Solid residence time distribution in a cross-flow dense fluidized bed with baffles

Chemical Engineering Science 200 (2019) 320–335

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Solid residence time distribution in a cross-flow dense fluidized bed with baffles Leina Hua a,⇑, Hu Zhao a,b, Jun Li a, Qingshan Zhu a, Junwu Wang a a b

State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, PR China University of Chinese Academy of Sciences, Beijing 100049, PR China

h i g h l i g h t s
The effects of baffles on a cross-flow gas-solid dense fluidized bed are studied.
Eulerian-Eulerian model is applied to simulate solid RTD for free and baffled beds.
The predicted lateral solid dispersion coefficient is comparable to literature data.

a r t i c l e

i n f o

Article history: Received 26 October 2018 Received in revised form 4 January 2019 Accepted 22 January 2019 Available online 23 February 2019 Keywords: Cross-flow dense fluidized bed Baffle Non-spherical particle Solid residence time distribution Solid dispersion coefficient Eulerian-Eulerian model

a b s t r a c t The residence time distribution (RTD) of particles in a gas-solid dense fluidized bed with a continuous solid flow operation and baffles needs much more focus for the purpose of the wide industrial applications. The available models of solid RTD are not adequate in this case due to the additional convective diffusion induced by the cross-flow of the solid feeding and the complex geometry caused by the baffles. To address this problem, this work applied a two-phase Eulerian-Eulerian model combined with the species transport equation to predict solid RTD in a dense fluidized bed. The effects of the continuous solid feeding and baffles were considered. The solid dispersion coefficient Ds in the species transport equation was calculated by an analytical solution from kinetic theory of granular flow. Ds only needs the value at a molecular level due to the fact that the computational fluid dynamics (CFD) model is able to reproduce the RTD procedure exactly same as the practical experiments. To validate the established CFD model, a series of 3D lab-scale cold flow experiments were conducted in the free and baffled beds for the bed material with various severe non-spherical shapes. The measurement included the solid hydrodynamic characteristics at the outlet of the outflow pipe and solid RTD of the system. The reasonable agreement between the CFD prediction and experimental data demonstrated the good performance of the CFD model. 1D plug flow with dispersion model and the tanks-in-series model were further used to fit the calculated RTD curves. The estimated lateral dispersion coefficient of solids locates in a rational range compared with the data collected extensively from the literature. The results showed that the lateral dispersion coefficient of solids decreases when the baffles are installed. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction As a crucial index of measuring the macroscopic mixing of particles, residence time distribution (RTD) of solids is always the focused topic in both the research and industrial fields (Gao et al., 2012; Hua and Wang, 2018; Nauman, 2008; Zhang and Xu, 2015). It determines the product directly because the inadequate residence time of the reactant in the reactor can lead to an overor under-reaction not beneficial to the industrial companies. The ⇑ Corresponding author. E-mail address: [email protected] (L. Hua). https://doi.org/10.1016/j.ces.2019.01.054 0009-2509/Ó 2019 Elsevier Ltd. All rights reserved.

research on solid RTD has been explored widely in a circulating fluidized bed (CFB) riser or other CFB parts due to the fact that the particles pass the CFB riser and downcomer continuously to form an external loop (Bi, 2004; Breault, 2006; Harris et al., 2003). In contrast, RTD of solids is seldom investigated in a batch-mode dense fluidized bed that is operated in bubbling, slugging or turbulent fluidization regime. This is because particles always reside in the system since the superficial gas velocity is not high enough to entrain most of particles out of the bed. In some industrial applications (Sette et al., 2015), when a continuous solid flow operation is added to a dense bed, however, RTD of particles becomes accessible. The particles have the path to enter and exit the reactor to

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Nomenclature inverse of relaxation time (s1) cross-sectional area of bed (m2) a parameter related to the variance of particle velocity (dimensionless) c0 tracer concentration at the injection point (arbitrary) c(Z, h) tracer concentration at (Z, h) (arbitrary) C(Z, h) dimensionless tracer concentration at (Z, h) (dimensionless) C(1, h) dimensionless tracer concentration at (1, h) (dimensionless) Cd0 drag coefficient of one particle (dimensionless) Cb model coefficient (dimensionless) dn diameter of the sphere with equivalent projected area (m) dp particle diameter (m) dv equivalent-volume diameter (m) D dispersion coefficient (m2/s) equivalent-area bed diameter (m) Dbed Ds dispersion coefficient of solids (m2/s) Dsa axial dispersion coefficient of solids (m2/s) Dsr lateral dispersion coefficient of solids (m2/s) Ds,molecular dispersion coefficient of solids due to molecular diffusion (m2/s) Ds,turbulent dispersion coefficient of solids due to turbulence (m2/ s)
D rate of strain tensor of the solid phase (s1) ess particle–particle restitution coefficient (dimensionless) E(t) RTD curve (s1) E(h) dimensionless RTD curve (dimensionless) Ebed(t) RTD curve of bed system (s1) Ebed(h) dimensionless RTD curve of bed system (dimensionless) Einput(t) RTD curve of input system (s1) Eoutput(t) RTD curve of output system (s1) f(Usj, t|Usj0, t0) the probability that a particle with the initial velocity Usj0 at time t0 will have the velocity Usj at time t (dimensionless) Fr constant in the equation of pfs (N/m2) g gravitational acceleration (m/s2) g0 radial distribution function (dimensionless) Gs solid feeding rate (kg/m2s) H bed height (m) I2D second invariant of the deviator of the strain rate tensor of solid phase (s2)
I unit tensor (dimensionless) K1 Stokes’ shape factor (dimensionless) K2 Newton’ shape factor (dimensionless) L distance between the injection and detection points of tracer (m) m exponent in the equation of pfs (dimensionless) n exponent in the equation of pfs (dimensionless) N number of ideal mixed tank (dimensionless) p gas pressure (Pa) ps solid phase pressure (Pa) pfs solid frictional pressure (Pa) Pe Peclet number (dimensionless) q12 gas-solid velocity covariance (m2/s2) Qs,sample time-averaged mass flow rate of solids in the sampling duration (g/s) r radial coordinate in a cylindrical coordinate system (m) Res particle Reynolds number (dimensionless) t time (s) tm mean residence time of RTD curve (s) mean residence time of bed RTD curve (s) tm,bed a A b

tm,input tm,output t u ug, us Ug Umf Up Ut v V Va x y Ytracer z

mean residence time of input RTD curve (s) mean residence time of output RTD curve (s) true value of the mean residence time (s) convective velocity in plug flow with dispersion model (m/s) velocity of the gas phase and solid phase, respectively (m/s) superficial gas velocity (m/s) minimum fluidization velocity (m/s) convective velocity of tracer (m/s) particle terminal velocity (m/s) volumetric flow rate of solids (m3/s) packed volume of particles in the vessel (m3) active volume of particles in the vessel (m3) length direction of bed (m) width direction of bed (m) mass fraction of tracer (%) depth direction of bed or vertical coordinate (m)

Greek letters b inter-phase drag coefficient (kg/m3s) bErgun drag coefficient calculated by Ergun correlation (kg/ m3s) bWen-Yu-Ganser drag coefficient calculated by Wen-Yu and Ganser correlation (kg/m3s) chs collisional dissipation rate of the granular fluctuating energy (kg/ms3) Dp pressure drop (KPa) Dt elapsed time (s) Dx displacement of a tracer particle (m) eg voidage (dimensionless) emf voidage at the minimum fluidization condition (dimensionless) es solid volume fraction (dimensionless) esmax solid volume fraction at the maximum packing state (dimensionless) esmf solid volume fraction at the minimum fluidization condition (dimensionless) esmin solid volume fraction beyond which frictional force occurs (dimensionless) h dimensionless time (dimensionless) hs granular temperature (m2/s2) ks solid bulk viscosity (Pas) lg gas viscosity (Pas) ls solid shear viscosity (Pas) lfs solid frictional viscosity (Pas) fr calculated by the gas-solid relative velocity and the gas turbulent kinetic energy (dimensionless) qg, qs gas and particle density, respectively (kg/m3) r2t variance of RTD curve (s2) r2t;bed , r2t;input , r2t;output variance of bed RTD, input RTD, and output RTD curve, respectively (s2) s space time (s) st1 ,sc2 the time scale of the energetic turbulent eddies of gas and the relaxation time of particles due to the interparticle collision time, respectively (s) st12 ,sF12 the time scale of the gas turbulent motion viewed by particles and the relaxation time of particles due to drag with the gas, respectively (s)     sg ,ss stress tensor for the gas and solid phase, respectively (Pa) u specularity coefficient (dimensionless) uf internal friction angle of particles (degree)

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ugs

w xn

L. Hua et al. / Chemical Engineering Science 200 (2019) 320–335

inverse tangent function (dimensionless) sphericity (dimensionless)
 positive root of tanxn ¼ 4xn Pe= 4x2n  Pe2 (dimensionless)

form a cross-flow. The available publications on this subject are quite scarce because of the more complex feature of a cross-flow bed. The vertical or horizontal heat exchanger tubes or other form baffles are often placed in the industrial reactors to change the properties of RTD curves, with a purpose of raising the conversion of reactant and improving the selectivity of the desired product (Cui et al., 2006; Kunii and Levenspiel, 1991). The effects of baffles on solid mixing in a batch-mode dense fluidized bed are mainly studied by the experimental techniques, and the data are still far from abundant (Geldart, 1986). Kato et al. (1985) installed the vertical or horizontal multi-tube internals in a 3D lab-scale rectangular column to investigate the influences of internals on the lateral dispersion coefficient of solids Dsr. Dsr was estimated by means of fitting the experimental data with the prediction of a 1D diffusion model (May, 1959). The results showed that Dsr in a baffled bed is much smaller than that in a free bed without internals. This indicates that the baffles are quite effective in controlling the solid mixing rate. Ramamoorthy and Subramanian (1981) also applied the springs and rods as the vertical internals to study the axial solid mixing in a lab-scale bubbling column. 1D diffusion model was used to calibrate the axial dispersion coefficient Dsa as well. They took some bed material colored as the tracer particles and arranged the distance between the tracer injection and detection large enough along the bed height for an axial measurement. But a similar conclusion was drawn by Ramamoorthy and Subramanian (1981) that Dsa decreases in the presence of baffles and further decreases with a reduction of internal voidage (defined as a ratio of internal volume to bed volume). The setting of the horizontal internals in a fluid coker stripper also showed a reduction of Dsa clearly in Cui et al. (2006)’s experimental study. These limited works show evidently that the baffles alter the solid motion in the bed essentially and then have a critical impact on the solid mixing. Unfortunately, solid RTD is not involved in the above studies due to a batch operation of a dense bed. Recently, Kong et al. (2018) investigated the effects of baffles on solid RTD of the nonspherical particles in a lab-scale cross-flow bubbling fluidized bed experimentally, in which NaCl salt was injected into the bed less than 1 s to implement a tracer stimulus response measurement. The baffle setting included two top baffles and one bottom baffle vertically arranged in a long and shallow bed. Their measurements showed that the added baffles in the bed could enhance the plug flow. Then it could be expected that in a cross-flow dense bed, the baffles are able to affect the solid mixing, and then the solid RTD and dispersion coefficient of solids significantly. The effects of the operating conditions, such as Ug and Gs, on RTD curves have become another most concerned point in the literature. For a cross-flow dense bed, another operating condition, solid feeding rate Gs, is added to the superficial gas velocity Ug. This may change the solid circulation pattern in the bed greatly (GuíoPérez et al., 2013a; Schlichthaerle and Werther, 2001; Sette et al., 2015). To evaluate a RTD curve quantitatively, mean residence time tm, Peclet number Pe, or dispersion coefficient of solids Ds are applied frequently (Hua and Wang, 2018). tm is the first moment of the measured RTD curve, and Pe or Ds has to be calibrated in terms of the various adequate models (Levenspiel,

Subscripts g gas phase s solid phase

2012). Bi et al. (1995) designed a coarse-fine cross-flow fluidized bed, in which the continuous feeding coarse particles move in a lateral direction through a long, narrow, and shallow bed and meanwhile are fluidized by the vertical flowing gas with fine particles. The measured tm of coarse particles increases both with Ug and Gs. By applying 1D plug flow with dispersion model (Levenspiel, 2012) to interpret the tested RTD curves and then to calculate Ds, they reported a decrease of Ds with Ug and an increase of Ds with Gs. Recently, Sette et al. (2015) and (2016) conducted a series of experiments in a large-scale gas-solid dense bed under the industrial conditions, in order to investigate the effects of the cross-flow induced by the solid continuous feeding. They found that tm decreases when the excess velocity, Ug – Umf, increases, while Ds is in direct proportion to Ug – Umf, where Umf is the minimum fluidization velocity of bed material. The opposite trends of Ds and tm with Ug in above two studies might be attributed to the significantly distinct fluidized bed materials for which Bi et al. (1995) used 8–10 or 10–13 mm-diameter coarse coal particles and 0.22 mm-diameter fine silica gel particles and Sette et al. (2015) and (2016) used 60 lm-diameter bronze particles. The experimental founding of Kong et al. (2018) reported that Pe increases when Ug increases in a cross-flow dense bed without baffles, indicating a nearly plug flow at a high gas velocity. In order to predict solid RTD in a gas-solid dense fluidized bed, various models have been proposed in the literature. The first option is to use the countercurrent backmixing model based on the bubble wake transport theory (Van Deemter, 1967). This model assumes that the dense bed is comprised of the two phases, the wake phase and emulsion phase. The tracer concentration in these two phases is transported by the bed hydrodynamics, such as solid velocity and volume fraction of the two phases (Grasa and Abanades, 2002; Shen et al., 1995). More importantly, a crucial parameter, the exchange coefficient of solids between the wake phase and emulsion phase, has to be determined prior to the calculation. A reasonable evaluation of this coefficient, however, is difficult (Lim et al., 1995). Therefore, some parameters in this model are frequently estimated by the empirical correlations or are considered to be adjustable to fit the experimental data. Another common alternative of RTD prediction is to use a plug flow with dispersion model. If a continuous feeding of solids is imposed to a batch-mode bed, a convective term has to be added to 1D or 2D diffusion model, which leads to a plug flow with dispersion model. To predict RTD by a plug flow with dispersion model, the estimation of Pe, defined as a ratio of the convective term to the diffusive term, is needed beforehand. In other words, Ds as the diffusive term must be known. Although more than a dozen of Ds empirical correlations, closely related to the operating conditions and particle properties, have been proposed (Niklasson et al., 2002; Winaya et al., 2007), most of them work only for the batch-mode beds. For a bed with a continuous solid flow or with the multi-internal geometry, the physical mechanisms behind solid mixing are more complex. Then these empirical correlations of Ds might not be applicable any more. In summary, with regard to the present RTD prediction models, the countercurrent backmixing model relies heavily on the empirical correlations, and a reasonable determination of Ds in a plug flow with dispersion

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model is difficult. Therefore, an adequate model is urgently demanded in order to predict solid RTD in a cross-flow and baffled dense fluidized bed. Nowadays computational fluid dynamics (CFD) technique has become so prevailing that it begins to work as a necessary tool to assist the experimental study in the field of fluidization engineering, due to its advantages in modeling the complex geometry flexibly and obtaining the flow details conveniently. To predict solid RTD, two different level CFD models have been attempted in the literature in terms of how the motion of particles is described. Eulerian-Eulerian model, regarding both the gas and particles as the continuum media, has the outstanding merits in simulating the large-scale reactors and long-time reactions, due to its pseudo-fluid assumption of particles. In contrast, EulerianLagrangian model is able to obtain the detailed information of all particles at a particle-scale by means of tracking each particle; however, this model is usually limited to the application on the small-scale domain, short duration, large-size particles or 2D problem in order to save the computational cost. Table 1 summaries the relevant CFD studies on RTD prediction in a gas-solid system in the literature. It can be seen in Table 1 that Eulerian-Eulerian model in combination with the species transport equation has shown a quite satisfying performance in capturing solid RTD in several applications of fluidized bed. Li et al. (2009) discussed the effects of the secondary gas injection on a bubbling fluidized bed, but gas RTD became their most concerned topic. Liu et al. (2012) conducted a CFD simulation of solid mixing and RTD in a FCC stripper with and without the V-form internals, respectively. Their results suggested that the V-baffled form can improve the stripping efficiency significantly. No comparison with experimental data, however, was done in their work. Geng et al. (2017), Zou et al. (2017a), and Zou et al. (2017b) also tried to predict solid RTD in a gassolid bubbling fluidized bed, but how to set solid dispersion coeffi-

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cient was not presented in their works, in spite of its critical effects. The applicability of the Eulerian-Eulerian model and species transport equation in modeling RTD has been confirmed in our previous work as well (Hua et al., 2014) for the case of a CFB riser. During the modeling, the estimation of Ds, an input parameter in the species transport equation, seems to be important since it determines the diffusion of the tracer particles in the system and then affects RTD curve. Hua et al. (2014) investigated the effects of Ds on solid RTD in a CFB riser. Different theoretical and empirical correlations were used to estimate Ds. The results showed that Ds has a minor influence on solid RTD compared with the convection effect induced by the high-velocity carrier gas. The behavior of particles in a low-velocity fluidized bed, however, differs very much from that in a high-velocity fluidized CFB riser. In this study, a reasonable choice of Ds is made for a gas-solid dense fluidized bed. The present work aims to establish a CFD model to predict solid RTD in a baffled dense fluidized bed with a continuous solid flow reasonably well. To validate the established CFD model, a 3D labscale dense fluidized bed is constructed to measure the solid hydrodynamics and RTD curve. The test bed material contains the non-spherical particles. A free bed without any internal and a baffled bed with three vertical internals are set to investigated the effects of baffles. The results of the CFD model are compared with the experimental data, including the variation of solid mass flow rate and solid RTD at the bed outlet. Further, Ds is estimated by 1D plug flow with dispersion model on the basis of the predicted RTD curve, and then is compared with the available data in the literature. The effects of baffles on solid RTD and Ds are discussed. 2. Experimental measurement A 3D lab-scale dense fluidized bed was constructed to study the effects of baffles on solid RTD. To obtain the RTD of solids, there-

Table 1 The relevant CFD studies on solid RTD in a gas-solid system. Researcher

Bed form

CFD model

Remarks

Li et al. (2009)

A batch-mode bubbling fluidized bed

To study the impacts of the secondary gas injection by means of gas RTD

Zhao et al. (2010)

The riser and downer of a CFB A CFB downer

Two-phase Eulerian-Eulerian model and species transport equation CFD-Discrete Element Method (DEM) Three-phase Eulerian-Eulerian model

Khongprom et al. (2012) Li and Guenther (2012)

A CFB riser

Liu et al. (2012)

A fluid catalytic cracking (FCC) stripper

Lan et al. (2013) Shi et al. (2015) Hua et al. (2014)

A CFB riser

Geng et al. (2017)

A cross-flow bubbling fluidized bed

Zou et al. (2017a)

A cross-flow bubbling fluidized bed

Zou et al. (2017b)

A cross-flow bubbling fluidized bed

Vollmari and KruggelEmden (2018)

A cross-flow dualchamber dense fluidized bed

A CFB riser

Two-phase Eulerian-Eulerian model and species transport equation Two-phase Eulerian-Eulerian model and species transport equation CFD-Particle-in-Cell (PIC)

Two-phase Eulerian-Eulerian model and species transport equation Two-phase Eulerian-Eulerian model and species transport equation Three-phase Eulerian-Eulerian model and species transport equation Two-phase Eulerian-Eulerian model and species transport equation CFD-DEM

2D simulation, a short RTD process; The effects of particle clustering on solid RTD in the riser and downer The tracer particles were regarded as another independent solid-phase in addition to the gas and primary solid-phase. This might need complex modeling for the interphase force and could lead to a high computational cost. The effects of gas-solid jet injection on the overall flow hydrodynamics and gas and solid backmixing The effects of different V-form internals on solid mixing and RTD in a counter-flow gas-solid dense bed Particles are grouped as a parcel-of-particle, which have the same properties, such as size, density, velocity, residence time, etc.; To simulate the lab-scale RTD experiments in (Bhusarapu et al., 2004) The effects of the tracer injection time and tracer sampling resolution on solid RTD; The relative importance of the convection and diffusion mechanism in a riser; The effects of solid dispersion coefficient in RTD simulation The effects of the tracer injection and operating conditions (i.e., gas velocity, solid mass flow rate and bed height) on solid RTD; To propose an semi-empirical approach to predict a RTD curve with a long tail The bed material contains the particles with two different sizes (20% coarse particles and 80% fine particles). A two-compartment bed structure was considered.

The effects of the particle shape (i.e., sphere, cube and cylinder), gas velocity and solid feeding velocity on solid RTD

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(a) experimental set-up

(b) simulation geometry

Fig. 1. The schematic illustration of a 3D lab-scale rectangular fluidized bed (unit: mm).

fore, a special continuous operation of particles was designed in this work. Fig. 1(a) illustrates the experimental set-up, in which all parts and their sizes are introduced. All experiments were conducted in a bed of 0.12 m width, 0.06 m thickness, and 0.5 m depth, as shown in Fig. 1(a). The free bed is defined here as the bed without any baffle. If the baffles are installed, three boards made of the transparent Plexiglas, the material same as that of the main bed body, are glued to the top, bottom, front and back walls of the bed, respectively. The specific locations of the three baffles are provided with the great details in Fig. 1(a). Due to an irregular shape and a size distribution, the test particles have to pass the screw feeder and U valve before entering the bed, in order to achieve a sufficient mixing. The inflow and outflow pipes with a long length and a large diameter were also designed to convey the particles as uniformly as possible. The length of the inflow and outflow pipes was 0.4 and 0.6 m, respectively. The diameter of these two pipes was 0.055 m. The tracer particles first entered into the inflow pipe from the tracer injector, and then flew into the main bed along the pipe. The tracer particles were collected at the end of the outflow pipe. In the experiments, the particles were observed to splash violently near the surface of bed, especially when with a high gas velocity. The active movement of particles in the near-surface area becomes the primary cause to push the particles out of the bed. A long and large-diameter outflow pipe therefore could guide the flow of all particles smoothly, and avoid the loss of tracer. For a baffled bed, the expected path of particles during the whole passing process is shown by the dashed arrow line in Fig. 1(a). The gas sparger at the base of the bed, a 6 mm-thick porous metal sintered plate which was agglomerated by particles of 50 lm diameter, could

effectively distribute the inputting gas uniformly. At the top of the bed, an expanding zone and a bag were used to settle and collect the entrained fine particles, respectively. Solid RTD was measured by a colored tracer method. The bed is considered to be operated at a steady state when it satisfies two requirements: (1) the time-averaged mass flow rate of solids at the outlet (i.e., the end part of the outflow pipe) is the same as that at the inlet (i.e., the beginning part of the inflow pipe); and (2) the size distribution of particles at the outlet is equivalent to that at the inlet. Then the colored particles began to be injected into the bed from the injector locating at the inflow pipe. At the same time, the tracer particles were collected and sampled at the end of the outflow pipe. To reduce the disturbance of the tracer particles to the main flow of the bulk bed material, the ratio of the tracer mass to the total bed inventory was controlled to be less than 5%. As for this study, 90 g tracer particles was used for all tests. The injection period was less than 5 s to implement a pulse-response RTD measurement. During the sampling, the particles were collected within 15 s to get the tracer concentration. The sampling frequency is the first and third 15 s of one minute in 10 min and only the first 15 s of one minute in the remaining period. To change the operating conditions or conduct the repeatability of test, all bed material had to be replaced to initiate a new test. This is because it takes a long time for all tracer particles to flow out of the bed when the solid mixing of the dense fluidized bed is quite heavy. In some cases, this task is even impossible if some dead zones exist. Replacing the bed material, therefore, can prevent the background pollution of the residual tracer. To prepare the tracer particles, some particles with a relatively narrow size distribution were sieved from the bulk material in the

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Fig. 2. The test particles observed by a scanning electron microscope (amplified by 30 times).

Table 2 The physical properties of the test particles and operating conditions. Particle property

Value

Particle composition Particle density (qs) Bulk particle density (packed closely) Mean Sauter particle diameter (dp) Maximum solid volume fraction (esmax) Wall friction angle Internal friction angle of particles (uf) Particle terminal velocity (Ut) Minimum fluidization velocity (Umf) Voidage at the minimum fluidization condition (emf) Sphericity (w) Superficial gas velocity (Ug) Mass flow rate of solids at the inflow pipe Tracer weight Tracer injection period

Calcium carbonate 3135 kg/m3 1429 kg/m3 341 lm 0.4558 16.1  20.8° 34.3  35.2° 1.1677 m/s 0.095 m/s 0.6305 0.325 0.4 and 0.45 m/s 5.33 g/s 90 g 5s

bed. The material of acid red 18 (C20H11N2Na3O10S3) first dissolved in the water, and then the sieved particles were painted by above solution for many times until reaching a uniform coloring state. Finally the colored particles were dried totally to be ready for the tests. This coloring method could ensure that the tracer particles have exactly same physical properties as the bulk bed material, which is one of the essential requirements of the RTD technique (Harris et al., 2003). The tracer concentration in each sample was defined here as the mass fraction of the tracer particles. The collected solid mixture was put into the water, and then the mass of tracer could be determined according to the concentration of acid red 18 in the solution. The fluidized particles in this study have a typical irregular shape and a size distribution, which are shown in Fig. 2. The basic physical properties of the test material and operating conditions are summarized in Table 2. The determination of particle sphericity by means of the characteristics at the minimum fluidization condition (i.e., the pressure drop, bed height, gas velocity and voidage) had been introduced in (Hua et al., 2015). The interested readers could refer to that work for more information. 3. Simulation setting The evident property of particles’ non-spherical shape makes the widely used interphase exchange models and the constitutive relations of the solid phase in Eulerian-Eulerian model to be not effective again. Due to the dominating importance of the interphase drag model in the multi-phase flow (Niemi et al., 2013;

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Wachem et al., 2001), we proposed a new gas-solid drag model in (Hua et al., 2015) to predict the behavior of the irregular particles in a conventional batch-mode dense fluidized bed. The works on the monodisperse or polydisperse particles with irregular shape in a gas-solid system are extensive via a multi-fluid EulerianEulerian framework, such as Peirano et al. (2001) and Deza et al. (2009) for the monodisperse particles, and Qiaoqun et al. (2005) and Fotovat et al. (2015) for binary particles. This study still used the same interphase drag model and Eulerian-Eulerian model as those in (Hua et al., 2015) to do the simulation. It has been shown in (Hua et al., 2015) that in a batch-mode dense fluidized bed the agreement between the CFD results and measured data, such as the axial profile of pressure, the radial distribution of solid volume fraction, and the bed expansion ratio, was reasonably good. In this study, the test bed material and experimental apparatus were not changed but for a cross-flow bed. Therefore, we herein mainly focused on the prediction of solid RTD in a cross-flow bed, in contrast with the only hydrodynamic simulation of a batch-mode bed in (Hua et al., 2015). The governing equations, the gas-solid interphase drag model, and the constitutive relations applied by Eulerian-Eulerian model are listed in Table A1 in the Appendix A. To calculate b in Table A1, dn, the diameter of the sphere with the equivalent project area, and dv, the equivalent-volume diameter of particles, have to be determined beforehand. To simplify the problem, we took dn  dv  dp, following the analysis in our previous study (Hua et al., 2015). To obtain the curve of the tracer concentration, Ytracer, as a function of time, the species transport equation was solved with the prescribed initial and boundary conditions. The initial conditions indicate a pulse or step injection of the tracer particles. The boundary conditions mean if the tracer particles are permitted to pass the boundary plane over and over again during one test (i.e., the open or closed boundary condition). In this study, the tracer particles were injected as a pulse in spite of a relatively long injection period. The boundary conditions at both the inlet and outlet were closed because no back-mixing of the tracer particles could occur through the long inflow and outflow pipes in practice. es and us in the species transport equation come from the transient solution of Eulerian-Eulerian model. As discussed in (Hua et al., 2014), however, the most difficult part of this approach lies in how to set the dispersion coefficient of the solid phase, Ds. In a single-phase flow, the dispersion coefficient of any species can be divided into two levels according to their formation mechanisms: molecular and turbulent. The ability of the molecular dispersion is within the range of the basic material properties and often is determined experimentally. The turbulent dispersion is caused by the turbulent characteristics of flow, and can be modelled by the turbulent kinetic energy and dissipation rate, Schmidt number, and Reynolds stress (Younis et al., 2005). Eulerian-Eulerian model regards the particles as a kind of the pseudo-fluid phase. Correspondingly, the dispersion coefficient of the solid phase in a gas-solid multiphase flow can be estimated from the molecular and turbulent levels, respectively. Since kinetic theory of granular flow has been the popular method to construct the basic material properties of the solid phase in Eulerian-Eulerian model, such as the shear and bulk viscosities, and so on, it naturally comes into the mind that kinetic theory of granular flow can also be used to estimate the solid dispersion coefficient at the molecular level, defined roughly as Ds,molecular (Gidaspow, 1994; Hsiau and Hunt, 1993; Savage and Dai, 1993; Simonin, 1996). On the other hand, various experiments of solid mixing and solid RTD in a dense fluidized bed have been conducted, and many empirical correlations of solid dispersion coefficient also are proposed to summarize its variation with the operating conditions and particle properties (Niklasson et al., 2002; Winaya et al., 2007). In this case, the estimated dispersion coefficient of solids, Ds,turbulent, generally incorporates several

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sophisticated factors except the turbulence, which may source from the specific conditions of experiment, the measurement techniques, and the modelling analysis. Consequently, the magnitude of Ds,turbulent is often several orders greater than Ds,molecular. In our simulation, we chose an analytical solution from kinetic theory of granular flow to estimate Ds (Simonin, 1996), as listed in Table A1. The applied Ds here is capable of handling the complicated cases in a gas-solid flow (Hua et al., 2014), in which the gas-solid interaction, the gas turbulence, and the collision between particles are considered. The reasons of using Ds at a molecular level are given as follows: (1) the CFD simulation can build a geometry exactly same as that of the practical experimental apparatus including various baffles; (2) the properties of the bulk bed material and tracer particles in the simulation can set to be totally identical with those in the experiments; and (3) the way of injecting and detecting the tracer particles, like the location of tracer injection and detection, the tracer injection period, and the tracer sampling frequency, can be designated flexibly by CFD simulation in terms of the experimental operation. Therefore, the discrepancy between the CFD simulation and practical RTD experiments in the specific environmental setting could be eliminated as far as possible. In the common treatment of simulating a dense fluidized bed, the turbulence of the solid phase plays a negligible role, due to the relatively low gas velocities in the system or the overcomplicated and controversial modelling concept and definition (Jung et al., 2006; Kashyap and Gidaspow, 2011). As a result, only Ds,molecular is needed. Moreover, the predicted RTD curve by the CFD simulation with Ds,molecular can further be modeled by some available RTD models, especially a plug flow with dispersion model, to obtain a calibrated Ds. Herein Ds describes the overall dispersion ability of solids in the system, so it is defined on a macroscopic scale. It can be seen that the concept of the calibrated Ds is compatible with that of above Ds,turbulent in the literature. Then it becomes reasonable to compare the calibrated Ds with the extensively measured Ds,turbulent in the literature, since the effects of the specific conditions of experiments, the measurement techniques, and the modelling analysis are considered concretely in the CFD simulation. It is worth noting that the species transport equation in Table A1 is able to take into account the impacts of the particles’ shape on Ytracer. On the one hand, Eulerian-Eulerian model predicts es and us by the proposed gas-solid drag model in Table A1; on the other hand, during the calculation of Ds, the gas-solid drag coefficient b has to be used, and then the effects of the irregular shape of particles are considered as well through Ds in such an indirect way. For Ds in Table A1, st12 , sF12 and sc2 are the time scale of the gas turbulent motion viewed by particles, the relaxation time of particles due to the drag with the gas, and the inter-particle collision time, respectively; q12 is the gas-solid velocity covariance;

(a) the free bed when Ug = 0.4 m/s

(b) the baffled bed when Ug = 0.4 m/s

(c) the free bed when Ug = 0.45 m/s

Table 3 Simulation parameters of Eulerian-Eulerian model. Simulation parameter

Value

Gas density (qg) Gas viscosity (lg) Particle-particle restitution coefficient (ess) Specularity coefficient (u) Internal friction angle of particles (uf) Solid volume fraction beyond which frictional force occurs (esmin) Momentum discretization

1.2 kg/m3 1.8  105 Pa s 0.99 1.0 34° 0.3695 (=esmf)

Volume fraction discretization Species tracer discretization Time step

Second-order upwind Quick First-order upwind 5.0  104 s

(d) the baffled bed when Ug = 0.45 m/s Fig. 3. The predicted and measured time-averaged mass flow rate of solid mixture at the outlet of the outflow pipe for the free and baffled beds when Ug = 0.4 and 0.45 m/s.

L. Hua et al. / Chemical Engineering Science 200 (2019) 320–335

and st1 is the time scale of the energetic turbulent eddies of the gas. As a preliminary attempt, the effects of the gas-induced turbulence in a gas-solid dense fluidized bed were omitted in this study, which also had been a widely accepted assumption in the literature (Almuttahar and Taghipour, 2008). Then both st12 and q12 took a value of zero. A 3D simulation was conducted in this study. The main body of the bed and the outflow pipe were exactly same as those of the experiments, but the circular cross-section of the inflow pipe was changed to a smaller square cross-section with the size of 0.02  0.02 m2, as shown in Fig. 1(b). By the experimental observation, we found that the solid mixture slides down to the bed along the inflow pipe smoothly and continuously but only occupies a quite small fraction of the 0.055 m-diameter cross-section in most time. To reduce the computational cost, the cross-section of the inflow pipe was shrunk. In contrast, the outflow pipe kept the original circular cross-section shape. One striking characteristic observed clearly in the experiments about the particle outflow is that the flow of particles in the outflow pipe is not continuous. There may be no particles within several seconds, and then the particles begin to rush to the outlet suddenly with a relatively large quantity in the next several seconds. If the shape of the outflow pipe is changed to a smaller one, therefore, the particles possibly have little opportunities to go outside. In addition, the length of the inflow and outflow pipes was cut to 0.1 m to save the computational cost. As for the boundary conditions, a uniform gas velocity was specified at the bottom inlet, an atmospheric pressure was prescribed at the top of the bed, the gas was assumed to be nonslip at all walls including the internal baffles, and the motions of particles at the wall were supposed to be partial-slip. The specularity coefficient, applied by Johnson and Jackson model (Johnson and Jackson, 1987) for the particles’ partial-slip boundary condition, has a significant impact on the behavior of particles, particularly for the baffled bed that has larger wall area (Hua et al., 2016). Based on the study of (Hua et al., 2016), we set specularity coefficient to be 1.0. Due to the continuous inflow and outflow of particles, the initial conditions of bed could be given casually to some degree. To achieve a steady state of the system early, the bed was initially filled with the loose packed particles with a height of 0.29 m. A grid of 63,025 was used to discretize the CFD geometry model, which had been verified by a grid independent study on a batch-mode dense fluidized bed (Hua et al., 2015). Other parameters used in the simulation are summarized in Table 3.

4. Results and discussion 4.1. Hydrodynamics of particles To measure solid RTD, the bulk bed material was sampled together with the tracer particles at the outlet of the outflow pipe in order to obtain the tracer concentration. So the variation of solid mixture mass (including the bed material and tracer particles) within every sampling period was recorded in the experiments. Fig. 3 illustrates the time-averaged mass flow rate of solid mixture at the outlet within every sampling period, Qs,sample, for the bed with and without baffles when Ug = 0.4 and 0.45 m/s, respectively. The measured data and CFD results are included for the comparison. As observed in Fig. 3, the time-averaged mass flow rate of solid mixture Qs,sample oscillates by a steady mode within the whole duration. Both the experimental data and CFD prediction show this characteristic. On the one hand, both the predicted and measured Qs,sample fluctuate around 5.33 g/s (i.e., the 1red dashed line in 1 For interpretation of color in Fig. 3, the reader is referred to the web version of this article.

327

Fig. 3) with a similar magnitude. 5.33 g/s is the particle mass flow rate imposed at the inlet for all operating cases. On the other hand, the particles flow out of the bed through the outflow pipe by an unstable mode, which can be illustrated by the drastic oscillation of Qs,sample. The maximum deviation of Qs,sample from 5.33 g/s can reach 50% in all cases. The not continuous flow of particles in the outflow pipe was clearly observed in the experiments by the naked eyes. This flow state of particles in the outflow pipe might account for the significant fluctuations of Qs,sample. Summarily, Fig. 3 illustrates a reasonably good agreement between the CFD prediction and measured data for the free and baffled beds. At the first 5 s, 90 g tracer particles were injected into the bed. As a result, the mass flow rate of solids at the inlet of the inflow pipe changed from 5.33 g/s to 23.3 g/s instantly and kept 23.3 g/s until the end of the tracer injection. This injection might disturb the flow field to some degree, especially for the simulation. In the practical experiments, a 0.055 m-diameter inflow pipe was used to guide the particles to flow smoothly, while in the simulation, a 0.02  0.02 m2 square cross-section was set to replace the circular cross-section of the inflow pipe. As a result, a smaller cross-sectional area leads to a larger particle inlet velocity and then a more significant perturbation to the bed. This might cause the extra-large value of the solid mass flow rate at the first sampling point for the CFD simulation. This disturbance, however, seems to be eliminated quickly since the data at the second sampling point start to present a similar mode as the rest data. In Fig. 3, the CFD prediction has more sets of data than the experimental measurement. The CFD simulation could provide a more intensive result since the abundant information of particle flow can be obtained with a high spatial and temporal resolution. As introduced in Section 2, however, the sampling of experiments does not cover the whole RTD interval in order to save the labor cost. The collection of the tracer particles were conducted by an off-line manual operation. And then a time-averaging operation was conducted within every measured 15 s. In order to capture the not continuous flow of particles in the outflow pipe in this device accurately, therefore, a high time-resolution sampling technique is urgently recommended for the experimental measurement. Although a low experimental sampling frequency of the tracer collection in Fig. 3 did not reflect this phenomena quite acutely, the fluctuation of Qs,sample still could tell this characteristic to the degree. In the experiments, we observed clearly that the movement of particles near the inlet of the outflow pipe in the free bed is more active than that in the baffled bed. It can be seen that the baffles alter the particle motion significantly. Fig. 4 illustrates the predicted contour of solid volume fraction by the present CFD model at t = 800 s for the free and baffled beds when Ug = 0.4 m/s, respectively, in order to show the effects of baffles on the particle hydrodynamics. t = 800 s could ensure that the system has achieved a steady state. A similar result is obtained for Ug = 0.45 m/s, which is not shown for brevity. On the one hand, the wide unblocked space of the free bed is quite helpful to the full development of bubbles. So the size of bubbles in the free bed becomes relatively large when bubbles rise along the bed central area. In contrast, the application of baffles increases the wall effect and then retards the solid circulation. The vertical baffles split the bed into four slim channels. The narrow space of these channels limits the formation of big bubbles. For Geldart group B particles, bubbles usually grow to the size of a smaller hydraulic diameter of the bed that is influenced by the baffles. Thus, an axial slug flow might take place instead. Consequently, a large number of the small-size bubbles become the predominating way to mix the particles. On the other hand, the free bed generally has a broader bed surface with a size comparable to the bed width or length. This makes the breakage of big bubbles

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(a) the free bed

(b) the baffled bed

Fig. 4. The predicted contour of solid volume fraction by the present CFD model at t = 800 s in the central vertical plane (i.e., a front view) for the free and baffled beds when Ug = 0.4 m/s.

pipe essentially. For the baffled bed, however, the particles can move out of the bed only through the last channel. Then the flow within the bed and the movement of particles toward the outflow pipe becomes more peaceful.

4.2. RTD curve of particles

(a) the free and baffled beds when Ug = 0.4 m/s

(b) the free and baffled beds when Ug = 0.45 m/s Fig. 5. The predicted and measured RTD curves of solids for the free and baffled beds when Ug = 0.4 and 0.45 m/s.

or the jetting of individual particles in the near-bed surface zone so fiercely. Since the motion of particles in the near-bed surface area becomes the primary cause to push the particles out of the bed, this characteristic affects the behavior of particles in the outflow

4.2.1. Comparison of the predicted and measured RTD The RTD calculation could start when the flow in the bed reaches a statistical steady state. Herein the volume-averaged volume fraction of solids in the whole computation domain was monitored until it did not vary significantly with the time. Then a total amount of 90 g tracer was injected into the bed through the inflow pipe within 5 s. The tracer concentration at the outlet of the outflow pipe began to be sampled at the very instant when the tracer started to be injected. Ideally, the simulation does not end until all injected tracer particles are collected. In the practical assignment, however, the collection rate of tracer particles becomes unendurable smaller at the long tail part of RTD curve, especially after more than 1000 s. In order to reduce the computational effort, therefore, the RTD curve is considered to be integrated and the simulation is no more continued when the collection rate of tracer particles changes slowly and the total collected tracer is close to 90 g. In order to keep consistent with the measurement, the CFD data exported by a high temporal resolution was treated specially by the same sampling frequency used in the experiments. Fig. 5 plots the predicted RTD curves of solids for the free and baffled beds with Ug = 0.4 and 0.45 m/s, respectively. The measured data are provided as well for the comparison. Obviously, the overall agreement in Fig. 5 seems to be satisfactory. The location and magnitude of the crest of RTD curves are both predicted reasonably well. The characteristics of a long tail also are captured rationally. The first point of experimental data in the case of the free bed with Ug = 0.45 m/s, however, looks weird, due to its surprisingly high value. This might be attributed to the unusual measurement of the very initial stage in this test, because the rest of data show a normal behavior. The lowresolution off-line tracer sampling method and the perturbation

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L. Hua et al. / Chemical Engineering Science 200 (2019) 320–335 Table 4 The predicted and measured mean residence time and space time for the free and baffled beds when Ug = 0.4 and 0.45 m/s. Gas velocity, m/s

Bed form

Mean residence time tm, s (Simulation)

Mean residence time tm, s (Experiment)

Space time s, s (Simulation)

Space time s, s (Experiment)

0.4 0.4 0.45 0.45

Free bed Baffled bed Free bed Baffled bed

321.51 328.08 297.44 306.39

324.74 331.87 298.58 316.60

343.50 345.42 319.79 321.51

308.35 405.45 292.14 379.26

induced by the tracer injection could be reflected in the first measured data set. The effects of baffles on solid RTD are significant in the cases investigated in this study, as illustrated in Fig. 5. The setting of baffles makes the crest of RTD curve move toward the central part of curve, indicating a weakened mixing. In most applications, the addition of baffled structure aims to alleviate the degree of the undesired back-mixing. The three vertical baffles parallel to the side wall are installed in this study, which is able to narrow the cross-section of particle passing and then elongate the path of all particles. As a result, all particles are enforced to pass the bed by a behavior pattern as similarly as possible. Although this step cannot produce a standard or approximate plug flow yet, it enhances the uniformity of the particle movement to some degree. This measure leads to a move of RTD curve from the almost perfect mixed flow toward a non-ideal plug flow. Table 4 lists the predicted and measured mean residence time tm for the free and baffled beds with Ug = 0.4 and 0.45 m/s. The space time of the fluidized particles was also calculated by the definition of s = V/v, where V is the packed volume of particles, and v is the volumetric flow rate of solids in and out of the vessel (Levenspiel, 2012). The baffles have the capability of increasing tm because they push the RTD curve toward the trend of plug flow. Table 4 tells the effects of the operating gas velocity as well. The increasing gas velocity can reduce tm, even with a magnitude greater than that of baffles. In this study, the free bed full of the irregular particles locates in the undeveloped slugging regime under the control of both of Ug = 0.4 and 0.45 m/s. Therefore, no more tests on gas velocity greater than 0.45 m/s were conducted because of a too fiercely fluctuating bed surface. In this regard, other type particles or the apparatus with other sizes should be designed to investigate the effects of gas velocity concretely. To estimate the space time s, the packed volume of particles V has to be given beforehand. In the simulation, V could be calculated accurately by monitoring the volume-averaged solid volume fraction in a long time period and then doing the time-averaged operation. In contrast, it is difficult in the experiments to measure the total volume of particles within the bed due to a continuous flow of solids. Therefore, we estimated V roughly by the measured pressure drop Dp acting on the whole bed.

V ¼ es AH p es ¼ qDHg

ð1Þ

s

where A is the cross-sectional area of bed, H is the height of bed surface during the operation, qs is the particle density, and g is the gravitational acceleration. The measured Dp was 2.24 and 2.81 KPa for the free and baffled beds, respectively, when Ug = 0.4 m/s, and was 2.12 and 2.63 KPa for these two bed forms when Ug = 0.45 m/s. Theoretically, s should be equal to tm if there are no internal dead spaces and bypass zones (Levenspiel, 2012). As for the CFD results, the general trend of tm < s in Table 4 indicates there are some dead or nearly stagnant zones in the bed. The percentage of active particle volume, Va = tm  v, from V lies in 93– 95% for the free and baffled beds. The baffled beds have a slightly

greater volume of dead spaces compared with the free beds, as a result of the complicated geometry of baffles. However, the experimental results of tm > s for the free bed under the both gas velocities indicate an improper estimation of s. Eq. (1) neglects the friction effects of wall and any possible acceleration force induced by the entering gas. These simplified assumptions may result in an inaccurate estimation of V in a dense fluidized bed. 4.2.2. Comparison of the predicted Ds and literature data As illustrated in Fig. 1, the particles have to move across the bed horizontally to get out, because the inflow pipe and outflow pipe are installed at the same height. In addition, the experimental findings of the gas-solid dense fluidized beds have proven that the axial component of solid dispersion coefficient Dsa is at least an order of magnitude larger than the radial component Dsr (Berruti et al., 1986; Sánchez-Prieto et al., 2017). Bellgardt and Werther (1986) also confirmed that the vertical mixing of solids is much faster than the horizontal mixing by the careful measurements in a large and deep pilot-scale bed. Therefore, Dsr becomes the most concerned parameter in the present work. Several models have been proposed to estimate Dsr for a batch-mode or cross-flow dense fluidized bed, which are summarized in Table B1 in the Appendix B. 1D or 2D Fickian-type diffusion model is adequate to describe the lateral behavior of particles in the low-velocity fluidization regimes, especially in a batch-mode bed. For a crossflow bed, however, 1D or 2D plug flow with dispersion model is more appropriate due to that fact that a convective term is added to account for the mixing caused by the cross-flow motion of particles. A plug flow with dispersion model is quite prevailing in modeling RTD, because it is able to incorporate the convection and diffusion mechanisms of flow in terms of a simple formulation (Kunii and Levenspiel, 1991). 1D plug flow with dispersion model can be expressed by a partial differential equation with a dimensionless form.

@C @C 1 @2C þ ¼ @h @Z Pe @Z 2

ð2Þ

where C(Z, h) = c(Z, h)/c0, in which c(Z, h) and c0 denote the tracer concentrations at (Z, h) and the injection point, respectively; Pe = uL/D, in which u is usually assigned to be a constant along the distance between the tracer injection and detection, L, and D 



is dispersion coefficient; h = t/t , in which t is time, and t = L/u; and Z = z/L, where z is the position coordinate for a 1D problem. Together with the appropriate initial and boundary conditions, Eq.

Table 5 The estimated Peclet number of solids on the basis of the CFD results for the free and baffled beds when Ug = 0.4 and 0.45 m/s. Gas velocity, m/s

Bed form

Pe

0.4 0.4 0.45 0.45

Free bed Baffled bed Free bed Baffled bed

1.06 3.6 0.36 3.4

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(2) can be solved analytically or numerically to get the RTD curve at the location of tracer detection (Hua and Wang, 2018).

EðhÞ ¼ C ð1; hÞ= 

R1 0

C ð1; hÞdh

Eðt Þ ¼ EðhÞ= t

ð3Þ

In order to take into account the effects of tracer injection, the measured curve of RTD at the end of the outflow pipe, Eoutput(t), can be regarded as a convolution of the RTD functions of both the injection process, Einput(t), and the bed system, Ebed(t) (Sheoran et al., 2018; Smolders and Baeyens, 2000). Hence, the relationship of them is,

Z Eoutput ðt Þ ¼

t

Einput ðt 0 ÞEbed ðt  t0 Þdt0

ð4Þ

0

And for a tracer injection process that keeps a constant tracer concentration, Einput(t) has a form of,

Einput ðtÞ ¼ 1=2t m;input ;

t < 2tm;input

ð5Þ

where the mean residence time of the tracer injection process, tm,inis the half of the tracer injection duration, i.e., 2.5 s in this study. Because the closed boundary conditions are imposed both at the inlet and outlet here, Peclet number of Ebed(t) can be estimated by means of the best-matching of either Eoutput(t) curve (Brenner, 1962) or the corresponding variance (Levenspiel, 2012) between the model prediction and CFD results. More details can be referred to the Appendix C. Table 5 lists the estimated Pe for all cases in this work. It can be seen in Table 5 that the fitted Pe for the baffled beds is generally greater than that for the free beds. This means that the solid mixing in the baffled beds is relatively milder than that in the free beds. Kong et al. (2018) investigated the effects of baffles on Pe in a cross-flow bubbling fluidized bed experimentally, where Pe was estimated by 1D plug flow with dispersion model. They figured out that Pe increases with more baffle settings, which is consistent with the trend in Table 5. Kong et al. (2018) also studied how the gas velocity affects Pe and reported an increase Pe with Ug. A higher velocity of gas can cause a more vigorous motion of particles and a shorter residence time of particles in the bed. The solid flow therefore has a trend towards a plug flow. In Table 4, a reduced tm with a higher gas velocity also suggests an enhancement of plug flow when increasing Ug. In Table 5, however, Pe decreases slightly with the gas velocity. This might be attributed to the fact: (1) the gas velocity does not change too much in the present experiments due to the experiment limitations; and (2) the estimation of Pe by 1D plug flow with dispersion model in put,

Fig. 6. The lateral dispersion coefficient of solids in a gas-solid dense fluidized bed.

the case of a quite small Pe value (especially less than 1.0) is not so accurate (Levenspiel, 2012). Further, Dsr can be estimated in terms of the Pe results in Table 5 once u and L are appropriately provided. To calculate Dsr for the free beds, L takes the value of the bed length, because the particles are assumed to move relatively fast in the inflow and outflow pipes and the solid mixing mainly occurs in the main bed body. Then u is 

calculated by L/tm,bed and tm is used here instead of t . This treat

ment is reasonable because tm = t if the closed-closed boundary conditions are imposed properly. Although there are the vast measured data of Dsr by various experimental techniques (Liu and Chen, 2012) as well as more than a dozen of Dsr empirical correlations in the literature (Niklasson et al., 2002; Winaya et al., 2007), the comparison between the literature data and the estimated Dsr in the present study is still challenging. At present, three kinds of mechanisms have been reported to account for the movement of particles in a batch-mode bed (Geldart, 1986): (1) the large-scale gross circulation of solids caused by the channeling of the rising bubbles, like the upward growth and radial coalescence of bubbles, and the downward movement of the emulsion phase (Gel’perin et al., 1970); (2) the local small-scale sufficient mixing of particles within the wake induced by upflow bubbles (Kunii and Levenspiel, 1991); and (3) the process that the particles entrained in the bubble wake or drift region are finally scattered into the freeboard by the bursting bubbles (Davidson and Harrison, 1971; Heertjes et al., 1967). It can be seen that 1D or 2D diffusion model generally regards the solid mixing in a batch-mode bed as a whole diffusion process. Then the estimated Dsr begins to lump various effects. For a system with a cross-flow operation, however, the convective term in Eq. (2) means the mixing mechanism induced by the cross-flow of solids, and the diffusive term in Eq. (2) indicates an integrated bubbleinduced mixing in the whole bed, just similar to the assumption adopted by 1D or 2D diffusion model. Therefore, it is reasonable to compare the Dsr estimated by Pe in Table 5 for a cross-flow bed with the available Dsr in the literature mostly calibrated from the data in the batch-mode beds. Fig. 6 presents the such comparison. It should be noted that almost all of data in Fig. 6 are measured in a batch-mode bed except for those in (Bi et al., 1995) and (Sette et al., 2016), since the works related to a cross-flow bed are quite few. It can be seen that the estimated Dsr in this work is in a good agreement with the literature data, both in the magnitude and trend. The estimated Dsr has the order of magnitude of 103, which locates at the central area of other measured data. With the increase of Ug – Umf, the estimated Dsr increases as well. This direct proportional relationship between Dsr and Ug – Umf has been confirmed by many other experimental findings (Bellgardt and Werther, 1986; Berruti et al., 1986; Borodulya et al., 1982; Du et al., 2002; Fan et al., 1986; Liu and Chen, 2011, 2012; Shi and Fan, 1984; Subbarao et al., 1985; Xiao et al., 1998; Yang et al., 2002). The effects of baffles on Dsr are also illustrated in Fig. 6. The estimated Dsr in the baffled beds is smaller than that in the free beds under the both gas velocities. This conclusion is consistent with Kato et al. (1985)’s viewpoint that the vertical or horizontal internals can reduce Dsr significantly. Ramamoorthy and Subramanian (1981) and Cui et al. (2006) also showed that axial dispersion coefficient of solids decreases in the presence of internals in a dense fluidized bed. Some of Du et al. (2002)’s data in Fig. 6, however, are beyond the normal scope of Dsr. Du et al. (2002) detected the tracer particles at 0.076 and 0.152 m downstream and 0.076 m upstream of the tracer injector, respectively, in a 0.203 m-bed diameter column. This detection setting is totally different from the treat-

L. Hua et al. / Chemical Engineering Science 200 (2019) 320–335

(a) the free bed

331

(b) the baffled bed

Fig. 7. The contour of solid dispersion coefficient calculated by the equations in Table A1 at t = 800 s in the central vertical plane (i.e., a front view) for the free and baffled beds when Ug = 0.4 m/s.

Fig. 7 plots the contours of Ds calculated by the equations in Table A1 at t = 800 s for the free and baffled beds when Ug = 0.4 m/s. Ds depends on the solid volume fraction, the granular temperature, and the gas-solid interphase drag coefficient. Consequently, Ds is a function of the space and time. As discussed before, Ds in Fig. 7 is at the molecular level, i.e., Ds,molecular. And the estimated Ds in Fig. 6 by 1D plug flow with dispersion model on the basis of the predicted RTD curves is at the turbulent level, i.e., Ds,turbulent. It can be seen that although Ds,molecular has a smaller order of magnitude than Ds,turbulent, which can differ up to 103 at least for the baffled beds, the predicted Ds,turbulent is quite comparable to the literature data. This demonstrates that the use of Ds,molecular in the species transport equation, as the most important and bold assumption in the present CFD model, is reasonable. Fig. 8. The RTD curves predicted by the tanks-in-series model for the free and baffled beds when Ug = 0.4 m/s.

ment in other experimental works. In addition, they also applied a 2D plug flow with dispersion model purposely to describe the axial mixing of solids, so Dsr was used as one of the three fitting parameters to match the measurement. It is also worth noting that Bi et al. (1995)’s data reported a different trend of Dsr with Ug – Umf. It looks like Dsr has an unexpected inverse relationship with Ug – Umf. Not only the coarse particles but also some fine particles were fluidized at the same time in their study to improve the fluidization quality of particles. In contrast, other researchers often applied the monodisperse particles rather than the polydisperse particles to simplify the problem. The significant discrepancy of particle behavior in these two situations might lead to the inconsistent trend of Dsr with Ug – Umf. In addition, some of Sette et al. (2016)’s data look relatively huge in Fig. 6 as well. Two kinds of the bed operating modes were investigated by Sette et al. (2016): the free bed without and with a continuous solid feeding. Then Dsr collected from their work contains one group of the batch-mode bed and three groups of the cross-flow bed. More importantly, they used Einstein’s equation to calculate Dsr, which incorporates the convection mixing induced by the cross-flow of particles into Dsr. As a result, Dsr is large in a cross-flow bed.

4.2.3. Modeling RTD by the tanks-in-series model As one of the popular RTD models, the tanks-in-series model is used often due to its easy application and stable performance (Guío-Pérez et al., 2014; 2013b; Kong et al., 2018; Sivashanmugam and Sundaram, 2000). The equations of the tanks-in-series model is written as (Levenspiel, 2012).

Eðt Þ ¼

tN - 1 NN etN=tm N tm ðN  1Þ!

ð6Þ

where the integer N denotes the number of the ideal mixed tanks. For the pulse-response RTD experiment conducted in this study, it would be better to take into account the duration of the tracer injection. To determine N, the mean and variance of the input and output RTD curves have to be calculated, and then we have,


2 tm;output  tm;input  N¼  r2t;output  r2t;input

ð7Þ

where tm,input and tm,output are the mean of the input and output RTD curves, respectively, and r2t;input and r2t;output are the variances of these two curves. The optimal values of N for the free and baffle beds with these two gas velocities are 1 and 2, respectively. Fig. 8 presents the results of the tanks-in-series model and the corresponding experimental data for Ug = 0.4 m/s.

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It is interesting to note that the best-fitting tank number of the baffled bed is 2 rather than 4, though there are actually four small compartments partitioned by the three baffles. As illustrated schematically in Fig. 1(a), there exist two identical motion patterns of solids that look like a U shape. This may explain why 2 is the optimal tank number for the baffled beds. When applying the tanks-in-series model, it is assumed that the entrance and exit of vessel are well approximated by the closed boundary conditions. In this study, this requirement of boundary conditions is well met.

ited due to its size setting and experimental layout. As a first step, the present work is to address the effects of baffles on the RTD of the monodisperse particles in a cross-flow dense bed. In the coal combustion, roasting of ilmenite and direct reduction of iron ores, the solid materials usually have a wide size distribution. The complete reaction time of a particle varies with its particle diameter. Therefore, the different residence times are required for the solid particles with a different size distribution. In the next research plan, we are going to study how the baffles, the gas velocity, and the solid feeding rate modulate the RTD of the polydisperse particles.

5. Conclusion In order to predict the solid RTD of the non-spherical particles in a lab-scale baffled dense bed with a cross solid flow, this work applied Eulerian-Eulerian model to describe the hydrodynamics of the gas and particles, and applied the species transport equation to simulate the behavior of the tracer particles. The impacts of the irregular particle shape on the system were considered by a proposed interphase drag force model, which used the sphericity to describe the particle shape. As one key parameter, solid dispersion coefficient Ds in the species transport equation has to be provided prior to the simulation. In this work, Ds was calculated by an analytical solution from kinetic theory of granular flow, which could account for the effects of the gas-solid interaction, the turbulence of interstitial gas and the collision between particles. This Ds could be considered as a material property induced only by the molecular motion. This treatment is reasonable due to the fact that the CFD model is able to reproduce the RTD procedure exactly same as the practical experiments, such as the physical properties of tracer particles, the non-ideal pulse injection of tracer particles, the irregular sampling duration and sampling frequency of tracer particles, the location and configuration of baffles, etc. Therefore, the discrepancy in the specific system geometry and RTD measurement technique between the experiments and CFD simulation can be eliminated effectively. In addition, the turbulence effects of the solid phase are not considered here in a gassolid dense bed. Therefore, Ds only needs the value at a molecular level. A 3D lab-scale rectangular bed with a continuous solid flow was constructed to validate the established CFD model. Some bulk particles were selected to be colored as the tracer particles. The three vertical baffles parallel to side walls were installed when a baffled bed was studied. The solid mixture mass and tracer concentration at the outlet of the outflow pipe were monitored to get solid RTD curves. The CFD model reproduced the oscillating characteristic of the solid mass flow rate at the outlet of the outflow pipe and calculated the RTD curves reasonably well for the free and baffled beds. On the basis of the predicted RTD curves, 1D plug flow with dispersion model was further used to estimate the lateral dispersion coefficient of solids Dsr in the whole bed. Through the comparison with the available literature data, the estimated Dsr showed a reasonable agreement in both magnitude and trend. This overall good agreement demonstrated that the setting of Ds in the CFD model is adequate. The results showed that in a gas-solid dense fluidized bed: (1) the installation of baffles can transform the solid RTD from a perfect well-mixed flow toward a non-ideal plug flow effectively and (2) the baffle setting can decrease Dsr. The primary purpose of the present study is to establish a CFD model to predict solid RTD in a cross-flow dense bed effectively. The role of the lab-scale experiments conducted here is to provide the detailed information for the model validation. It should be noted that the direct application of the present experimental results in the real industrial-scale reactors is quite lim-

Declaration of interest The author declares that there is no conflict of interest.

Acknowledgements This work was financially supported by the National Natural Science Foundation of China, China [grant number 21306202], the State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, China [grant number MPCS-2012-A-02], and the National Key R&D Program of China, China [grant number 2017YFE0106500].

Appendix A. The governing equations, gas-solid interphase drag model, and constitutive relations in Eulerian-Eulerian model Table A1.

Table A1 The equations of Eulerian-Eulerian model. Conservation of mass for gas and solid phase:     @ @t eg qg þ r  eg qg ug ¼ 0 @ @t ðes qs Þ þ r  ðes qs us Þ ¼ 0 eg þ es ¼ 1

Conservation of momentum for gas and solid phase:    
 @
@t eg qg ug þ r  eg qg ug ug ¼ eg rp þ r  sg þ eg qg g  b ug  us
 @
@t ðes qs us Þ þ r  ðes qs us us Þ ¼ es rp þ r  ss þ es qs g  b us  ug Stress tensor for gas and solid phase:

s
g ¼ eg lg f½rug þ ðrug ÞT   23 ðr  ug Þ
Ig nh

i

o

s
s ¼ ½ - ps þ es ks ðr  us Þ
I þ es ls rus þ ðrus ÞT  23 ðr  us Þ
I Gas-solid interphase drag coefficient:   b ¼ 1  ugs bWen - Yu - Ganser þ ugs bErgun es eg qg jug us j bWen - Yu - Ganser ¼ 34 C d0 e2:7 g dp 2:0es qg jug us j 180e2s lg bErgun ¼ 2 2 þ wdp w dp eg

ðes esmf Þ ugs ¼ arctanh½1802:0 þ 0:5 p

i 0:4305K 2 1:0 þ 0:1118ðRes K 1 K 2 Þ0:6567 þ 1þ C d0 ¼ Re24 3305 s K1 Res K 1 K 2  1 v K 1 ¼ 13 ddnv þ 23 w0:5  2:25 Ddbed 0:5743

K 2 ¼ 101:8148ðlogwÞ eg q dp jug us j Res ¼ g l g

Solid phase pressure: ps ¼ es qs hs ½1 þ 2g 0 es ð1 þ ess Þ þ pfs Solid shear viscosity: qffiffiffiffi pffiffiffiffiffiffi 2 dp phs  ls ¼ 9610eqs ðs1þe 1 þ 45 g 0 es ð1 þ ess Þ þ 45 es qs dp g 0 ð1 þ ess Þ hps þ lfs ss Þg 0

Frictional stress model:

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Appendix C. Estimate Peclet number by the E(t) curve and variance

Table A1 (continued) pfs sinuf

lfs ¼ 2pffiffiffiffiffi ; I2D I (

pfs

¼

2D

h i
: D;
D
¼ 1 ru þ ðru ÞT  1 ðr  u Þ
I ¼  12 D s s s 2 3

Þ F r ððees esmin n smax es Þ 0

m

es > esmin Fr = 0.05 N/m2, m = 2, n = 5 es 6 esmin

Brenner (1962) proposed a numerical treatment to solve Ebed(t) curve for the situation imposed with the closed-closed boundary conditions and the step injection of tracer, which is given as follows,

Solid bulk viscosity:

qffiffiffiffi ks ¼ 43 es qs dp g 0 ð1 þ ess Þ hps Radial distribution function:  1=3 1 es g 0 ¼ 1  esmax

Ebed ðhÞ ¼ 2exp

Conservation of the granular fluctuating energy (algebraic formulation):  
s : rus  c  3bhs 0 ¼ ps
I þ s hs

Collisional dissipation rate of the granular fluctuating energy: 12 1e2 g chs ¼ ðd pssffiffipffiÞ 0 qs e2s h3=2 s p

Species transport equation for tracer particles: @ @t ðes qs Y tracer Þ þ r  ðes qs us Y tracer Þ ¼ r  ðes qs Ds rY tracer Þ Dispersion coefficient of solid phase: 
 sF 1 Ds ¼ 13 st12 q12 þ sF12 hs 1 þ 23 s12c 2  1=2 st12 ¼ st1 1 þ C b f2r

sF12 ¼ qsbes qffiffiffiffi sc2 ¼ 24deps g0 hps


Pe 2

 2  P ð1Þnþ1 x2n x  Pe h 1 exp  Pen h n¼1 Pe2 2 4 4

þxn þPe

n Pe xn is positive root oftanxn ¼ 4x4x2 Pe 2

ðC:1Þ

n

Ebed ðt Þ ¼ Ebed ðhÞ=t m;bed ;

h ¼ t=t m;bed

Eq. (C.1) is applicable to Pe <16. Compared with Eq. (2), tm,bed is used to transform h to t, and Ebed(h) to Ebed(t), due to the fact that 

for a closed-closed system, t = tm. With the help of Eqs. (5) and (C.1), Eoutput(t) curve can then be predicted by Eq. (4) for a system with a relatively long tracer injection. Here Pe is the only adjustable parameter to obtain a good match with the experimental data. With regard to the second approach, the mean and variance of the RTD curve resulted from the bed system, Ebed(t), are calculated by Eq. (C.2) if the tracer injection time cannot be neglected.

tm;bed ¼ tm;output  t m;input

r2t;bed ¼ r2t;output  r2t;input tm ¼

r2t ¼

Appendix B. The models to estimate the lateral dispersion coefficient of solids In Table B1, x, y, and z denote the length, width and depth direction of a bed in a Cartesian coordinate system, respectively, and r is the radial direction in a cylindrical coordinate system.

R1 0

tEðt Þdt

0

ðt  t m Þ2 Eðt Þdt

R1

ðC:2Þ

Under the condition of the closed-closed boundary conditions and the pulse injection of tracer, Levenspiel (2012) deduced that when Pe <100.

r2t;bed ¼

 2 2
 1  ePe Pe Pe2

ðC:3Þ

Then Pe could be determined once the variances of the measured RTD curve, Eoutput(t), and the tracer input RTD curve, Einput(t), are obtained by Eq. (C.2).

Table B1 The available models in the literature to calibrate lateral dispersion coefficient of solids in a gas-solid dense fluidized bed. Model

Equation

1D diffusion model

@C @t

¼

2D diffusion model

@C @t

¼ Dsr

Einstein’s equation

Dsr ¼ 12 ðDDxtÞ

Stochastic diffusion model

2 Dsr @@xC2



@2 C @x2

2

þ @@yC2



Objective

Researcher

Dsr: lateral dispersion coefficient of solids

lateral mixing in a batch-mode bed

/

lateral mixing in a batch-mode bed lateral mixing in a batch-mode or crossflow bed lateral mixing in a batch-mode bed

Borodulya et al. (1982)Shi and Fan (1984) Kato et al. (1985)Bellgardt and Werther (1986)Xiao et al. (1998)Yang et al. (2002) Winaya et al. (2007)Sánchez-Prieto et al. (2017) Berruti et al. (1986)Niklasson et al. (2002) Liu and Chen (2011)Liu and Chen (2012) Sette et al. (2014a)Sette et al. (2014b)

Dx: displacement of a tracer particle;Dt: elapsed time

2

@f ðU sj ;t jU sj0 ;t 0 Þ @t

Variable

¼ ab

@ 2 f ðU sj ;t jU sj0 ;t 0 Þ @U 2sj

@U f ðU sj ;t jU sj0 ;t 0 Þ þa sj @U sj

1D plug flow with dispersion model

@C @t

@ C þ U p @C @x ¼ Dsr @x2

2D plug flow with dispersion model

@C @t


@C  Dsr @ @2 C þ U p @C @z ¼ Dsa @z2 þ r @r r @r

2

f(Usj, t|Usj0, t0): the probability that a particle with the initial velocity Usj0 at time t0 will have the velocity Usj at time t;a: inverse of relaxation time;b: a parameter related to the variance of particle velocity Up: convective velocity of tracer

Dsa: axial dispersion coefficient of solids

lateral mixing in a cross-flow bed lateral mixing in a batch-mode bed

Mostoufi and Chaouki (2001)Pallarès and Johnsson (2006)Sette et al. (2016)

Fan et al. (1986)

Shi and Fan (1985)Bi et al. (1995) Schlichthaerle and Werther (2001)Sette et al. (2015) Du et al. (2002)

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