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CFD simulation of transient gas to particle heat transfer for fluidized and spouted regimes
Accepted Manuscript CFD simulation of transient gas to particle heat transfer for fluidized and spouted regimes Mohsen Fattahi, Seyyed Hossein Hosseini, Goodarz Ahmadi PII:
S1359-4311(15)00539-6
DOI:
10.1016/j.applthermaleng.2015.05.071
Reference:
ATE 6674
To appear in:
Applied Thermal Engineering
Received Date: 11 December 2014 Revised Date:
25 May 2015
Accepted Date: 27 May 2015
Please cite this article as: M. Fattahi, S.H. Hosseini, G. Ahmadi, CFD simulation of transient gas to particle heat transfer for fluidized and spouted regimes, Applied Thermal Engineering (2015), doi: 10.1016/j.applthermaleng.2015.05.071. 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.
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CFD simulation of transient gas to particle heat transfer for fluidized and spouted regimes
a
Department of Chemical Engineering, Faculty of Engineering, University of Kurdistan, Sanandaj 66177, Iran b
c
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Mohsen Fattahia, Seyyed Hossein Hosseinib*, Goodarz Ahmadic
Department of Chemical Engineering, Ilam University, Ilam 69315–516, Iran
Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699-5725, USA
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Abstract
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The transient gas to particle heat transfer for fluidized and spouted regimes was studied using an Eulerian–Eulerian two-fluid model (TFM) in conjunction with the kinetic theory of granular flows (KTGF). The inlet gas temperature was transient which increased with time. Effect of modelling parameter of specularity coefficient on the simulation results was studied. It was found that specularity coefficient is a critical parameter which affects on the particles behaviour
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and the value of 0.025 leads to creation of incoherent spouting similar to the measurements. The temporal variation of particles concentration and temperature distribution in the spouted and fluidized regimes were compared with the corresponding experimental data for three distinct
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regions of spout, fountain and annulus. It was shown that the CFD model predicted the particles
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distribution properly for both regimes. In addition, the CFD predictions for particle temperature distribution for the spouted regime were in a better agreement with the measurements, when compared with that for the fluidized regime. Keywords: CFD, Spouted bed, Transient heat transfer, Temperature distribution, Fluidized regime.
*
. Corresponding author: Tel: +98-913-7944470; E-mail address: [email protected] (S. H. Hosseini)
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1. Introduction The original spouted bed was developed by Gishler [1] in 1954 for drying moist wheat
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particles as an alternative to a slugging fluidized bed. Spouted beds are widely used in various industrial operations such as coating, heterogeneous catalysis, gasification of biomass and coal, spray drying and drying of grains [2–7].
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Accurate predictions of flow field and temperature distributions of gas-solid fluidized beds are not possible by the classical models, because of the complex interactions between the phases
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[8]. The knowledge of heat transfer characteristics in the spout channel, fountain, and annulus in spouted beds are critical for design and operation of reactors for processing heat sensitive granular materials [9].
With increasing computational power, recent advances in numerical algorithms, and deeper understanding of multiphase flow processes, the computational fluid dynamics (CFD) has
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emerged as a powerful tool for predicting the properties of dense gas–solid flows [10]. Inclusion of heat transfer into the CFD model enhances its capabilities to provide information of the phasic temperature distributions.
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Although numerous CFD studies on the hydrodynamics of fluidized and spouted beds were
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reported in the literature [11–26], very few if any investigations includes the heat transfer in the spouted beds. For many applications of spouted beds such as drying equipment, chemical reactors, and biological processes, knowledge of temperature distribution is critical. Therefore, there is a need to provide a computation model that can predict the temperature distribution in these systems. Information concerning heat transfer for particles reported in the literature is rather scarce. The previous experimental studies have typically neglected the effect of intra-particle
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temperature gradients by assuming equal gas and particle temperatures. This type of approach most often leads to an overall average bed heat transfer coefficient and the spatial resolution of gas and particle heat transfer characteristics are lost [9].
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It worth mentioning that using the TFM approach several papers concerning wall-to-bed heat transfer in gas-solid and liquid-solid bubbling fluidized beds were published in the literature [27, 28, 29]. In addition, some researchers have investigated the heat transfer between a bubbling
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fluidized bed and an immersed body by the TFM [30, 31] and the DEM [32, 33]. For instance Dong et al. [34] studied the effects of different tube shapes on the flow characteristics and local
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heat transfer coefficients by TFM. The hybrid Euler-Lagrange approach, which takes the middle ground between the TFM and the DEM, has the potential to handle larger systems compare to those that can be analyzed by the DEM.
Recently, the hybrid Euler-Lagrange was used for
analyzing the behaviour of large scales gas-solid systems [35, 36, 37]. This approach should be
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evaluated for heat transfer analysis in dense gas-solid fluidization systems in the future. In addition, some researchers have studied a few heat transfer parameters in gas-solid fluidized beds. Behjat et al. [38] performed the numerical simulation of hydrodynamics and heat
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transfer in the gas-solid fluidized bed reactors using the TFM. They predicted the gas and solid phase temperature distributions in the reactor, and included the heat generated from the particles.
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Comparison between the computed gas and particles temperature distributions with the experimental measurements, however, was not performed. Dehnavi et al. [39] investigated the effect of drag function on the bed pressure drop and gas temperature along the ethylene polymerization reactor. They showed that the computed gas temperature profiles along the bed produced by different drag models are closer.
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Swasdisevi et al. [40] investigated the flow and heat transfer of gas-particle mixture in a twodimensional spouted bed with draft plates. They founded that the gas-to-particle heat transfer occurs mainly in the central or spout region. The validation of the heat transfer analysis,
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however, was not presented.
Recently, Hamzehei and Rahimzadeh [41] and Hamzehei et al. [42] studied a bubbling fluidized bed using both experimental and numerical approaches. Comparison of their CFD results with
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their experimental data showed good agreement for the gas temperature profile. The solid temperature distribution in the bed however was not measured in these studies.
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Kmiec et al. [8] applied an Eulerian-Eulerian two-fluid model to study the heat and mass transfer in a cylindrical spouted bed dryer with a draft tube. They found that the proposed model predicts the mass transfer rate very well, but over-predicts the heat transfer rate. From the above review of the literature, it is found that the investigations of heat transfer in
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spouted and fluidized beds are limited in that the experimentally verified phasic particle and gas temperature distributions are not available. In particular, almost all the existing results are provided by numerical simulation and the phasic experimental data are lacking. In the present
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study a pseudo two-dimensional bed described by Brown and Lattimer [9] is used to study the detailed phasic particle temperature distributions and solid concentration in the bed under
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fluidized and spouted regimes. The influence of specularity coefficient, which significantly affect the particle-wall boundary condition, is also studied.
2. CFD modeling
A two-fluid model based on the Eulerian–Eulerian approach is proposed to investigate the hydrodynamics and heat transfer in a pseudo two-dimensional gas–solid bed for fluidized and
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spouted regimes. In this approach continuous gas phase and the dispersed particles phase are treated as interpenetrating continua.
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2.1. Governing equations The governing equations and the associated constitutive equations of the Eulerian-Eulerian TFM that are used in the simulation of spouted beds are summarized in this section. The solid-
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phase contains mono-dispersed spherical granular particles with uniform diameter. The fluctuation of particles and the inter-particle collisions are considered using the kinetic theory of
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granular flow. The virtual mass and lift effects, which are expected to negligible, are neglected. The particle rotational effect and the associated lift become important for particles of large diameters that are not considered in the present study. The gas–solid interphase exchange coefficient, β, is modeled using the expression of Gidaspow et al. [43]. The solid shear viscosity is composed of collisional, kinetic and frictional effects. For very dense
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flows, the friction created between the particles generates a large amount of stress.
The
expression of Johnson and Jackson [44] is used to model the frictional viscosity in the dense
by Lun et al. [45].
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region. The bulk viscosity and the solid pressure are evaluated using the expressions suggested
by
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The continuity equation for the qth phase without any mass transfer between the phases is given
∂ (α ρ )+∇.(αq ρq υ q )=0 (1) ∂t q q
where αq , ρq and υ q , 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:
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ACCEPTED MANUSCRIPT ∂ (α ρ υ )+∇.(αg ρg υ g υ g )=-αg ∇p+∇.τ̿g +β(υ s -υ g )+αg ρg g (2) ∂t g g g Solid phase:
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∂ (α ρ υ )+∇.(αs ρs υ s υ s )=-αs ∇p-∇ps +∇.τ̿s +β(υ g -υs )+αs ρs g (3) ∂t s s s
where αs = 1 − αg .
Gas phase:
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∂ (α ρ H )+∇.(αg ρg υg Hg )=∇.αg Kg,eff ∇.Tg -hgs (Ts -Tq ) ∂t g g g
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The conservation of energy for the gas and solid phases are given by
Solid phase:
∂ (α ρ H )+∇.(αs ρs υ s Hs )=∇.αs Ks,eff ∇.Ts +hsg (Ts -Tq ) ∂t s s s
(4)
(5)
given as:
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The transport equation for granular temperature, Θ (fluctuation kinetic energy of particles), is
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3 ∂ ̅ τ̿g :∇υ s +∇.(ks ∇Θs )-γ -3βΘs (6) (ρ α Θ )+∇.(αs ρs υ s Θs ) =-∇ps I+ s 2 ∂t s s s 2.1.2. The constitutive equations for hydrodynamic: Solid and gas phase stress tensors:
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2 T τ̿s =αs µs ∇υ s +(∇υ s ) +(αs λs - αs µs )∇.υ s I ̅ (7) 3
T 2 τ̿g =αg µg ∇υ g +(∇υ g ) - ∇.υ g I̅ (8) 3
Radial distribution function:
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g0 =(1-
-2.5αs,max
αs αs,max
)
(9)
Collisional energy dissipation: ρs α2s Θ3/2 s (10)
Solid pressure [45]:
Ps =αs ρs Θs +2ρs (1+es )α2s g0 Θs
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d s √π
(11)
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γs =
12(1-e2s )g0
Solid shear viscosity (Gidaspow et al. [46]):
2 10ds ρs !πΘs 4 4 Θs 2 1+ αs g0,ss (1+ess ) (12) µs = αs ρs ds g0 (1+ess ) + π 96(1+ess )g0,ss 5 5
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1
Frictional viscosity (Johnson and Jackson [44]): µs,fric =Fr.
(αs − αs,min )n
αs, max − αs
p "#$ % (13)
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where Fr, n and p are empirical parameter the model.
The diffusivity of granular temperature is given as [46]: 2 Θs 2 6 1+ αs g0 (1+es ) +2αs ρs ds g0 (1+es ) (14) 384(1+es )g0 5 π s
1
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kΘs =
150ρs d !πΘs
Solid bulk viscosity is given by [45]
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4 Θs λ= αs ρs ds g0,ss (1+ess )' (15) 3 π Gas–solid drag coefficient is given as (Gidaspow et al. [43]):
β=(1-φgs )βErgun +φgs βWen-Yu (16)
where
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2 +1.75
αg ds
αs ρg )υ s -υ g )
for αg ≤0.8
ds
αs αg ρg )υ s -υ g ) 3 βWen-Yu = CD α-2.65 g 4 ds
for αg >0.8
24 +1+0.15(αg Res )0.687 ,, CD = *αg Res CD =0.44,
Res <1000 Res ≥1000
Arctan-150×1.75(0.2-αs ). +0.5 π
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φgs =
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βErgun =150
α2s µg
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2.1.3. Constitutive equations for the thermal energy balance 2.1.3.1. Interphase heat transfer
The heat transfer coefficient is related to the Nusselt number, Nus , by the following equation: 6kg /s /g Nus
d2s
(17)
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hsg =
An empirical correlation for the interphase heat transfer coefficient was proposed by Gunn [47] which relates the Nusselt number with the particle Reynolds number and Prandtl number. That
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is,
Nus =7-10αg +5α2g 1+0.7Rep (Pr)3
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2
1
+(1.33-2.4αg +1.2α2g )(Rep )0.2 (Pr)3 (18)
where
Pr=
1
Cpq µq kq
2.1.3.2. Effective conductivities of the gas and particles phases
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The thermal conductivity of the particulate phase depends on the particle material and also on the nature of contacts of the particles in the fluidized bed. For a mathematical description of the heat-transfer rate in the two-fluid continuum model it is necessary to separate the overall bulk
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thermal conductivity into a thermal conductivity of the gas phase and the solids phase, [27]. For the gas phase, the effective thermal conductivity is given by [27],
1-!αs 1 Kg (19) αg
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Kg,eff = 0
The effective thermal conductivity of the solids phase can be expressed as:
where
kg -ωA+(1-ω)Γ. (20)
(A-1) B A (B-1) (B+1) 6 ln 2 B B 21- B3 2 21- 3 21- B3 A A A 2
4
A
and for spherical particles
/ 8 , ; = 1.25( )=>/@ , A = 7.26 × 10CD 89 /9
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7=
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Γ=
!αs 1
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Ks,eff =
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in which A represents the ratio of the particle contact area to the total particle surface area.
In experiments conducted by Brown and Lattimer [9], the inlet gas temperature is not constant and increases with time (Fig. 1). On the other hand Kg is a function of temperature, thus, Kg is also depend on time and this matter has been considered in the present model. Accordingly, Fig. 2 shows the variation of Kg with temperature, and Fig. 3 shows the variation of Kg with time as a
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cubic polynomial, which is implemented in FLUENT by means of user-defined functions (UDF) written in C-code.
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2.2. Simulation conditions The corresponding physical and numerical parameters selected for the present simulation are listed in Table 1. According to Hosseini et al. [48], Hosseini [49] and Shu et al. [50] particle-
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wall restitution coefficient of 0.2 is used in this study. In these earlier work [48, 49, 50] it was shown that the particle-wall restitution coefficient plays a significant role on the CFD results in
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the dilute zone of particles such as draft tube embedded in cylindrical spouted bed and downer reactor. 2.3. Simulation procedure
In the present simulations, a three-dimensional model in the Cartesian framework is used. Also in order to save computational time, half of the bed is considered assuming the symmetry
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condition. The CFD code FLUENT is utilized to simulate the hydrodynamics and heat transfer of the bed that was operated by Brown and Lattimer [9]. The set of governing equations
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described in Section 2.1 is solved by a finite control volume technique. The phase coupled SIMPLE algorithm, which is an extension of the SIMPLE algorithm for multiphase flow, was
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used for the pressure–velocity coupling and correction. The momentum and volume fraction equations were discretized by a first-order upwind scheme. The adaptive time step in the range of 0.00001–0.0005 was used with about 100 iterations per time step. A convergence criterion of 10-3 for each scaled residual component was specified for the relative error between two successive iterations. The grid structures for the system under study are illustrated in Fig. 4.
2.4. Simulation system
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As mentioned before, the experimental data for the bed of Brown and Lattimer [9] for fluidized and spouted regimes is used to validate the present CFD model. Brown and Lattimer [9] performed a set of experiments in a pseudo two-dimensional fluidized bed, shown in Fig. 5,
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with cross-section dimensions of 56.4×4.95 mm2, static and overall bed heights of 49.75 mm and 280 mm, respectively. They used the new technique of infrared radiation spectroscopy to measure the rate of heat transfer in the entire bed. The advantage of this approach is it is non-
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intrusive and can provide experimental data for the gas–solid flows. It is worth mentioning that the method is not limited to steady-state condition and the detailed localized and time-dependent
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data needed for the validation of CFD models can be measured.
The base of the bed is a stainless steel distributor plate with a single slit jet of 1.6 mm wide and depth of 4.95 mm. The outside of the plate was coated by a low thermal conductivity silicon rubber insulation to reduce thermal energy losses. The back and side walls consisted of
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polymethyl–methacrylate coated in 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 particles are related to the Geldart B classification with mean diameter of 550 µm.
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2.5. Initial and boundary conditions
A Dirichlet boundary condition at the bottom of the bed is used to specify a uniform gas inlet
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velocity. The outflow condition with zero velocity gradients is specified at the top of the freeboard. A no-slip 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 [44] wall boundary condition is used for the tangential velocity, and granular temperature of the solid phase at the wall, with specularity coefficient in range of 0.025–0.2. As mentioned before, the inlet air temperature varies with time. The walls are assumed adiabatic. Initially, the particle concentration in the
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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.
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3. Results and discussion 3.1. Grid independency test
The grid independency was investigated by comparing the simulation results for three grid sizes.
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Fig. 6 shows the contour plot of gas volume fraction and time average particle volume fraction in axial direction of the bed. This figure shows that the simulation results obtained by fine (27,520
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cells) and medium grids (21,280 cells) are identical, but the coarse grid (13,200 cells) results show a discrepancy with the finer grid sizes. Therefore, the medium grid was used in the rest of the simulations. The mesh size is taken to be 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. 3.2. Spouted regime
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3.2.1. Effect of specularity coefficient on heat transfer The effect of important term in solid boundary condition, namely specularity coefficient, on
the results of heat transfer distribution is investigated, here. The specularity coefficient, E, is a
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parameter used in computational modelling to determine whether the wall is smooth (E → 0) or
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rough (E → 1) determining the solids tangential velocity imposed at the wall. The specularity
coefficient varies with a number of factors including the material of the wall, the type of particles, and the geometry of the walls. However, there are no generic correlations available in the literature, which relates the specularity coefficients to different factors. Thus, developing such a correlation would be very useful for the numerical simulation and design of gas-solid systems.
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Fig. 7 shows the effect of specularity coefficient on gas volume fraction through the spouted bed. Here, sensitivity of the results to different values of specularity coefficient (0.025-0.2) is studied. The experimental measurements by Brown and Lattimer [9] are also shown in the figure for
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comparison. It is observed that the value of specularity coefficient significantly affects the fountain height and particles behaviour through the bed. Particularly, the fountain height increases by decreasing the value of specularity coefficient. Fig. 7 also shows that an appropriate
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value for this coefficient is 0.025. According to the simulation results, particle concentration in the fountain is higher than that in the spout, but lower than the annulus, which is in agreement
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with the experimental observations. The CFD results for the hydrodynamics and heat transfer in the bed are sensitive to the value of specularity coefficient. This could be in part because of low thickness of the bed. In addition, Fig. 7 shows that the specularity coefficient of 0.025 leads to incoherent spouting, which is consistent with the experimental finding of Brown and Lattimer
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[9].
Fig. 8 shows the contour plot of gas volume fraction for spouted regime at φ=0.025 for different times. This figure shows that the behaviour of particles is unsteady and incoherent spouting is
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clearly seen. The predicted gas volume fraction shown in Fig. 8 are consistent with the numerical results of Zhao et al. [21] that were obtained using the DEM approach and simulation results of
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Hosseini et al. [25] performed using the TFM approach of MFIX code for a thin slot rectangular spouted bed. According to Fig. 8 the spout geometry and the resulting fountain height are not steady and change periodically with time. Moreover, the shapes of bed free surface of both experimental and numerical results are concave, which further confirms the validity of the present CFD model.
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Fig. 9 shows the effect of different values of specularity coefficient on simulation results of temperature distribution inside the bed. As seen in Fig. 9, with increase of specularity coefficient, the dead zones that appeared in the annulus portion increases, while the fountain height
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decreases. In addition, the incoherent spouting occurs for φ=0.025, while for other larger values of this coefficient stable spouting appears as seen in Fig. 7. This in turn results for the nonuniform temperature distribution of particles in the spout region for φ=0.025 and uniform
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temperature distributions for other values of specularity coefficient. Unless stated otherwise the specularity coefficient of 0.025 is used in the rest of simulations.
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3.2.2. Fountain height
Lan et al. [26] suggested that the fountain height is a key parameter that can be used to determine the accuracy of the numerical models for describing the hydrodynamic behaviour of spouted beds. In addition, Hosseini et al. [24] in their simulation of cylindrical spouted beds showed that
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by selecting appropriate value of particle-particle restitution coefficient the CFD model can predict accurate results for the fountain height. The predicted fountain heights versus spouting velocity as a dimensionless quantity are plotted in Fig. 10 and compared with experimental
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measurements of [51]. Good agreement of the CFD results with the experimental data [51] in
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terms of fountain height is found.
3.2.3. Temperature distribution Fig. 11 compares the simulation and experimental results of the time dependent particle temperature variations for the spouted regime. It is noticeable that the gas-to-particle heat exchange rate affects on the particle surface temperature entire the bed. A small fraction of gas phase passes through the annulus, making particle-to-particle exchange a significant heat transfer factor, while gas-to-particle exchange is insignificant. The high particles residence time in the
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annulus region leads to uniformity between the gas and particle temperatures in this region. Therefore, a uniform particle temperature distribution with value of 296 °K is clearly observed in the annulus zone for both simulation and experiment.
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Fig. 11 also shows that the highest particle temperature is not at the gas inlet region but at the height slightly above it. The particles entrain from the fountain to the top of the annulus as shown in Figs. 7 and 8. Also, the timeline of particles temperature distribution shown in Fig. 11
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reveals that particle temperature in the bed increases with time. This figure also shows that in spout and fountain regions, where the particles concentration is low, the particles temperature are
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higher. Similarly, moving from spout to fountain base, because of the increase in the particle concentration at fountain base, the particle temperature decreases. Also moving from the fountain base to fountain top, the particle temperature increases due to decrease of particle
3.3. Fluidized regime
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concentration. Fig. 11 shows good agreement between the CFD results and experimental data.
Brown and Lattimer [9] reported that for gas inlet velocity of 1.6Umf the fluidized regime
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forms in the bed. Similar to the study that was done for spouted regime, the time-dependency of thermal conductivity of gas phase for transient inlet air temperature for fluidized regime is also
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studied. A UDF was developed and implemented in the computational model for this purpose. Fig. 12 compares the particle velocity vector fields for spouted and fluidized regimes, respectively, at Ug=3Umf and Ug=1.6Umf. As shown in Figs. 7, 8 and 11a, in spouted regime (Ug=3Umf) three distinct regions, namely, the spout, the fountain and the annulus, are formed. The solids above the bed surface are entrained by the spout and then rains down on the annulus are designated as the fountain. However, in fluidized regime (Ug=1.6Umf) fountain is not formed and particles move in the radial direction at the top of the bed as shown in Fig. 12.
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A comparison between simulation and experimental results of gas volume fraction under fluidized regime is shown in Fig. 13. As mentioned before, the fountain is not formed in this case. However, similar to the spouted regime, for fluidized regime the particle flow is unsteady
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and the instability in this case is much more than the spouted regime. This is due to the breakage of the central bubble that is attained to the top of the bed and then reformation of bubble from the nozzle as a cyclic process.
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Fig. 14 shows the contour plot of gas volume fraction for the fluidized regime with φ=0.025 and Ug=1.6Umf for different times. This figure clearly shows that the behaviour of particles
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periodically changes with time. The cause of this matter is the central bubble that is attained to the top of the bed breakages and in a cyclic process the reformation of bubble from the nozzle continues, which affect on the particles behaviour.
Fig. 15 compares the simulation and experimental results of the time dependent particle
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temperature distributions for the fluidized regime. The low particle temperature regions are surrounding the jet inlet due to negligibly small convective gas-to-particle heat transfer in those regions. The highest particle temperature is recorded at the top surface of the bed. This is due to
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two main reasons with respect to the thermal-dynamics of the bed and to the measurement technique used. The first and main reason is convective gas-to-particle heat transfer occurs at the
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top of the bed from the highly turbulent bubble eruptions. Secondly, the contact intensity between the high rate of gas and a particle occurs in the top of the bed as the particle velocity decreases to zero, which leads to increase of the intra-particle temperature in that region. As can be seen in Fig. 15, simulation results of transient particle temperature distribution for fluidized regime and experimental observations show the same trend. From Figs. 11 and 15, the CFD
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predictions for particle temperature distribution for the spouted regime were in a better agreement with the measurements, when compared with that for the fluidized regime.
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4. Conclusion In the present study, the transient gas to particle heat transfer for spouted and fluidized regimes was studied using an Eulerian–Eulerian two-fluid model including the kinetic theory of
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granular flows. Heat transfer models for effective conductivities of the gas and solids phases were added as compiled executable code using the UDF programming. Time dependent inlet air
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temperature led to increase of the particles temperature distribution through the bed. A critical hydrodynamic modeling parameter (specularity coefficient) was evaluated in terms of fountain height and production of three distinct regions in spouted regime. It was found that this coefficient has a great effect on particles behaviour through the bed and formation of
studied.
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incoherent spouting. In addition, effect of this parameter on the heat transfer of CFD model was
The simulation results indicated that the CFD results for hydrodynamics and heat transfer distribution because are very sensitive to the value of specularity coefficient. The sensitivity
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could be due to the low thickness of the bed.
Using the specularity coefficient of φ=0.025
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seems to lead to the optimal results for the cases studied. Simulation results of the time dependent particle temporal variations in the spouted regime were compared with the experimental data and good agreements were found.
The temperature
distributions in three distinct regions of spouted bed were also investigated. The CFD results suggests that the highest particle temperatures were not at the base of the gas inlet but rather at a slightly distance about the inlet nozzle in the spout and fountain zones. prediction was in agreement with experimental observations.
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This numerical
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CFD results for the temperature distribution in the spouted regime showed that the particle temperature in spout and fountain regions are not uniform and increase from spout to fountain base and from fountain base to fountain top due to incoherent behaviour of the particles inside
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the bed.
Finally, the particles temperature distributions and gas volume fraction for the fluidized regime
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were also investigated and a closed agreement was found with the experimental observations.
CD
: drag coefficient [-]
ds
: particle diameter [m]
Dp
: Depth of the bed [m] : Vessel heigh [m]
W
: width of the bed [m]
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H
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Nomenclature
: particle-particle restitution coefficient [-]
g
: acceleration due to gravity [m/s2]
H0 IG I2D
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g0
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es
: radial distribution coefficient [-]
: Static bed depth [mm] : stress tensor [-] : second invariant of the deviatoric stress tensor [-]
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: diffusion coefficient for granular energy [kg/m s]
β
: gas/solid momentum exchange coefficient [kg/m3s ]
βErgun
: gas/solid momentum exchange coefficient by Ergun equation [kg/m3s ]
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kΘs
βWen-Yu : gas/solid momentum exchange coefficient calculated by Wen-Yu equation[kg/m3s ] : pressure [N/m2]
Ps
: solids pressure [N/m2]
t
: Time [s]
U
: superficial gas velocity [m/s]
HI
: velocity [m/s]
Umf
: minimum fluidization velocity [m/s]
T
: Temperature [K]
K
: thermal conductivity [w/m.k]
Cp
: specific heat capacity [J/Kg.K]
Nu
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h
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P
: Enthalpy [J/Kg]
: Nusselt number [-]
Re
: Reynolds number [-]
Pr
: Prandtle number [-]
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Greek letters /I
: volume fraction [-] : the collisional dissipation of energy [kg/s3m]
J
: granular temperature [m2/s2]
τ́ I
%
: density [kg/m3] : stress tensor [N/m2]
: angle of internal friction [deg]
E
: specularity coefficient [-]
Subscripts : collision
fr
: friction
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g
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col
kin
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MI
: shear viscosity [kg/m s]
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LI
: solid bulk viscousity [kg/m s]
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K
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γs
: kinetic
: gas
p
: particle
q
: phase type (solid or gas)
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: solids
T
: stress tensor
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References
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Chem. Eng. 61 (1983) 267–268.
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Dryer with a Draft Tube. In Proceedings of the 12th International Drying Symposium (IDS 2000), Elsevier Science B.V., Amsterdam, 2000.
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Tall-Form Spray Dryer, Trans. Inst. Chem. Eng. 80 (2002) 163-175.
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bed, Exp. Therm. Fluid Sci. 44 (2013) 883-892.
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[9] S.L. Brown, B.Y. Lattimer, Transient gas-to-particle heat transfer measurements in a spouted
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behaviour in spouted beds, Chem. Eng. Res. Des. 79 (2001) 593–599. [12] W. Du, X. Bao, J. Xu, W. Wei, Computational fluid dynamics (CFD) modeling of spouted
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bed: Assessment of drag coefficient correlations, Chem. Eng. Sci. 61 (2006) 1401–1420. [13] W. Du, X. Bao, J. Xu, W. Wei, Computational fluid dynamics (CFD) modeling of spouted
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bed: Influence of frictional stress, maximum packing limit and coefficient of restitution of particles, Chem. Eng. Sci. 61 (2006) 4558–4570. [14] W. Du, W. Wei, J. Xu, Y. Fan, X. Bao, Computational fluid dynamics (CFD) modeling of fine particle spouting, Int. J. Chem. React. Eng. 4 (2006) A21. [15] Z. G. Wang, H. T. Bi, C. J. Lim, Numerical simulations of hydrodynamic behaviors in conical spouted beds, China Particuology 4 (2006) 194–203.
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[16] Z. H. Wu and A. S. Mujumdar, CFD modeling of the gas-particle flow behavior in spouted beds, Powder Technol. 183 (2008) 260–272.
D spouted bed, Chem. Eng. Sci. 60 (2005) 1267–1276.
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[17] S. Takeuchi, X. S. Wang, M. J. Rhodes, Discrete element study of particle circulation in a 3-
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conical-base spouted beds, Powder Technol. 184 (2008) 141–150.
[19] W. Zhong, Y. Xiong, Z. Yuan, M. Zhang, DEM simulation of gas-solid flow behaviors in
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spout-fluid bed, Chem. Eng. Sci. 61 (2006) 1571–1584.
[20] X.-L. Zhao, S.-Q.Li, G.-Q. Liu, Q. Song, Q. Yao, Flow patterns of solids in a two dimensional spouted bed with draft plates: PIV measurement and DEM simulations,
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Powder Technol. 183 (2008) 79–87.
[21] X.-L. Zhao, S.-Q.Li, G.-Q. Liu, Q. Yao, J. S. Marshall, DEM simulation of the particle dynamics in two-dimensional spouted beds, Powder Technol. 184 (2008) 205–213.
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[22] S. Azizi, S.H. Hosseini, M. Moraveji, G. Ahmadi, CFD modeling of a spouted bed with a
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porous draft tube, Particuology 8 (2010) 415–424. [23] S.H. Hosseini, M. Zivdar, R. Rahimi, CFD simulation of gas–solid flow in a spouted bed with a nonporous draft tube, Chem. Eng. Process. 48 (2009) 1539–1548. [24] S.H. Hosseini, G. Ahmadi, M. Olazar, CFD simulation of cylindrical spouted bed by the kinetic theory of granular flow, Powder Technol. 246 (2013) 303–316.
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[25] S.H. Hosseini, G. Ahmadi, B. Saeedi Razavi, W. Zhong, Computational Fluid Dynamic Simulation of Hydrodynamic Behavior in a Two-Dimensional Conical Spouted Bed,
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Energy Fuels 24 (2010) 6086–6098. [26] X. Lan, C. Xu, J. Gao, M. Al-Dahhan, Influence of solid-phase wall boundary condition on CFD simulation of spouted beds, Chem. Eng. Sci. 69 (2012) 419–430.
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Bubbling Fluidized Beds, AIChE J. 52 (2005) 58-74.
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[27] D.J. Patil, J. Smit, M.S Annaland, J.A.M. Kuipers, Wall-to-Bed Heat Transfer in Gas-Solid
[28] L.M. Armstrong, S. Gu, K.H. Luo, Study of wall-to-bed heat transfer in a bubbling fluidised bed using the kinetic theory of granular flow, Int. J. Heat Mass Transfer 53 (2010) 4949– 4959.
Comparison
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[29] Y. Lu, T. Zhang, X. Dong, Bed to wall heat transfer in supercritical water fluidized bed: with
the
gas-solid
fluidized
bed,
http://dx.doi.org/10.1016/j.applthermaleng.2014.09.052
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[30] A. Schmidt, U. Renz, Eulerian computation of heat transfer in fluidized beds, Chem. Eng.
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Sci. 54 (1999) 5515–5522.
[31] N.H. Dong , L.M. Armstrong , S. Gu, K.H. Luo, Effect of tube shape on the hydrodynamics and tube-to-bed heat transfer in fluidized beds, Appl. Therm. Eng. 60 (2013) 472–479 [32] F.P.D. Maio, A.D. Renzo, D. Trevisan, Comparison of heat transfer models in DEM-CFD simulations of fluidized beds with an immersed probe, Powder Technol. 193 (2009) 257– 265.
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[33] Y. Zhao, M. Jiang, Y. Liu, J. Zheng, Particle-Scale Simulation of the Flow and Heat Transfer Behaviors in Fluidized Bed with Immersed Tube, AIChE J. 55 (2009) 3109–3124.
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[34] N.H. Dong, L.M. Armstrong, S. Gu, K.H. Luo, Effect of tube shape on the hydrodynamics and tube-to-bed heat transfer in fluidized beds, Appl. Therm. Eng. 60 (2013) 472–479.
[35] W. Adamczyk, G. Węcel, M. Klajny, P. Kozołub, A. Klimanek, R. Białecki, Modeling of
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particle transport and combustion phenomena in a large-scale circulating fluidized bed
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boiler using a hybrid Euler-Lagrange approach, Particuology 16 (2014) 29–40. [36] D. Snider, S. Clark, P. O'Rourke, Euleriane-Lagrangian method for three-dimensional thermal reacting flow with application to coal gasifies, Chem. Eng. Sci. 66 (2011) 1285– 1295.
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[37] W. Adamczyk, P. Kozołub, G. Węcel, A. Klimanek, R. Białecki, T. Czakiert, Modeling oxy-fuel combustion in a 3D circulating fluidized bed using the hybrid Euler-Lagrange approach, Appl. Therm. Eng. 71 (2014) 266–275.
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[38] Y. Behjat, S. Shahhosseini, S.H. Hashemabadi, CFD modeling of hydrodynamic and heat
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transfer in fluidized bed reactors, Int. Commun. Heat Mass Transfer 35 (2008) 357–368. [39] M.A. Dehnavi, S. Shahhosseini , S.H. Hashemabadi, S.M. Ghafelebashi, CFD simulation of hydrodynamics and heat transfer in gas phase ethylene polymerization reactors, Int. Commun. Heat Mass Transfer 37 (2010) 437–442. [40] T. Swasdisevi, W. Tanthapanichakooni, T. Charinpanitkul, T. Kawaguchi, T. Tanaka and Y. Tsugi, Prediction of gas–particle dynamics and heat transfer in a two-dimensional spouted bed, Adv. Powder Technol. 16 (2005) 275–293.
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[41] M. Hamzehei, H. Rahimzadeh, Experimental and numerical study of hydrodynamics with heat transfer in a gas-solid fluidized-bed reactor at different particle sizes, Ind. Eng. Chem.
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Res. 48 (2009) 3177–3186.
[42] M. Hamzehei, H. Rahimzadeh, and G. Ahmadi, Computational and experimental study of heat transfer and hydrodynamics in a 2D gas-solid fluidized bed reactor, Ind. Eng. Chem. Res.
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49 (2010) 5110–5121.
[43] L. Huilin, D. Gidaspow, J. Bouillard, L. Wentie, Hydrodynamic Simulation of Gas-Solid
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Flow in a Riser Using Kinetic Theory of Granular Flow, Chem. Eng. J. 95. (2003) 1–13.
[44] P. C. Johnson and R. Jackson, Frictional-Collisional Constitutive Relations for Granular Materials, with Application to Plane Shearing, J. Fluid Mech. 176 (1987) 67–93.
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[45] C.K.K. Lun, S.B. Savage, D.J. Jeffrey, N. Chepurniy, Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field, J.
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Fluid Mech. 140 (1984) 223–256.
[46] D. Gidaspow, R. Bezburuah, J. Ding, Hydrodynamics of circulating fluidized beds. Kinetic
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theory approach, fluidization VII, Proceedings of the Seventh Engineering Foundation Conference on Fluidization, (1992) 75–82.
[47] D.J. Gunn, Transfer of Heat or Mass to Particles in Fixed and Fluidised Beds, Int. J. Heat Mass Transfer, 21 (1978) 467- 476.
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[48] S.H. Hosseini, G. Ahmadi, M. Olazar, CFD Study of Particle Velocity Profiles inside a Draft Tube in a Cylindrical Spouted Bed with Conical Base, J. Taiwan Inst. Chem. Eng. 45
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(2014) 2140–2149.
[49] S.H. Hosseini, Influences of Geometric Factors on CFD Results of a Draft Tube Cylindrical Spouted Bed, Prog. Comput. Fluid Dy. Accepted paper.
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[50] Z. Shu, G. Peng, J. Wang, N. Zhang, S.-G. Li, W. Lin, Comparative CFD Analysis of
Res. 53 (2014) 3378−3384.
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Heterogeneous Gas–Solid Flow in a Countercurrent Downer Reactor, Ind. Eng. Chem.
[51] S.L. Brown, Master of Science Thesis, Hydrodynamics and Transient Heat Transfer Characteristics in Fluidized and Spouted Beds, Virginia Polytechnic Institute and State
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University, 2012.
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Captions
Table 1. The physical and numerical parameters.
Fig. 3. Variation of air thermal conductivity with time.
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Fig. 4. Grid structures for the computational domain.
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Fig. 2. Variation of air thermal conductivity with temperature [47].
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Fig. 1. Transient inlet air temperature ramp for the spouted and fluidized regimes [9].
Fig. 5. Schematic of the 2D flat bottom rectangular column setup with dimensions in mm [9].
Fig. 6. Effect of mesh size on CFD results, (a) contour plot of gas volume fraction, (b) particle
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volume fraction at axis of the bed
Fig. 7. Effect of specularity coefficient on simulation results of gas volume fraction
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Fig. 8. Contour plot of particle concentration for different times for spouted regime.
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Fig. 9. Effect of specularity coefficient on simulation results of particle temperature distribution.
Fig. 10. The predicted fountain heights versus spouting velocity
Fig. 11. Comparison of simulation result of time dependent particle temporal variations in the spouted regime with experimental observations, (a) Experimental, (b) Simulation.
Fig. 12. Comparison of solid velocity vectors in Ug=3Umf for spouted and in Ug=1.6Umf for fluidized regimes, (a) spouted regime, (b) Fluidized regime.
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Fig. 13. Comparison of simulation and experimental results of gas volume fraction in fluidized regime, (a) Simulation, (b) Experimental.
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Fig.14. Contour plot of particle concentration for different times for fluidized regime.
Fig. 15. Comparison of simulation result of time dependent particle temporal variations in the
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fluidized regime with experimental observations, (a) Experimental, (b) Simulation.
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Table 1. Physical and numerical parameters
Description
Simulation
d (μm)
Particle diameter
550
ρ (Kg/m3)
Particle density
2500
ρ (Kg/m3)
Gas density
1.22
μ (Kg/m.s)
Gas viscosity
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symbol
.
×
Minimum fluidization velocity
0.24
Ug,s
Superficial velocity in spouted regime
3Umf
Ug,f
Superficial velocity in fluidized regime
1.6Umf
ϕ (-)
Sphericity
1
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U (m/s)
Static bed depth
49.75
ϕ (-)
Internal friction angle of particles
28.5
-
Geldart Group
B
es (-)
Particle–particle restitution coefficient
0.8
ew (-)
Wall to particle restitution coefficient
0.2
α, (−)
Maximum volume fraction of particles
0.62
(-)
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H0 (mm)
Specularity coefficient
0.025-0.2
Bed height
280
Bed width
56.4
Bed depth
4.95
Ks (W/m.k)
Thermal conductivity of solids
1.05
Cp,s (KJ/Kg.k)
Specific heat capacity of solids
0.84
Cp,g (KJ/Kg.k)
Specific heat capacity of gas
1.006
H (mm) W (mm)
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D (mm)
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Highlights ► CFD study of transient heat transfer in pseudo 2D bed for spouted & fluidized states ► Specularity coefficient affects the voidage and particles temperature in the bed
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► In spouted state incoherent spouting was occured by proper specularity coefficent ► Highest particle temperatures were discerned in spout and fountain regions
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► Highest particle temperatures were discerned along the jet in fluidized state
S1359-4311(15)00539-6
DOI:
10.1016/j.applthermaleng.2015.05.071
Reference:
ATE 6674
To appear in:
Applied Thermal Engineering
Received Date: 11 December 2014 Revised Date:
25 May 2015
Accepted Date: 27 May 2015
Please cite this article as: M. Fattahi, S.H. Hosseini, G. Ahmadi, CFD simulation of transient gas to particle heat transfer for fluidized and spouted regimes, Applied Thermal Engineering (2015), doi: 10.1016/j.applthermaleng.2015.05.071. 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.
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CFD simulation of transient gas to particle heat transfer for fluidized and spouted regimes
a
Department of Chemical Engineering, Faculty of Engineering, University of Kurdistan, Sanandaj 66177, Iran b
c
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Mohsen Fattahia, Seyyed Hossein Hosseinib*, Goodarz Ahmadic
Department of Chemical Engineering, Ilam University, Ilam 69315–516, Iran
Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699-5725, USA
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Abstract
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The transient gas to particle heat transfer for fluidized and spouted regimes was studied using an Eulerian–Eulerian two-fluid model (TFM) in conjunction with the kinetic theory of granular flows (KTGF). The inlet gas temperature was transient which increased with time. Effect of modelling parameter of specularity coefficient on the simulation results was studied. It was found that specularity coefficient is a critical parameter which affects on the particles behaviour
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and the value of 0.025 leads to creation of incoherent spouting similar to the measurements. The temporal variation of particles concentration and temperature distribution in the spouted and fluidized regimes were compared with the corresponding experimental data for three distinct
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regions of spout, fountain and annulus. It was shown that the CFD model predicted the particles
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distribution properly for both regimes. In addition, the CFD predictions for particle temperature distribution for the spouted regime were in a better agreement with the measurements, when compared with that for the fluidized regime. Keywords: CFD, Spouted bed, Transient heat transfer, Temperature distribution, Fluidized regime.
*
. Corresponding author: Tel: +98-913-7944470; E-mail address: [email protected] (S. H. Hosseini)
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1. Introduction The original spouted bed was developed by Gishler [1] in 1954 for drying moist wheat
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particles as an alternative to a slugging fluidized bed. Spouted beds are widely used in various industrial operations such as coating, heterogeneous catalysis, gasification of biomass and coal, spray drying and drying of grains [2–7].
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Accurate predictions of flow field and temperature distributions of gas-solid fluidized beds are not possible by the classical models, because of the complex interactions between the phases
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[8]. The knowledge of heat transfer characteristics in the spout channel, fountain, and annulus in spouted beds are critical for design and operation of reactors for processing heat sensitive granular materials [9].
With increasing computational power, recent advances in numerical algorithms, and deeper understanding of multiphase flow processes, the computational fluid dynamics (CFD) has
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emerged as a powerful tool for predicting the properties of dense gas–solid flows [10]. Inclusion of heat transfer into the CFD model enhances its capabilities to provide information of the phasic temperature distributions.
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Although numerous CFD studies on the hydrodynamics of fluidized and spouted beds were
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reported in the literature [11–26], very few if any investigations includes the heat transfer in the spouted beds. For many applications of spouted beds such as drying equipment, chemical reactors, and biological processes, knowledge of temperature distribution is critical. Therefore, there is a need to provide a computation model that can predict the temperature distribution in these systems. Information concerning heat transfer for particles reported in the literature is rather scarce. The previous experimental studies have typically neglected the effect of intra-particle
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temperature gradients by assuming equal gas and particle temperatures. This type of approach most often leads to an overall average bed heat transfer coefficient and the spatial resolution of gas and particle heat transfer characteristics are lost [9].
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It worth mentioning that using the TFM approach several papers concerning wall-to-bed heat transfer in gas-solid and liquid-solid bubbling fluidized beds were published in the literature [27, 28, 29]. In addition, some researchers have investigated the heat transfer between a bubbling
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fluidized bed and an immersed body by the TFM [30, 31] and the DEM [32, 33]. For instance Dong et al. [34] studied the effects of different tube shapes on the flow characteristics and local
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heat transfer coefficients by TFM. The hybrid Euler-Lagrange approach, which takes the middle ground between the TFM and the DEM, has the potential to handle larger systems compare to those that can be analyzed by the DEM.
Recently, the hybrid Euler-Lagrange was used for
analyzing the behaviour of large scales gas-solid systems [35, 36, 37]. This approach should be
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evaluated for heat transfer analysis in dense gas-solid fluidization systems in the future. In addition, some researchers have studied a few heat transfer parameters in gas-solid fluidized beds. Behjat et al. [38] performed the numerical simulation of hydrodynamics and heat
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transfer in the gas-solid fluidized bed reactors using the TFM. They predicted the gas and solid phase temperature distributions in the reactor, and included the heat generated from the particles.
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Comparison between the computed gas and particles temperature distributions with the experimental measurements, however, was not performed. Dehnavi et al. [39] investigated the effect of drag function on the bed pressure drop and gas temperature along the ethylene polymerization reactor. They showed that the computed gas temperature profiles along the bed produced by different drag models are closer.
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Swasdisevi et al. [40] investigated the flow and heat transfer of gas-particle mixture in a twodimensional spouted bed with draft plates. They founded that the gas-to-particle heat transfer occurs mainly in the central or spout region. The validation of the heat transfer analysis,
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however, was not presented.
Recently, Hamzehei and Rahimzadeh [41] and Hamzehei et al. [42] studied a bubbling fluidized bed using both experimental and numerical approaches. Comparison of their CFD results with
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their experimental data showed good agreement for the gas temperature profile. The solid temperature distribution in the bed however was not measured in these studies.
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Kmiec et al. [8] applied an Eulerian-Eulerian two-fluid model to study the heat and mass transfer in a cylindrical spouted bed dryer with a draft tube. They found that the proposed model predicts the mass transfer rate very well, but over-predicts the heat transfer rate. From the above review of the literature, it is found that the investigations of heat transfer in
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spouted and fluidized beds are limited in that the experimentally verified phasic particle and gas temperature distributions are not available. In particular, almost all the existing results are provided by numerical simulation and the phasic experimental data are lacking. In the present
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study a pseudo two-dimensional bed described by Brown and Lattimer [9] is used to study the detailed phasic particle temperature distributions and solid concentration in the bed under
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fluidized and spouted regimes. The influence of specularity coefficient, which significantly affect the particle-wall boundary condition, is also studied.
2. CFD modeling
A two-fluid model based on the Eulerian–Eulerian approach is proposed to investigate the hydrodynamics and heat transfer in a pseudo two-dimensional gas–solid bed for fluidized and
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spouted regimes. In this approach continuous gas phase and the dispersed particles phase are treated as interpenetrating continua.
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2.1. Governing equations The governing equations and the associated constitutive equations of the Eulerian-Eulerian TFM that are used in the simulation of spouted beds are summarized in this section. The solid-
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phase contains mono-dispersed spherical granular particles with uniform diameter. The fluctuation of particles and the inter-particle collisions are considered using the kinetic theory of
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granular flow. The virtual mass and lift effects, which are expected to negligible, are neglected. The particle rotational effect and the associated lift become important for particles of large diameters that are not considered in the present study. The gas–solid interphase exchange coefficient, β, is modeled using the expression of Gidaspow et al. [43]. The solid shear viscosity is composed of collisional, kinetic and frictional effects. For very dense
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flows, the friction created between the particles generates a large amount of stress.
The
expression of Johnson and Jackson [44] is used to model the frictional viscosity in the dense
by Lun et al. [45].
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region. The bulk viscosity and the solid pressure are evaluated using the expressions suggested
by
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The continuity equation for the qth phase without any mass transfer between the phases is given
∂ (α ρ )+∇.(αq ρq υ q )=0 (1) ∂t q q
where αq , ρq and υ q , 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:
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∂ (α ρ υ )+∇.(αs ρs υ s υ s )=-αs ∇p-∇ps +∇.τ̿s +β(υ g -υs )+αs ρs g (3) ∂t s s s
where αs = 1 − αg .
Gas phase:
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∂ (α ρ H )+∇.(αg ρg υg Hg )=∇.αg Kg,eff ∇.Tg -hgs (Ts -Tq ) ∂t g g g
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The conservation of energy for the gas and solid phases are given by
Solid phase:
∂ (α ρ H )+∇.(αs ρs υ s Hs )=∇.αs Ks,eff ∇.Ts +hsg (Ts -Tq ) ∂t s s s
(4)
(5)
given as:
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The transport equation for granular temperature, Θ (fluctuation kinetic energy of particles), is
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3 ∂ ̅ τ̿g :∇υ s +∇.(ks ∇Θs )-γ -3βΘs (6) (ρ α Θ )+∇.(αs ρs υ s Θs ) =-∇ps I+ s 2 ∂t s s s 2.1.2. The constitutive equations for hydrodynamic: Solid and gas phase stress tensors:
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2 T τ̿s =αs µs ∇υ s +(∇υ s ) +(αs λs - αs µs )∇.υ s I ̅ (7) 3
T 2 τ̿g =αg µg ∇υ g +(∇υ g ) - ∇.υ g I̅ (8) 3
Radial distribution function:
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g0 =(1-
-2.5αs,max
αs αs,max
)
(9)
Collisional energy dissipation: ρs α2s Θ3/2 s (10)
Solid pressure [45]:
Ps =αs ρs Θs +2ρs (1+es )α2s g0 Θs
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d s √π
(11)
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γs =
12(1-e2s )g0
Solid shear viscosity (Gidaspow et al. [46]):
2 10ds ρs !πΘs 4 4 Θs 2 1+ αs g0,ss (1+ess ) (12) µs = αs ρs ds g0 (1+ess ) + π 96(1+ess )g0,ss 5 5
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1
Frictional viscosity (Johnson and Jackson [44]): µs,fric =Fr.
(αs − αs,min )n
αs, max − αs
p "#$ % (13)
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where Fr, n and p are empirical parameter the model.
The diffusivity of granular temperature is given as [46]: 2 Θs 2 6 1+ αs g0 (1+es ) +2αs ρs ds g0 (1+es ) (14) 384(1+es )g0 5 π s
1
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kΘs =
150ρs d !πΘs
Solid bulk viscosity is given by [45]
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4 Θs λ= αs ρs ds g0,ss (1+ess )' (15) 3 π Gas–solid drag coefficient is given as (Gidaspow et al. [43]):
β=(1-φgs )βErgun +φgs βWen-Yu (16)
where
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2 +1.75
αg ds
αs ρg )υ s -υ g )
for αg ≤0.8
ds
αs αg ρg )υ s -υ g ) 3 βWen-Yu = CD α-2.65 g 4 ds
for αg >0.8
24 +1+0.15(αg Res )0.687 ,, CD = *αg Res CD =0.44,
Res <1000 Res ≥1000
Arctan-150×1.75(0.2-αs ). +0.5 π
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φgs =
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βErgun =150
α2s µg
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2.1.3. Constitutive equations for the thermal energy balance 2.1.3.1. Interphase heat transfer
The heat transfer coefficient is related to the Nusselt number, Nus , by the following equation: 6kg /s /g Nus
d2s
(17)
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hsg =
An empirical correlation for the interphase heat transfer coefficient was proposed by Gunn [47] which relates the Nusselt number with the particle Reynolds number and Prandtl number. That
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is,
Nus =7-10αg +5α2g 1+0.7Rep (Pr)3
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2
1
+(1.33-2.4αg +1.2α2g )(Rep )0.2 (Pr)3 (18)
where
Pr=
1
Cpq µq kq
2.1.3.2. Effective conductivities of the gas and particles phases
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The thermal conductivity of the particulate phase depends on the particle material and also on the nature of contacts of the particles in the fluidized bed. For a mathematical description of the heat-transfer rate in the two-fluid continuum model it is necessary to separate the overall bulk
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thermal conductivity into a thermal conductivity of the gas phase and the solids phase, [27]. For the gas phase, the effective thermal conductivity is given by [27],
1-!αs 1 Kg (19) αg
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Kg,eff = 0
The effective thermal conductivity of the solids phase can be expressed as:
where
kg -ωA+(1-ω)Γ. (20)
(A-1) B A (B-1) (B+1) 6 ln 2 B B 21- B3 2 21- 3 21- B3 A A A 2
4
A
and for spherical particles
/ 8 , ; = 1.25( )=>/@ , A = 7.26 × 10CD 89 /9
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7=
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Γ=
!αs 1
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Ks,eff =
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in which A represents the ratio of the particle contact area to the total particle surface area.
In experiments conducted by Brown and Lattimer [9], the inlet gas temperature is not constant and increases with time (Fig. 1). On the other hand Kg is a function of temperature, thus, Kg is also depend on time and this matter has been considered in the present model. Accordingly, Fig. 2 shows the variation of Kg with temperature, and Fig. 3 shows the variation of Kg with time as a
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cubic polynomial, which is implemented in FLUENT by means of user-defined functions (UDF) written in C-code.
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2.2. Simulation conditions The corresponding physical and numerical parameters selected for the present simulation are listed in Table 1. According to Hosseini et al. [48], Hosseini [49] and Shu et al. [50] particle-
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wall restitution coefficient of 0.2 is used in this study. In these earlier work [48, 49, 50] it was shown that the particle-wall restitution coefficient plays a significant role on the CFD results in
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the dilute zone of particles such as draft tube embedded in cylindrical spouted bed and downer reactor. 2.3. Simulation procedure
In the present simulations, a three-dimensional model in the Cartesian framework is used. Also in order to save computational time, half of the bed is considered assuming the symmetry
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condition. The CFD code FLUENT is utilized to simulate the hydrodynamics and heat transfer of the bed that was operated by Brown and Lattimer [9]. The set of governing equations
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described in Section 2.1 is solved by a finite control volume technique. The phase coupled SIMPLE algorithm, which is an extension of the SIMPLE algorithm for multiphase flow, was
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used for the pressure–velocity coupling and correction. The momentum and volume fraction equations were discretized by a first-order upwind scheme. The adaptive time step in the range of 0.00001–0.0005 was used with about 100 iterations per time step. A convergence criterion of 10-3 for each scaled residual component was specified for the relative error between two successive iterations. The grid structures for the system under study are illustrated in Fig. 4.
2.4. Simulation system
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As mentioned before, the experimental data for the bed of Brown and Lattimer [9] for fluidized and spouted regimes is used to validate the present CFD model. Brown and Lattimer [9] performed a set of experiments in a pseudo two-dimensional fluidized bed, shown in Fig. 5,
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with cross-section dimensions of 56.4×4.95 mm2, static and overall bed heights of 49.75 mm and 280 mm, respectively. They used the new technique of infrared radiation spectroscopy to measure the rate of heat transfer in the entire bed. The advantage of this approach is it is non-
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intrusive and can provide experimental data for the gas–solid flows. It is worth mentioning that the method is not limited to steady-state condition and the detailed localized and time-dependent
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data needed for the validation of CFD models can be measured.
The base of the bed is a stainless steel distributor plate with a single slit jet of 1.6 mm wide and depth of 4.95 mm. The outside of the plate was coated by a low thermal conductivity silicon rubber insulation to reduce thermal energy losses. The back and side walls consisted of
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polymethyl–methacrylate coated in 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 particles are related to the Geldart B classification with mean diameter of 550 µm.
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2.5. Initial and boundary conditions
A Dirichlet boundary condition at the bottom of the bed is used to specify a uniform gas inlet
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velocity. The outflow condition with zero velocity gradients is specified at the top of the freeboard. A no-slip 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 [44] wall boundary condition is used for the tangential velocity, and granular temperature of the solid phase at the wall, with specularity coefficient in range of 0.025–0.2. As mentioned before, the inlet air temperature varies with time. The walls are assumed adiabatic. Initially, the particle concentration in the
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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.
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3. Results and discussion 3.1. Grid independency test
The grid independency was investigated by comparing the simulation results for three grid sizes.
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Fig. 6 shows the contour plot of gas volume fraction and time average particle volume fraction in axial direction of the bed. This figure shows that the simulation results obtained by fine (27,520
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cells) and medium grids (21,280 cells) are identical, but the coarse grid (13,200 cells) results show a discrepancy with the finer grid sizes. Therefore, the medium grid was used in the rest of the simulations. The mesh size is taken to be 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. 3.2. Spouted regime
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3.2.1. Effect of specularity coefficient on heat transfer The effect of important term in solid boundary condition, namely specularity coefficient, on
the results of heat transfer distribution is investigated, here. The specularity coefficient, E, is a
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parameter used in computational modelling to determine whether the wall is smooth (E → 0) or
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rough (E → 1) determining the solids tangential velocity imposed at the wall. The specularity
coefficient varies with a number of factors including the material of the wall, the type of particles, and the geometry of the walls. However, there are no generic correlations available in the literature, which relates the specularity coefficients to different factors. Thus, developing such a correlation would be very useful for the numerical simulation and design of gas-solid systems.
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Fig. 7 shows the effect of specularity coefficient on gas volume fraction through the spouted bed. Here, sensitivity of the results to different values of specularity coefficient (0.025-0.2) is studied. The experimental measurements by Brown and Lattimer [9] are also shown in the figure for
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comparison. It is observed that the value of specularity coefficient significantly affects the fountain height and particles behaviour through the bed. Particularly, the fountain height increases by decreasing the value of specularity coefficient. Fig. 7 also shows that an appropriate
SC
value for this coefficient is 0.025. According to the simulation results, particle concentration in the fountain is higher than that in the spout, but lower than the annulus, which is in agreement
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with the experimental observations. The CFD results for the hydrodynamics and heat transfer in the bed are sensitive to the value of specularity coefficient. This could be in part because of low thickness of the bed. In addition, Fig. 7 shows that the specularity coefficient of 0.025 leads to incoherent spouting, which is consistent with the experimental finding of Brown and Lattimer
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[9].
Fig. 8 shows the contour plot of gas volume fraction for spouted regime at φ=0.025 for different times. This figure shows that the behaviour of particles is unsteady and incoherent spouting is
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clearly seen. The predicted gas volume fraction shown in Fig. 8 are consistent with the numerical results of Zhao et al. [21] that were obtained using the DEM approach and simulation results of
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Hosseini et al. [25] performed using the TFM approach of MFIX code for a thin slot rectangular spouted bed. According to Fig. 8 the spout geometry and the resulting fountain height are not steady and change periodically with time. Moreover, the shapes of bed free surface of both experimental and numerical results are concave, which further confirms the validity of the present CFD model.
13
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Fig. 9 shows the effect of different values of specularity coefficient on simulation results of temperature distribution inside the bed. As seen in Fig. 9, with increase of specularity coefficient, the dead zones that appeared in the annulus portion increases, while the fountain height
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decreases. In addition, the incoherent spouting occurs for φ=0.025, while for other larger values of this coefficient stable spouting appears as seen in Fig. 7. This in turn results for the nonuniform temperature distribution of particles in the spout region for φ=0.025 and uniform
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temperature distributions for other values of specularity coefficient. Unless stated otherwise the specularity coefficient of 0.025 is used in the rest of simulations.
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3.2.2. Fountain height
Lan et al. [26] suggested that the fountain height is a key parameter that can be used to determine the accuracy of the numerical models for describing the hydrodynamic behaviour of spouted beds. In addition, Hosseini et al. [24] in their simulation of cylindrical spouted beds showed that
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by selecting appropriate value of particle-particle restitution coefficient the CFD model can predict accurate results for the fountain height. The predicted fountain heights versus spouting velocity as a dimensionless quantity are plotted in Fig. 10 and compared with experimental
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measurements of [51]. Good agreement of the CFD results with the experimental data [51] in
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terms of fountain height is found.
3.2.3. Temperature distribution Fig. 11 compares the simulation and experimental results of the time dependent particle temperature variations for the spouted regime. It is noticeable that the gas-to-particle heat exchange rate affects on the particle surface temperature entire the bed. A small fraction of gas phase passes through the annulus, making particle-to-particle exchange a significant heat transfer factor, while gas-to-particle exchange is insignificant. The high particles residence time in the
14
ACCEPTED MANUSCRIPT
annulus region leads to uniformity between the gas and particle temperatures in this region. Therefore, a uniform particle temperature distribution with value of 296 °K is clearly observed in the annulus zone for both simulation and experiment.
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Fig. 11 also shows that the highest particle temperature is not at the gas inlet region but at the height slightly above it. The particles entrain from the fountain to the top of the annulus as shown in Figs. 7 and 8. Also, the timeline of particles temperature distribution shown in Fig. 11
SC
reveals that particle temperature in the bed increases with time. This figure also shows that in spout and fountain regions, where the particles concentration is low, the particles temperature are
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higher. Similarly, moving from spout to fountain base, because of the increase in the particle concentration at fountain base, the particle temperature decreases. Also moving from the fountain base to fountain top, the particle temperature increases due to decrease of particle
3.3. Fluidized regime
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concentration. Fig. 11 shows good agreement between the CFD results and experimental data.
Brown and Lattimer [9] reported that for gas inlet velocity of 1.6Umf the fluidized regime
EP
forms in the bed. Similar to the study that was done for spouted regime, the time-dependency of thermal conductivity of gas phase for transient inlet air temperature for fluidized regime is also
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studied. A UDF was developed and implemented in the computational model for this purpose. Fig. 12 compares the particle velocity vector fields for spouted and fluidized regimes, respectively, at Ug=3Umf and Ug=1.6Umf. As shown in Figs. 7, 8 and 11a, in spouted regime (Ug=3Umf) three distinct regions, namely, the spout, the fountain and the annulus, are formed. The solids above the bed surface are entrained by the spout and then rains down on the annulus are designated as the fountain. However, in fluidized regime (Ug=1.6Umf) fountain is not formed and particles move in the radial direction at the top of the bed as shown in Fig. 12.
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A comparison between simulation and experimental results of gas volume fraction under fluidized regime is shown in Fig. 13. As mentioned before, the fountain is not formed in this case. However, similar to the spouted regime, for fluidized regime the particle flow is unsteady
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and the instability in this case is much more than the spouted regime. This is due to the breakage of the central bubble that is attained to the top of the bed and then reformation of bubble from the nozzle as a cyclic process.
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Fig. 14 shows the contour plot of gas volume fraction for the fluidized regime with φ=0.025 and Ug=1.6Umf for different times. This figure clearly shows that the behaviour of particles
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periodically changes with time. The cause of this matter is the central bubble that is attained to the top of the bed breakages and in a cyclic process the reformation of bubble from the nozzle continues, which affect on the particles behaviour.
Fig. 15 compares the simulation and experimental results of the time dependent particle
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temperature distributions for the fluidized regime. The low particle temperature regions are surrounding the jet inlet due to negligibly small convective gas-to-particle heat transfer in those regions. The highest particle temperature is recorded at the top surface of the bed. This is due to
EP
two main reasons with respect to the thermal-dynamics of the bed and to the measurement technique used. The first and main reason is convective gas-to-particle heat transfer occurs at the
AC C
top of the bed from the highly turbulent bubble eruptions. Secondly, the contact intensity between the high rate of gas and a particle occurs in the top of the bed as the particle velocity decreases to zero, which leads to increase of the intra-particle temperature in that region. As can be seen in Fig. 15, simulation results of transient particle temperature distribution for fluidized regime and experimental observations show the same trend. From Figs. 11 and 15, the CFD
16
ACCEPTED MANUSCRIPT
predictions for particle temperature distribution for the spouted regime were in a better agreement with the measurements, when compared with that for the fluidized regime.
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4. Conclusion In the present study, the transient gas to particle heat transfer for spouted and fluidized regimes was studied using an Eulerian–Eulerian two-fluid model including the kinetic theory of
SC
granular flows. Heat transfer models for effective conductivities of the gas and solids phases were added as compiled executable code using the UDF programming. Time dependent inlet air
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temperature led to increase of the particles temperature distribution through the bed. A critical hydrodynamic modeling parameter (specularity coefficient) was evaluated in terms of fountain height and production of three distinct regions in spouted regime. It was found that this coefficient has a great effect on particles behaviour through the bed and formation of
studied.
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incoherent spouting. In addition, effect of this parameter on the heat transfer of CFD model was
The simulation results indicated that the CFD results for hydrodynamics and heat transfer distribution because are very sensitive to the value of specularity coefficient. The sensitivity
EP
could be due to the low thickness of the bed.
Using the specularity coefficient of φ=0.025
AC C
seems to lead to the optimal results for the cases studied. Simulation results of the time dependent particle temporal variations in the spouted regime were compared with the experimental data and good agreements were found.
The temperature
distributions in three distinct regions of spouted bed were also investigated. The CFD results suggests that the highest particle temperatures were not at the base of the gas inlet but rather at a slightly distance about the inlet nozzle in the spout and fountain zones. prediction was in agreement with experimental observations.
17
This numerical
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CFD results for the temperature distribution in the spouted regime showed that the particle temperature in spout and fountain regions are not uniform and increase from spout to fountain base and from fountain base to fountain top due to incoherent behaviour of the particles inside
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the bed.
Finally, the particles temperature distributions and gas volume fraction for the fluidized regime
SC
were also investigated and a closed agreement was found with the experimental observations.
CD
: drag coefficient [-]
ds
: particle diameter [m]
Dp
: Depth of the bed [m] : Vessel heigh [m]
W
: width of the bed [m]
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H
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Nomenclature
: particle-particle restitution coefficient [-]
g
: acceleration due to gravity [m/s2]
H0 IG I2D
AC C
g0
EP
es
: radial distribution coefficient [-]
: Static bed depth [mm] : stress tensor [-] : second invariant of the deviatoric stress tensor [-]
18
ACCEPTED MANUSCRIPT
: diffusion coefficient for granular energy [kg/m s]
β
: gas/solid momentum exchange coefficient [kg/m3s ]
βErgun
: gas/solid momentum exchange coefficient by Ergun equation [kg/m3s ]
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kΘs
βWen-Yu : gas/solid momentum exchange coefficient calculated by Wen-Yu equation[kg/m3s ] : pressure [N/m2]
Ps
: solids pressure [N/m2]
t
: Time [s]
U
: superficial gas velocity [m/s]
HI
: velocity [m/s]
Umf
: minimum fluidization velocity [m/s]
T
: Temperature [K]
K
: thermal conductivity [w/m.k]
Cp
: specific heat capacity [J/Kg.K]
Nu
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TE D
EP
AC C
h
SC
P
: Enthalpy [J/Kg]
: Nusselt number [-]
Re
: Reynolds number [-]
Pr
: Prandtle number [-]
19
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Greek letters /I
: volume fraction [-] : the collisional dissipation of energy [kg/s3m]
J
: granular temperature [m2/s2]
τ́ I
%
: density [kg/m3] : stress tensor [N/m2]
: angle of internal friction [deg]
E
: specularity coefficient [-]
Subscripts : collision
fr
: friction
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g
EP
col
kin
SC
MI
: shear viscosity [kg/m s]
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LI
: solid bulk viscousity [kg/m s]
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K
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γs
: kinetic
: gas
p
: particle
q
: phase type (solid or gas)
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ACCEPTED MANUSCRIPT
s
: solids
T
: stress tensor
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University, 2012.
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Captions
Table 1. The physical and numerical parameters.
Fig. 3. Variation of air thermal conductivity with time.
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Fig. 4. Grid structures for the computational domain.
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Fig. 2. Variation of air thermal conductivity with temperature [47].
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Fig. 1. Transient inlet air temperature ramp for the spouted and fluidized regimes [9].
Fig. 5. Schematic of the 2D flat bottom rectangular column setup with dimensions in mm [9].
Fig. 6. Effect of mesh size on CFD results, (a) contour plot of gas volume fraction, (b) particle
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volume fraction at axis of the bed
Fig. 7. Effect of specularity coefficient on simulation results of gas volume fraction
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Fig. 8. Contour plot of particle concentration for different times for spouted regime.
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Fig. 9. Effect of specularity coefficient on simulation results of particle temperature distribution.
Fig. 10. The predicted fountain heights versus spouting velocity
Fig. 11. Comparison of simulation result of time dependent particle temporal variations in the spouted regime with experimental observations, (a) Experimental, (b) Simulation.
Fig. 12. Comparison of solid velocity vectors in Ug=3Umf for spouted and in Ug=1.6Umf for fluidized regimes, (a) spouted regime, (b) Fluidized regime.
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Fig. 13. Comparison of simulation and experimental results of gas volume fraction in fluidized regime, (a) Simulation, (b) Experimental.
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Fig.14. Contour plot of particle concentration for different times for fluidized regime.
Fig. 15. Comparison of simulation result of time dependent particle temporal variations in the
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fluidized regime with experimental observations, (a) Experimental, (b) Simulation.
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Table 1. Physical and numerical parameters
Description
Simulation
d (μm)
Particle diameter
550
ρ (Kg/m3)
Particle density
2500
ρ (Kg/m3)
Gas density
1.22
μ (Kg/m.s)
Gas viscosity
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symbol
.
×
Minimum fluidization velocity
0.24
Ug,s
Superficial velocity in spouted regime
3Umf
Ug,f
Superficial velocity in fluidized regime
1.6Umf
ϕ (-)
Sphericity
1
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Static bed depth
49.75
ϕ (-)
Internal friction angle of particles
28.5
-
Geldart Group
B
es (-)
Particle–particle restitution coefficient
0.8
ew (-)
Wall to particle restitution coefficient
0.2
α, (−)
Maximum volume fraction of particles
0.62
(-)
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H0 (mm)
Specularity coefficient
0.025-0.2
Bed height
280
Bed width
56.4
Bed depth
4.95
Ks (W/m.k)
Thermal conductivity of solids
1.05
Cp,s (KJ/Kg.k)
Specific heat capacity of solids
0.84
Cp,g (KJ/Kg.k)
Specific heat capacity of gas
1.006
H (mm) W (mm)
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D (mm)
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Highlights ► CFD study of transient heat transfer in pseudo 2D bed for spouted & fluidized states ► Specularity coefficient affects the voidage and particles temperature in the bed
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► In spouted state incoherent spouting was occured by proper specularity coefficent ► Highest particle temperatures were discerned in spout and fountain regions
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► Highest particle temperatures were discerned along the jet in fluidized state