Ma and Ahmadi and Arastoopour predict slightly lower particle concentrations near the wall compared those observed in the experiments. It should be noted that all boundary conditions are the same for all RDF models used. Fig. 6 also shows that the simulation results for RDFs of Ogawa et al. [38] and Lun and Savage [32] are quite similar; however, the CFD results with the use of Lun and Savage [32] expression are in a better agreement with the experimental data in term of fountain height and incoherent spouting. The Models of MA and Ahmadi and Arastoopour, however, better predict the high level of particle concentration region near the top of the fountain compared to the models of Ogawa et al. [38] and Lun and Savage [32]. The experimental contours for gas volume fraction of [31] shows that incoherent spouting occurs and also the bed surface, upper zone of annulus, is not flat but concave. This trend is predicted by the Lun and Savage [32] correlation for the RDF reasonably well. Fig. 6 also shows that dead zones predicted by all RDF expressions, except for Carnahan and Stirling [37] correlation, are roughly the same. 3.2.2. Temperature distribution The results of particle temperature distribution predicted by different RDFs are compared with the experimental observation of Brown and Lattimer [31] in Fig. 7. It is seen that the temperature differences across the bed is about 2 °C. The experimental data for the particle temperature distribution shown in Fig. 7 indicates that the highest particle temperature is not at the gas inlet zone but at certain height slightly above the gas inlet zone. Also the temperature is decreasing from spout toward the fountain base, because of increasing particle concentration. In addition, the temperature is increasing from the fountain base to the fountain top. This is in part because of the decrease in the particle concentration. Experimental result also clearly shows the dead zones near the wall surrounding the jet inlet. While the temperature contours predicted by various correlations appears to be quite different, the actual temperature field vary only slightly. All correlations shows that the particles temperature varies from the maximum value in fountain region and decrease with distance from the bed center consistent with the experiment of Brown
Fig. 6. Contour plot of voidage distribution obtained by different radial distribution functions.
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Fig. 7. Contour plot of particle temperature distribution obtained by different radial distribution functions.
and Lattimer [31] shown in Fig. 7. The minimum temperature at lower elevation occurs between the fountain and the walls, but at higher elevation occurs at the wall. Similar to the CFD results for voidage distribution, only the temperature distribution obtained by Carnahan and Stirling [41] expression differs from that observed by the experiment. The results of Ma and Ahmadi [39] and Arastoopour [44] for temperature distribution are similar. The correlations of Ogawa et al. [38] Ma and Ahmadi [39] and Arastoopour [44] for the RDF lead to better agreement with the experimental data in the fountain and around the spout regions compared with that of Lun and Savage [32]. However, these correlations over-predict the temperature in annulus region near the wall at higher elevation. The Lun and Savage [32] correlation shows that the particles temperature varies from maximum value in spout to minimum in the annulus region. Figs. 6 and 7 suggests that the RDF of Lun and Savage [32] lead to slightly better results in predicting particles temperature distribution and particles behavior in the dead zone in the bed among the other forms of RDFs. 3.2.3. Particle velocity distribution Fig. 8 shows the predicted particles velocity in the bed for different forms of RDFs. As can be seen from this figure, except for Carnahan and Stirling [37], all RDFs models lead to similar particle velocity distributions. The expression of Lun and Savage [32], however, leads to the very low magnitude of particles velocity in region surrounding
the jet, as the particle temperature in that region does not change or varies slight with little gas to particle heat transfer. This figure also reveals that beyond the region surrounding the jet inlet, the results predicted by Ogawa et al. [38] and Ma and Ahmadi [39] correlations similar to Lun and Savage [32] model lead to the maximum velocity magnitude in spout region of about 1.15 m/s. As can be seen in Fig. 8, the particles velocity obtained by the Carnahan and Stirling [37] model in the zone surrounding the jet inlet is high which also lead to the over-prediction of the particles temperature in this region shown in Fig. 7. For different RDF models, the predicted vertical particle velocities at different levels of y = 0.02 and 0.03 m from the bottom of the bed are plotted, respectively, in Fig. 9 (a, b). For the sake of clarity, the CFD results obtained from the Carnahan and Stirling [37] correlation, which differs significantly from the other expressions, was not shown in Fig. 9. It is seen that all models show that the particles velocity reaches to their maximum at the center of spout and sharply decreases to zero at the intersection of spout and annular regions. The decreasing trend is remarkable in the spout region, while in the annulus region, the speed of particles moving downward is very small. The DEM results obtained by Deb and Tafti [47] for the same spouted bed are also reproduced in Fig. 9 (a, b) for comparison. It is seen that the CFD results using all RDF are consistent with the DEM of Deb and Tafti in the fountain region. The Lun and Savage model, however, leads to
Please cite this article as: S.H. Hosseini et al., CFD Study of hydrodynamic and heat transfer in a 2D spouted bed: Assessment of radial distribution function, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.06.027
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Fig. 8. Contour plot of particle velocity obtained by different radial distribution functions.
Fig. 10. particle temperature distribution at y = 0.02 m obtained by different radial distribution functions.
particle velocities that are more comparable with the DEM of Deb and Tafti in the annular region for both levels. That is Fig. 9 (a, b) shows that the speed of particles moving downward by Lun and Savage RDF is smaller than those observed by the other RDFs. In the top left corner of Fig. 9a the slight downward movement of the particles predicted by Lun and Savage model is shown. As mentioned before, Lun and Savage [32] expression leads to high solid shear viscosity at closest packing, thus leading to the lowest particles velocity in the annulus region. The vertical particle velocities predicted by the other correlations are larger due to lower solid viscosity in the annulus region. Fig. 10 represent the particles temperature at y = 0.02 m for different RDFs. As can be seen from this figure, all expressions predict the maximum solid temperature at the spout portion and Ma and Ahmadi [39] and Ogawa et al. [38] expressions lead to the highest temperature of particles. Note that the particles temperature distribution predicted by almost all expressions except Lun and Savage [32] is greatly differ from experimental observation (see Fig. 7). 3.3. Effect of angle of base of the bed Fig. 9. Vertical velocity of particles at y = 0.02 m obtained by different radial distribution functions.
For many applications of spouted beds such as drying and coating of particles, in order to have uniform products, behavior of particles
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4. By changing the angle of the spouted bed base, the dead zone highly decreased and the particles temperature distribution was more uniform in the bed. References
Fig. 11. Effect of changing the angle of base of the bed to 60◦ on CFD results, (a) Contour plot of voidage distribution, (b) Contour plot of particle temperature distribution.
should be stable and uniform. Hosseini et al. [25] investigated different angle of base of the conical spouted bed. They showed that conical angle in spouted beds is a critical parameter in appearance of unstable spouting. Liu et al. [48] also showed that unstable spouting would occur when the conical angle is less than 30° by CFD tool. In this study the conical angle of the bed is changed to 60◦ and behavior of particles at this condition is investigated. Fig. 11a shows the voidage distribution and temperature distribution of particles for a bed with angle base of 60◦ using Lun and Savage’s [32] RDF. As seen in Fig. 11 by changing the angle of base of the bed to 60◦ , incoherent spouting is not occurred and stable spouting is reached. In addition, the figure shows that with changing the angle of the base of the bed to 60◦ dead zone is highly reduced. 4. Conclusions In the present study, the hydrodynamics and the transient gas to particle heat transfer in a 2D rectangular spouted bed were investigated using an Eulerian–Eulerian two-fluid model (TFM) in conjunction with the kinetic theory of granular materials. The Gidaspow drag model was used to obtain the interphase interaction of gas and solid phases. The boundary condition of the Johnson–Jackson was applied to characterize the particle-wall collisions and sliding. The empirical relation for interphase heat transfer coefficient of Gunn [46] was also used. In addition, frictional viscosity of Johnson and Jackson [34] was applied in the model. The inlet gas temperature was not constant and increased with time. The influence of different forms of RDF on the CFD results of the spouted beds was evaluated. The main conclusions are summarized as follows: 1. A quantitative comparison between different forms of RDF was made. It was found that there are some differences in the model predictions for the closest packing region in the annulus near the wall. 2. While all model predictions for particle velocity in the bed, under the same conditions listed in Table 1, are reasonable, the RDF Lun and Savage’s [32] lead to slightly better results in the annular region near the wall. The incoherent spouting was predicted reasonably by Lun and Savage’s [32] expression. 3. For prediction of particles temperature in the bed, under the same conditions listed in Table 1, CFD model including Lun and Savage’s [32] RDF showed better agreement with the experimental data in the region near the wall. The models of Ma and Ahmadi [39] and Arastoopour [44], however, do better prediction in the fountain region.
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Please cite this article as: S.H. Hosseini et al., CFD Study of hydrodynamic and heat transfer in a 2D spouted bed: Assessment of radial distribution function, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.06.027