A structure-dependent multi-fluid model (SFM) for heterogeneous gas–solid flowContents lists available at SciVerse ScienceDirect
Chemical Engineering Science
A structure-dependent multi-fluid model (SFM) for heterogeneous gas–solid flow
Kun Hong a,b, Zhansheng Shi a,b, Wei Wang a,n, Jinghai Li a
a The EMMS Group, State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, China b University of Chinese Academy of Sciences, Beijing 100049, China
H I G H L I G H T S
A structure-dependent multi-fluid model (SFM) was proposed for gas–solid flow.
The SFM unifies the bubble-based and cluster-based EMMS models. The SFM unifies the classic two-fluid model and EMMS models. A new version of bubble-based EMMS model was validated for wide range of flow regimes.
a r t i c l e i n f o
Article history: Received 6 March 2013 Received in revised form 2 May 2013 Accepted 23 May 2013 Available online 4 June 2013
Keywords: Multiphase flow Fluidization Bubble Structure Mathematical modelling EMMS
09/$ - see front matter & 2013 Elsevier Ltd. A x.doi.org/10.1016/j.ces.2013.05.050
esponding author. Tel.: +86 10 8254 4837; fax ail address: wangwei@home.ipe.ac.cn (W. Wa
a b s t r a c t
For the heterogeneous gas–solid flow in a fluidized bed, meso-scale structures, such as bubble and cluster, have significant effects on the hydrodynamics, mass/heat transfer and reaction rate. These structures can be described with certain kinds of bimodal probability density distribution of solids concentration, i.e., the dilute-dense two-phase structures. To keep the physical nature of these meso- scale structures in mathematical formulation, a structure-dependent, multi-fluid model (SFM) was proposed. Then, the SFM was reduced to the conventional two-fluid model (TFM) as well as the hydrodynamic equations of the bubble-based and cluster-based EMMS (energy-minimization multi- scale) models by assuming different simplifications of structures. Thus, the SFM unifies these different models. A new version of bubble-based EMMS model was presented thereby and validated with comparison to experimental data. This bubble-based EMMS model was found to be applicable to wide flow regimes ranging from bubbling, turbulent to fast fluidization.
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1. Introduction
Gas–solid flow is inherently aggregative, featuring heteroge- neous structures over a wide range of scales with respect to time and space. For example, in a gas–solid fluidized bed, the flow structure varies with material properties and operating conditions. With increasing gas velocity, as shown in Fig. 1, it may take the form of bubble, cluster or streamer, and characterize the flow regimes from uniform expansion (corresponding to Umf), bubbling fluidization (corresponding to Umb), turbulent fluidization, fast fluidization to pneumatic transport (corresponding to Upt) in succession. In mathematics, these flow structures can be described with certain bimodal probability density distribution of voidage, one apex corresponding to the gas-rich, dilute phase and the
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other to the solid-rich, dense phase (However, for uniform expan- sion and pneumatic transport, their distributions are further reduced to unimodal curve) (Li and Kwauk, 1994; Lin et al., 2001). It is also interesting to note that these meso-scale struc- tures were found to form and propagate by the same mechanism, and hence belong to the same family of nonuniform solutions (Glasser et al., 1998).
The meso-scale structure significantly affects the hydrody- namics, mass/heat transfer and reaction rate (Agrawal et al., 2001; Dong et al., 2008; Holloway and Sundaresan, 2012; Li and Kwauk, 1994; Yang et al., 2003). Thus, there are many researches focusing on predicting flow structures as well as their effects. Starting from an assumption of idealized bubble, bubble-based model (Davidson, 1961; Toomey and Johnstone, 1952) has been widely developed since the early 1950s. At nearly the same time, there are also many efforts extending the classic fluid dynamics for single phase flow to two-phase flow, to predict the complex behavior in fluidized beds (Anderson and Jackson, 1967; Murray,
K. Hong et al. / Chemical Engineering Science 99 (2013) 191–202192
1965; Soo, 1967). The statistical behavior of a large number of particles with similar diameter and density is normally assumed to mimic a continuum with a continuous density and velocity distribution. If all particles are characterized by a mean diameter and density, the two-phase hydrodynamic model can be termed the two-fluid model (TFM) which is now widely applied in industrial simulations in literature (Gidaspow, 1994; Hartge et al., 2009; Loha et al., 2012; Zhang et al., 2010).
The pioneering work of Gidaspow's group (Ding and Gidaspow, 1990; Tsuo and Gidaspow, 1990) has shown that the classic TFM allows capturing heterogeneous structures like bubbles and clus- ters in a fluidized bed. Recent studies further showed that coarse- grid simulations need to take into account meso-scale structures within each grid for modeling of fine-particle fluidization (Agrawal et al., 2001; Benyahia, 2012a; O'Brien and Syamlal, 1993; Parmentier et al., 2012; Wang and Li, 2007; Wang et al., 2010; Yang et al., 2003; Zhang and VanderHeyden, 2002). For predicting bed expansion in a bubbling fluidized bed, Wang et al. (2009) found that the grid size used in TFM should be of the order of three particle diameters to be comparable to the discrete element method (DEM) results. For circulating fluidized beds, Benyahia (2012b) demonstrated that fine-grid resolution with grid size of 1 mm (about 18 times particle diameter) helps to predict the S-shaped axial profile of voidage, though the predicted solids flux is still much larger than experimental data. Ullah et al. (2013) found through drift-flux analysis that refining grid resolution may help to predict bi-stable state corresponding to the S-shaped axial profile of voidage, whereas quantitative accuracy of the TFM still need more elaborate studies. Indeed it still remains a hot dispute as to the feasibility of using the classic TFM to capture the effects of two-phase flow structures over the whole range of scales (Lu et al., 2009; Syamlal and Pannala, 2011).
The most direct way to resolve the meso-scale structure is direct numerical simulation (DNS). However, DNS requires huge, even unaffordable computational resources, and therefore is not suitable for industrial applications. Between the DNS and TFM, various Eulerian–Lagrangian approaches, such as CFD-DEM (Tsuji et al., 1993; Xu and Yu, 1997) and discrete particle model (DPM) (Deen et al., 2007; Hoomans et al., 1996), have been developed. These approaches provide finer resolution of the gas–solid two-phase flow than TFM. However, their averaged description of the gas phase still requires sub-grid modeling as is in TFM (Benyahia and Sundaresan, 2012; Xu et al., 2007). In addition, tracking trajectories of all particles as well as their collisions restricts their application out of large-scale simulation of fluidized beds, though recent progress in coarse-grained methods, e.g., the multiphase particle-in-cell (MP-PIC) method (Andrews and O'Rourke, 1996) and smoothed particle hydrodynamics (SPH) (Monaghan and Kocharyan, 1995; Xiong et al., 2011), may greatly reduce the computing load.
For the sake of industrial application, we restrict our method in this article onto the Eulerian–Eulerian continuum approach. In this case, a practical way is to incorporate the structure in terms of its bimodal probability distribution, as mentioned above, into the Eulerian continuum modeling. Thus, we may reformulate the normally used TFM into a seemingly four-fluid model character- ized by the dilute-phase gas, dilute-phase solid, dense-phase gas and dense-phase solid, though it is essentially different from a commonly termed multi-fluid model, as we will discuss in later section. This simplification has its rationality based on experi- ments, but also has limitation due to its phenomenological nature. For example, a practical problem is how to select the characteristic structure, bubble, cluster or whatever, for possible wide range of flow regimes. Our previous work (Lu et al., 2009; Wang and Li, 2007; Yang et al., 2003) on the energy-minimization multi-scale (EMMS) model was based on cluster characterization (Li and Kwauk, 1994), and its application was restricted in high-velocity circulating fluidized bed (CFB). Recently we extended the EMMS model to a bubble-based version and found it works well for low- velocity bubbling fluidization (Shi et al., 2011a, 2011b). The clear importance of the meso-scale structure in formulating these EMMS-based models drives us to propose a structure-dependent multi-fluid model (SFM) (Hong et al., 2012). This first version of the SFM was based on cluster structure and has proved to unify the classic TFM and EMMS models with different structures. In this article, we try to unify these two types of structural descriptions, i.e., bubble and cluster, under the umbrella of the SFM, and thereby aim to unify our past efforts on the cluster-based EMMS (Li and Kwauk, 1994) and the bubble-based EMMS models (Shi et al., 2011a, 2011b). Hopefully, such unification may allow one to simulate gas–solid flows in different flow regimes ranging from bubbling to fast fluidization with satisfactory accuracy and affordable computing cost.
2. Structure-dependent multi-fluid model (SFM)
Following the above discussion, we may resolve a monodis- perse gas–solid flow into a mixture of particle-rich, dense phase (denoted by subscript c) and gas-rich, dilute phase (denoted by subscript f) and then derive a structure-dependent multi-fluid model (SFM). Following the conventional TFM approach, we may refine it by defining four continua of structural sub-elements, namely the dense-phase gas (denoted by gc), dense-phase solid (denoted by sc), dilute-phase gas (denoted by gf) and dilute-phase solid (denoted by sf), as shown in Fig. 1.
As presented in Hong et al. (2012), both the dense and dilute phases are assumed uniformly dispersed and homogeneous inside; the mass exchange terms (g, s) are taken to account for
K. Hong et al. / Chemical Engineering Science 99 (2013) 191–202 193
dynamic transformation of gas/solids between the dilute and dense phases. That can be described by the following balance equations for four structural sub-elements:
Continuity equation for the dense-phase gas:
∂ ∂t
Continuity equation for the dense-phase solid:
∂ ∂t
Continuity equation for the dilute-phase gas:
∂ ∂t
ð1−f Þεgf ρg þ∇ ð1−f Þεgf ρgugf Þ
¼ −Γg ; ð3Þ
∂ ∂t
ð1−f Þεsf ρs þ∇ ð1−f Þεsf ρsusf Þ
¼−Γs; ð4Þ
∂ ∂t
ðf εgcρgugcÞ þ ∇ ðf εgcρgugcugcÞ ¼−f∇pþ∇ ðf τgcÞ þf ρgg−fFdc þ Γgui
g ; ð5Þ
∂ ∂t
ðf εscρsuscÞ þ ∇ ðf εscρsuscuscÞ ¼ −∇psc þ ∇ ðf τscÞ þf εscðρs−ρgÞgþ fFdc þ Fdi þ Γsui
s; ð6Þ
∂ ∂t
ð1−f Þεgf ρgugf þ∇ ð1−f Þεgf ρgugfugf
¼−ð1−f Þ∇pþ ∇ ½ð1−f Þτgf þ ð1−f Þρgg −ð1−f ÞFdf−Fdi−Γgui
g ;
∂ ∂t
¼ −∇psf þ∇ ð1−f Þτsf
þð1−f Þεsf ðρs−ρgÞg þ ð1−f ÞFdf−Γsui s ð8Þ
here, for brevity, we only list the final forms of equations, whereas greater detail can be referred to Hong et al. (2012). Our first attempt at formulating structure-dependent, multi-fluid conserva- tion equations dates back to the Appendix in Wang and Li (2007), followed by Dong et al. (2008) and Hong et al. (2012). Lu et al. (2008) have presented a model with dilute-dense structure and two granular temperatures. We can further find that this set of SFM equations (Eqs. (1)–(8)) is not just a natural extension of the TFM to that for four fluids. In fact, the SFM differs from a four-fluid model in that SFM has meso-scale drag force (Fdi) caused by structures and meso-scale mass exchange (g, s) which are important even without reactions, reflecting the dynamic trans- formation of two-phase structures.
This set of governing equations (Eqs. (1)–(8)) follows the type-B hydrodynamic model of Gidaspow (1994) in the sense that all the gas pressure drop is exerted on the gas phase. In literature, there still exist some disputes about the applicability of various types of two-fluid models (Zhou et al., 2010). In principle, these models can be transformed into each other, if the respective drag models are appropriately used in line with the momentum conservation equations. To determine the independent variables (i.e., p, f, εgf, εgc, ugf, ugc, usf, usc), the remaining parameters (i.e., τsf, τsc, psf, psc, g, s, Fdf, Fdc, Fdi) should be closed appropriately. For example, inside the dilute and dense phases, the drag forces (Fdf, Fdc) can be closed with Wen and Yu (1966) correlation, as follows:
Fdf ¼ 3 4 Cdf
1−εgf dp
ρg Uslip;f Uslip;f
1−εgc dp
ρg Uslip;c Uslip;c
ð10Þ
where Uslip,f and Uslip,c are superficial slip velocity in the dilute and dense phases, respectively. The solid pressure and viscosities (psf, psc, τsf, τsc) can be derived from the kinetic theory of granular flows (KTGF) (Gidaspow, 1994). If the meso-scale structures are further assumed to be uniformly dispersed in the given volume in forms of bubbles or clusters, as shown in Fig. 1, the meso-scale drag force (Fdi) can also be closed with the same drag function but with respective structural properties (e.g., bubble diameter or cluster diameter). We can see that, accordingly, it is the closure of the meso-scale drag force that may fashion different SFMs. It should also be noted that the meso-scale structure is inherently dynamic and hard to measure. Thus, the mass exchange terms (g, s) are hard to quantify. In addition, direct solution of SFM requires higher computational demand than that of TFM. Thus, we will try to tackle two issues in the following sections: one is how to characterize the meso-scale inter-phase drag (Fdi), and the other is how to simplify the SFM to the form of classic TFM but with corrected closures, thus avoiding the modeling of Γ and reducing the computational loading.
3. Closure of meso-scale drag force Fdi
Following the conventional characterization of bubble and cluster in a fluidized bed, we may introduce two kinds of meso- scale drag (Fdi), as follows:
3.1. Cluster-based Fdi
If the dense phase is assumed to exist in form of spherical clusters with diameter dcl, and is uniformly dispersed in the dilute phase, then the meso-scale drag force (Fdi) can be closed by
Fdi ¼ 3 4 Cdi
f dcl
where Cdi is closed by Wen and Yu (1966) correlation
Cdi ¼ Cdi0ð1−f Þ−4:65 ð12Þ with
Cdi0 ¼ 24=Rei þ 3:6=Re0:313i ; ð12aÞ and
Rei ¼ ρgdclUslip;i=μg : ð12bÞ here, the inter-phase superficial slip velocity (Uslip,i) is defined by (Ugf−Uscεgf/εsc)(1−f). Then, the closure issue changes to the deter- mination of cluster diameter, whereas our previous versions normally adopted the original correlation of Li and Kwauk (1994) where the cluster diameter is inversely proportional to the energy consumption of Nst. Table 1 lists some empirical or semi-empirical correlations in literature. In principle, they can also be used to replace the Nst-related correlation.
3.2. Bubble-based Fdi
If the dilute phase can be characterized by uniformly dispersed bubbles with diameter of db, we can get Fdi as follows:
Fdi ¼ 3 4 Cdb
1−f db
Cdb ¼ Cdb0f −0:5 ð14Þ
Table 1 Some correlations for cluster diameter in literature.
Author Correlation
Li and Kwauk (1994) dcl ¼ dp ½Us=ð1−εmax Þ−ðUmf þεmf Us=ð1−εmf ÞÞ⋅g Nstρs=ðρs−ρg Þ−ðUmf þεmf Us=ð1−εmf ÞÞ⋅g
Zou et al. (1994) dcl ¼ 1:8543½ε−1:5g ð1−εgÞ0:25=ðεg−εmf Þ2:411:3889dp þ dp Gu and Chen (1998) dcl ¼ dp þ ð0:27−10dpÞεs þ 32εs6
Harris et al. (2002) dcl ¼ εs=ð40:8−94:5εsÞ, where εs is cross-sectionally averaged voidage Subbarao (2010)
dcl ¼ 1−εg εg−εgc
1=3 2u2 t
g ð1þ u2 t =u
2 srÞ−1 þ dp , where usr ¼ 0:35ðgDt Þ1=2
Table 2 Some correlations for bubble diameter in literature.
Author Correlation
Horio and Nonaka (1987) db ¼ ½−γM þ ðγ2M þ 4dbm=Dt Þ0:52 Dt=4,
γM ¼ 2:56 10−2ðDt=gÞ0:5=Ugc , dbm ¼ 2:59g−0:2½ðUg−UgcÞπD2 t =40:4
Werther (1983) dbðhÞ ¼ 0:853½1þ 0:272ðUg−Umf Þ1=3ð1þ 0:0684hÞ1:21 Mori and Wen (1975) dbðhÞ ¼ dbm−ðdbm−db0Þe−0:3h=Dt ; dbm ¼ 0:65½ðUg−Umf ÞπD2
t =40:4
K. Hong et al. / Chemical Engineering Science 99 (2013) 191–202194
with
Cdb0 ¼ 38Re−1:5i 0oRei ≤1:8 2:7þ 24=Rei Rei41:8
( ð14aÞ
and
Rei ¼ ρcdbUslip;i=μc :ð14bÞ
here, ρc is the mixture density of the dense phase defined by ρc¼εscρs+εgcρg. Uslip,i is defined by f(Uf−Uc), where Uf and Uc are mean velocities of the dilute and dense phases, respectively. This definition of meso-scale slip velocity differs a little bit from that in the cluster-based EMMS model, because the continuous phase changes from the dilute phase to the dense phase. And their quantitative difference is (1−f)ρgUgc/ρc, which is normally negligi- ble for gas–solid systems. Similarly, the closure of Fdi switches to the determination of bubble diameter. In literature, bubble has been extensively studied ever since the classic work of Davidson (1961). There are many correlations proposed and some of them are listed in Table 2. Compared to cluster diameter, bubble diameter arouses less disputes and hence is easier to characterize. The visual bubbles are normally irregular and in constantly dynamic transformation, which may deviate much from spherical assumption. Thus, the diameter of bubble here can be viewed as drag-equivalent definition. More elaborate characterization may be based on DNS simulation, which is however beyond the scope of this work. In this article, we adopted Horio and Nonaka's (1987) correlation which was reported to apply to both Geldart group A and B particles.
4. Unification of the EMMS model and TFM
As we have presented in our previous work (Hong et al., 2012), the SFM may revert to the hydrodynamic equations of the classic TFM and cluster-based EMMS model if corresponding simplifica- tions are introduced. Here, we will extend such unification to the bubble-based EMMS model. To present the results as a whole, we will also include some of derivations that have appeared in our previous paper (Hong et al., 2012). For more details, readers are referred to Hong et al. (2012).
4.1. Restoration to the cluster-based EMMS model
As presented in Hong et al., (2012), combining the dilute and dense continuity equations, we get the mass balance equations for the gas and solids, as follows:
Ug ¼ fUgc þ ð1−f ÞUgf ; ð15Þ
Us ¼ fUsc þ ð1−f ÞUsf ð16Þ
For a lumped-parameter model of a steady-state reactor, the force balance equations in SFM can be simplified as follows:
For the dense-phase solid,
for the dilute-phase solid,
for the dense-phase gas,
for the dilute-phase gas,
ð1−f ÞFdf þ Fdi ¼ −ð1−f Þ∇pþ ð1−f Þρgg: ð20Þ
Eliminating the gas pressure gradient from Eqs. (19) and (20) yields the pressure drop balance equation of the EMMS model,
Fdf þ Fdi 1−f
¼ Fdc: ð21Þ
For transient flow, different inertial effects should be included in both the dilute and dense phases. Then, we can obtain the hydrodynamic equations of the EMMS model at the sub-grid level (Lu et al., 2009; Wang and Li, 2007), whose detail is referred to Hong et al. (2012).
Eqs. (15)–(18) and (21) recover the hydrodynamic equations of the classic, cluster-based EMMS model, if Li and Kwauk's (1994) correlation in Table 1 and Eq. (11) are used to characterize the cluster diameter and the meso-scale drag force on it. If we choose instead the bubble to characterize the structure (e.g., Horio and Nonaka's (1987) correlation in Table 2) and thus replace Eq. (11) with Eq. (13), then we can transform the SFM into the recently proposed, bubble-based EMMS model (Shi et al., 2011a).
K. Hong et al. / Chemical Engineering Science 99 (2013) 191–202 195
4.2. Restoration to the bubble-based EMMS model
For a bubble–emulsion structure, the mass balance equation of gas keeps the same as Eq. (15). If the bubble is assumed to be void of solids, as is the case in Shi et al. (2011a), then the mass balance equation of solids, Eq. (16), can be further simplified by using εsf¼0, that is,
Us ¼ fUsc: ð22Þ If one neglects the meso-scale drag force, Fdi, in the force balance for the dense-phase solid, as assumed in Shi et al. (2011a), Eq. (17) reduces to
Fdc ¼−εscðρs−ρgÞg: ð23Þ Combining Eqs. (19) and (23), we can derive the pressure gradient by
∇p¼ ðεscρs þ εgcρgÞg¼ ρcg: ð24Þ Substituting Eq. (24) into Eq. (20), we can reformulate the force
balance of bubble as follows:
−Fdi ¼ ð1−f Þðρc−ρgÞg: ð25Þ In practice, Eqs. (15), (22), (23) and (25) recover the balance
equations of our bubble-based EMMS model (Shi et al., 2011a). From the above derivation, we can see that the SFM may revert
to the cluster-based EMMS and bubble-based EMMS models under different assumptions of structure parameters and thus unify them. Indeed the meso-scale structure has always been the critical part of the EMMS model. We may expect more variants of the EMMS model will be proposed because of different descriptions of the meso-scale structure together with more in-depth under- standing. Furhtermore, the SFM may be treated as a criterion to examine various variants of the EMMS model. For example, Wang et al. (2008a) proposed an alternative method with added mass force to close the cluster diameter. Shi et al. (2011a) neglected the meso-scale drag in the force balance for the dense-phase solid. More efforts are under way for a possible universal version of the EMMS model.
4.3. Closure of EMMS: Stability condition
Though we can derive a set of conservation equations of the EMMS model from the SFM, the number of equations is less than that of independent variables (e.g., db, f, εgf, εgc, Ugf, Ugc, Usf and Usc
for the bubble-based EMMS). For such underdetermined problem, Li and Kwauk (1994) proposed a stability condition of Nst-min to close it.
According to Li and Kwauk (1994), in concurrent-up gas–solid riser flow, the gas tends to flow upward with less resistance, whereas particles tend to be arrayed with minimal potential energy. The compromise between them results in a tendency of minimal energy consumption for suspending and transporting particles with respect to unit mass of particles, that is,
Nst ¼ f Fdc Ugc þ ð1−f ÞFdf Ugf þ Fdi Ugf ð1−f Þ
ð1−εgÞρs -min: ð26Þ
Nst ¼ ρs−ρg ρs
Ug− εgf−εg 1−εg
f ð1−f ÞUgf
g-min: ð27Þ
For bubbling fluidization where solids entrainment is negligi- ble, it can be reformulated as in Shi et al. (2011a), as follows:
Nst ¼ f Fdc Ugc þ ð1−f ÞFdf Ugf
ð1−εgÞρs þ ð1−f Þεgf Ug g=εg-min: ð28Þ
These two stability conditions are actually the same in physics and the only difference lies in the energy consumption term
related to meso-scale drag because of their different characteriza- tion of the meso-scale structure. For general purpose, Ge and Li (2002) indicated that the stability condition is better defined by a dimensionless ratio, Nst/NT-min, where NT is the total mass- specific energy consumption. In summary, the EMMS model consists of a set of conservation equations, a definition of meso- scale structure and a stability condition, whereas the concrete equations may vary and depend on the parameters for meso-scale structure.
4.4. Reduction to the TFM
Combining the mass/momentum equations of the dilute and dense phases in the SFM, we can derive a set of equations similar to that of TFM, as detailed in Hong et al. (2012). Here, we only list these equations as follows:
∂ ∂t
∂ ∂t
∂ ∂t
ðεgρgugÞ þ ∇ ðεgρgugugÞ ¼−∇pþ ∇ τge þρgg−βBeðug−usÞ þ ∇ τDg ; ð31Þ
∂ ∂t
ðεsρsusÞ þ∇ ðεsρsususÞ ¼ −∇pse þ∇ τse þ εsðρs−ρgÞg þβBeðug−usÞ þ ∇ τDs; ð32Þ
where
εk ¼ f εkc þ ð1−f Þεkf ; ; ð33Þ and
uk ¼ f εkcukc þ ð1−f Þεkfukf
εk : ð34Þ
The structure-dependent solids stress (τe), diffusion stress (τD) and solid pressure (pse) are the same as in Hong et al. (2012), which are usually negligible compared to the drag force. That is also the strategy what we have followed in previous simulations for the sake of simplicity (Lu et al., 2009; Wang and Li, 2007). The effective drag coefficient (βBe) can be written as
βBe ¼ fFdc þ Fdi þ ð1−f ÞFdf
ug−us ; ð35Þ
which exactly reverts to our definition in the EMMS-based multi-scale CFD approaches. Thus, our multi-scale CFD that is characterized by coupling TFM and EMMS drag can be viewed as a simplified solution of the SFM.
5. Validation of the bubble-based EMMS model
As discussed above, both the cluster-based EMMS and bubble- based EMMS models can be derived from the SFM in terms of Eqs. (15)–(20) and thus, be unified under the umbrella of the SFM with different structures, i.e., bubble and cluster. In literature (Ge and Li, 2002; Li and Kwauk, 1994; Wang et al., 2008b), there are many efforts to validate the cluster-based EMMS model for riser flows. Thus, in the following section, we will only focus on the validation of the bubble-based EMMS model, but for more flow regimes covering from bubbling fluidization to riser flow.
As mentioned above, our previous version of the bubble-based EMMS model neglects the contribution of the meso-scale drag, Fdi, in the force balance equation for the dense phase. Here, for the sake of completeness, we will include all components of the drag
K. Hong et al. / Chemical Engineering Science 99 (2013) 191–202196
force and reorganize the three force balance equations (Eqs. 17, 18 and 21) as follows:
3 4 Cdf
ρgU 2 slip;f
3 4 Cdc
ρgU 2 slip;c
3 4 Cdb
1−f db
ρcU 2 slip;i ¼ f ðεg−εgcÞðρs−ρgÞg: ð38Þ
Thus, Eqs. (15), (16), (36)–(38) form the new version of the bubble-based EMMS model. Following Ge and Li (2002), we used a global search scheme to determine the eight parameters (db, f, εgf, εgc, Ugf, Usf, Ugc, Usc) in the model, as follows:
1.
Fig
K Jo
For the given operating conditions (Ug, Us), εgf is assumed to be equal to εmax, i.e., 0.9997 (Ge and Li, 2002; Matsen, 1982); initial value of Nst/NT is assigned to be unity;
2.
Sweep εgc and f within the range of [εmf, εmax] and [0, 1], respectively;
3.
4.
With the definition of Uslip,c, Uslip,f in Table A1, calculate Ugf, Usf,
Ugc, Usc from Eqs. (15) and (16);
5.
Table 2, and then Uslip,i1 from Eq. (38);
6.
Calculate Uslip,i2 from its definition in Table A1, and then the
difference between two slip-velocities, ΔUslip,i¼Uslip,i1−Uslip,i2.
7.
Compare the value of ΔUslip,i with the convergence criterion, if
converged, then store all related parameters and continue the traversal;
8.
Find the optimal root through minimization of Nst/NT among all possible roots satisfying the convergence criterion of ΔUslip,i.
To measure the different components of the EMMS drag, in following discussions, we will additionally introduce kc and ki that
. 2. Comparison of predicted volume fraction of bubbles and experimental data.
le 3 terial properties of the experiments in literature cited in Fig. 2.
uthors ρs (kg/m3) dp (mm)
rishna and van Baten (2001) 1480 0.06 hnsson et al. (1991) 2600 0.15
0.46
represent the ratio of fFdc and Fdi, respectively, to the overall drag force exerted on the dense-phase solid. As the sum of kc and ki equals unity, we can easily get kc¼ f+(1−εgf)(1−f)/(1−εgc) and ki¼1 −kc. If the solids content in bubbles is neglected (i.e., εgf¼1), one can obtain kc¼ f and ki¼1−f.
In a bubbling fluidized bed, the volume fraction of bubbles is an important factor affecting the inter-phase heat/mass transfer, since the gas in bubbles interacts less with particles than that in the emulsion. Fig. 2 gives the comparison of model prediction and experimental data in the literature listed in Table 3. Generally, the bubble-based EMMS model predicts a reasonable trend which is similar to that in Shi et al. (2011a) and also in agreement with experimental data. Following Shi et al. (2011a), we have also compared the model prediction with some other parameters such as bubble velocity, emulsion fraction and so on. All these results are in fair agreement with what has been reported in Shi et al. (2011a). For brevity, we will not present these comparisons repeatedly here. In addition, it is interesting to note in Fig. 2 that neglecting the meso-scale drag (Fdi) in the force balance for dense- phase solids, as practiced in Shi et al. (2011a), does not cause big discrepancy from current model. One of the reasons may lie in that ki¼1−f is less than 0.5 for most cases as shown in Fig. 2. Thus, compared to fFdc, Fdi has less effect on the dense-phase solids. More elaborate analysis are still needed to clarify this issue.
In summary, our previous version of the bubble-based EMMS model (Shi et al., 2011a) can be viewed as a simplified version of the current work. This new model can well predict the effects of meso-scale bubbles in bubbling fluidized beds. In following sec- tion, we will further validate it by performing CFD simulations with SFM and this bubble-based EMMS drag.
6. Simplified solution of SFM: Coupling of TFM and bubble-based EMMS drag
For transient simulation of a fluidized bed, the inertial effects should be included into the force balance equations. Similar to the derivation of Shi et al. (2011a), the bubble-based EMMS model for the unsteady state consists of ten structure variables (e.g., db, f, εgf, εgc, Ugf, Usf, Ugc, Usc, asc, asf), which are closed by seven equations and one stability condition. The relevant parameters and defini- tions are summarized in Appendix. For a given set of conditions (Ug, Gs, εg, ρg, ρs, dp, μg), such nonlinear programming problem needs to be solved to determine the structural parameters and the structure-dependent drag coefficient (βe),
βe ¼ ε2g Uslip
¼ ε2g Uslip
ðρs−ρgÞ f ð1−εgcÞðg þ ascÞ þ ð1−f Þð1−εgcÞðg þ asf Þ
: ð39Þ
following Wang and Li (2007), βe is measured with the hetero- geneity index, Hd,
Hd ¼ βe βWY
0.0022 0.38 Air-FCC 0.02 0.68 Air-silica sand 0.18
K. Hong et al. / Chemical Engineering Science 99 (2013) 191–202 197
where βWY is Wen and Yu (1966) correlation with
βWY ¼ 3 4 Cd0
εgεs dp
ð41Þ
here, the inertial term asf is approximated to equal the gravita- tional acceleration (Wang and Li, 2007). Similar to our previous version (Wang and Li, 2007), three transient force balance equa- tions can be rewritten as follows:
f Fdc ¼ 3 4 Cdc
f ð1−εgcÞ dp
ρgU 2 slip;c ¼ f ð1−εgcÞðρp−ρgÞðg þ ascÞkc; ð42Þ
Fdi ¼ 3 4 Cdb
1−f db
ρcU 2 slip;i ¼ f ð1−εgcÞðρp−ρgÞðg þ ascÞki; ð43Þ
Fdf ¼ 3 4 Cdf
1−εgf dp
ρgU 2 slip;f ¼ ð1−εgf Þðρp−ρgÞðg þ asf Þ; ð44Þ
as the sum of kc and ki equals unity, one can easily get the following expressions,
kc ¼ f þ ð1−f Þ ð1−εgf Þðasf þ gÞ ð1−εgcÞðasc þ gÞ ; ð45Þ
ki ¼ 1−kc: ð46Þ Similarly, we adopt the global search scheme to solve Eqs. (15),
(16), (33), (42) and (43) as follows:
1.
Gs
Fig. bed (Ven (Li a
For a given system with specified conditions (Ug, Gs, εg), traverse over trial value of εgc within the range of [εmf, εg];
2.
Calculate the volume fraction of dense phase f from Eq. (33);
e 4 mary of material properties and operating parameters for three testing cases.
levant parameters Zhu et al. (2008)
6.510−5 m 1780 kg/m3
1.225 kg/m3
0.2 m/s 0 kg/(m2s)
3. Variation of Hd with εg for three operation conditions: bubbling fluidized with Ug¼0.2 m/s (Zhu et al., 2008), turbulent fluidized bed with Ug¼0.6 m/s derbosch, 1998) and fast fluidized bed with Ug¼1.52 m/s and Gs¼14.3 kg/(m2 s) nd Kwauk, 1994).
3.
Traverse over trial values for asc with the range of [−9.8, amax], here amax is a big value dependent on specific cases;
4.
5.
With the definition of Uslip,c, Uslip,f, calculate Ugf, Usf, Ugc, Usc
from Eqs. (15) and (16);
6.
then Uslip,i from Eq. (43);
7.
between two slip-velocities ΔUslip,i;
Compare the value of ΔUslip,i with the convergence criteria, if
converged then store this value and continue the traversal until finished, else directly jump back to the step 3;
9.
Find the optimal root through minimization of Nst/NT among all possible roots satisfying hydrodynamic convergence with respect to ΔUslip,i;
10.
Calculate the effective coefficient βe and heterogeneity index Hd from Eqs. (39) and (40), respectively.
By comparison, our previous version of the unsteady-state, bubble-based EMMS model (Shi et al., 2011a) introduces a correla- tion for the added mass force (Wang et al., 2008a; Zhang and VanderHeyden, 2002) to replace the correlation of bubble dia- meter. It also neglects the solids content inside bubbles and the meso-scale drag force (Fdi) in the force balance of the dense-phase solid. As shown in the next sectionit can be found that the new model predicts Hd-1 for two homogeneous ends corresponding to εg¼εmf and εg¼1, whereas the previous model in Shi et al. (2011a) fails to predict this tendency.
6.1. Model results
Fig. 3 shows the variation of heterogeneous index (Hd) against voidage under three operating conditions that cover flow regimes from bubbling to so-called fast fluidization. The material proper- ties and operating parameters are listed in Table 4. The fitting functions of relevant heterogeneous index are summarized in Table 5.
In comparison to the drag of Wen and Yu (1966), Hd shows a remarkable drop within most of the range of voidage and converges to unity near the two ends, where the heterogeneous distribution gradually gives way to the uniform, minimum fluidi- zation and free, isolated sedimentation.
To investigate the grid dependency of this bubble-based EMMS drag, we follow the scheme of Agrawal et al. (2001) by performing a series of simulations over doubly periodic domain. The simula- tion settings are almost the same as in our previous work (Lu et al., 2009) except that different drag coefficients, as listed in Table 5, are used. For brevity, more detail on simulation setting parameters is referred to Lu et al. (2009).
Fig. 4 shows the variation of the time-mean, domain-averaged slip velocity with the dimensionless grid size. When the drag correlation for the cases of Zhu et al. (2008) and Venderbosch
Venderbosch (1998) Li and Kwauk (1994)
9.010−5 m 5.410−5 m 1375 kg/m3 930 kg/m3
1.225 kg/m3 1.1795 kg/m3
1.789410−5 Pa s 1.887210−5 Pa s 0.4 0.4 0.6 m/s 1.52 m/s 0 kg/(m2s) 14.3 kg/(m2s)
K. Hong et al. / Chemical Engineering Science 99 (2013) 191–202198
(1998) are used, the dimensionless slip velocities increase to around 2.7 as the grid is refined to about ten times the particle diameter. When the drag correlation obtained for the case of Li and Kwauk (1994) is used, however, the grid dependency is much reduced, whereas the dimensionless axial slip velocity remains around 4.7 for the whole range of grid size. These results mean that, for the case of Li and Kwauk (1994), we may apply the EMMS drag model in Table 5 over a wide choice of grid size, thus greatly reducing the computing load by allowing to perform coarse-grid simulation. However, for the other two cases in Table 5, we need to carefully refine the grid size down to the level of ten times the particle diameter, to guarantee meaningful simulations. In the following section, we will further evaluate our EMMS drag models by simulation of practical fluidized beds.
6.2. CFD simulation setup
The numerical simulation was carried out by using the Eulerian multiphase model of Fluents 6.3.26, and the EMMS drag model was used through User-Defined Functions (UDF). The kinetic theory of granular flows (KTGF) (Gidaspow, 1994) was still used to close the solid pressure and viscosities, thus, neglecting the structure-dependent stress and pressure.
2D simulations were performed to save computational resources, though 3D configuration can be expected to give more
Fig. 4. Effect of grid resolution on time-averaged axial slip velocity for three drag models in Table 5 over a periodic domain (ρp¼1500 kg/m3, dp¼75 μm, ut¼0.2184 m/s, εmf¼0.4, domain size¼1.56 cm2).
Table 5 The fitting function of heterogeneous index (Hd) for three testing cases.
Fitting function of heterog
Zhu et al. (2008) Hd ¼ 0:15763þ 1:21795=½1 Hd ¼ e−4:0019þ5:4957εg−1:7412ε2g
Hd ¼ −37:73207 þ 38:7433 Hd ¼ 1
Venderbosch (1998) Hd ¼ 0:2478þ 1:11947=½1þ Hd ¼ e−3:7591þ5:3010εg−1:7907ε2g
Hd ¼ 1
Li and Kwauk (1994) Hd ¼ 0:06409þ 96:85546=½ Hd ¼ e−4:6495þ3:66287εg−0:55601
Hd ¼ e−265:08307þ457:73731εg−19
Hd ¼ 1
reasonable results (Zhang et al., 2008). The 2D schematics of three selected fluidized beds are shown in Fig. 5. For the system of Zhu et al. (2008), the bed is 2.464 m in height and 0.267 m in inner diameter. The disengaging section was neglected to save computa- tion time. Particles with minimum fluidization voidage are initially packed up to a height of h0¼1.2 m. The gas uniformly flows into the bottom of bed with Ug¼0.2 m/s and leaves from the top outlet, where atmospheric pressure is prescribed. The other two cases adopted similar settings. For the case of Venderbosch (1998), the main section of the bed is 0.75 m in height and 0.05 m in diameter, above which an expanded bed section (ID 0.1 m, 0.3 m long) is also included to keep particles from serious entrainment. It should be noted that in 2D numerical simulations, the diameter of the expanded bed is enlarged to keep the same superficial gas velocity as in real 3D geometry. The initial height of packed particles with εs¼0.5625 is 0.2 m. The gas velocity at the bottom inlet is 0.6 m/s. For Li and Kwauk's (1994) system, the riser is 10.5 m in height and 0.09 m in diameter. The gas velocity is Ug¼1.52 m/s at the bottom of riser. Initially, the solids in the riser were distributed with a solids volume fraction of 0.058. For all three cases, the initial velocities of the gas and solids inside the beds were assumed to be zero. To ensure particle conservation, the solid mass flow rate is monitored at the outlet and the entrained solids are recirculated into the solid inlet. The no-slip boundary condition was prescribed for the gas phase, whereas the partial-slip boundary condition developed by Johnson and Jackson (1987) was used for the solids with a specularity coefficient of 0.0001. Simulation was first performed for 20 s of physical time, and then the time-averaged analysis was carried out over the remaining 10 s. The setting parameters for simulations in Fluents 6.3.26 were summarized in Table 6.
6.3. CFD simulation results and discussion
Fig. 6 shows the snapshots of solids concentration under different grid resolutions for three simulation cases. In general, finer flow structures can be captured with grid refining. For the bubbling fluidized bed (Zhu et al., 2008), big void area surrounded by continuous solids can be discerned, though these voids are shapeless and can hardly be termed conventional bubbles. For the turbulent fluidized bed (Venderbosch, 1998), void area obviously occupies more space in the bed and the gas becomes the continuous phase, suspending dense particle clusters. For the fast fluidized bed (Li and Kwauk, 1994), the snapshots reveal obvious clusters and/or streamers across the whole bed. In all, the simplified SFM with the bubble-based EMMS drag predicts various heterogeneous structures in gas–solid fluidized beds, which agrees qualitatively with experimental findings.
eneous index (Hd) Range (εmf≤εg≤1)
þ ðεg=0:41554Þ25:37383 εmf≤εgo0.490
0.490≤εgo0.9937
ðεg=0:42992Þ12:5451 εmf≤εgo0.576
1þ ðεg=0:30852Þ17:70284 εmf≤εgo0.553 ε2g 0.553≤εgo0.978 2:62209ε2g 0.978≤εgo0.9997
0.9997≤εg≤1
Fig. 5. 2D schematic drawing for (a): bubbling fluidized bed (Zhu et al., 2008), (b): turbulent fluidized bed (Venderbosch, 1998) and (c): circulating fluidized bed (Li and Kwauk, 1994).
Table 6 Simulation settings used in Fluents 6.3.26.
Viscous model Laminar Unsteady formulation First-order implicit Pressure-velocity coupling Phase coupled SIMPLE Momentum discretization Second-order upwind Volume fraction discretization Quick Granular temperature discretization Second-order upwind Granular viscosity Gidaspow Granular bulk viscosity Lun et al. Frictional viscosity Schaeffer Angle of internal friction 301 Frictional pressure Based KTGF Solids pressure Lun et al. Radial distribution Lun et al. Particle-particle coefficient restitution, es 0.9 Particle-wall coefficient restitution, ew 0.99 Specularity coefficient, φ 0.0001 Close packing density, εs,max 0.63 Time step, Δt 510−4 s Max. iterations per time step 30
K. Hong et al. / Chemical Engineering Science 99 (2013) 191–202 199
Fig. 7 gives the comparison of axial profiles of solids concentration against experimental data for three cases. Generally, the simulation predicts quite well the cases of bubbling and circulating fluidized beds, whereas we seem to over-predict the bed expansion in the turbulent fluidized bed. Solids entrainment may contribute to such discrepancy though further analysis needs to be performed in future work.
The circulating solids flux is an important factor to characterize the overall flow performance of a CFB. The solids flux at the outlet was monitored and statistically averaged. Table 7 summarizes the calculated solids flux against experimental data for Li and Kwauk's
(1994) system. The predicted values of solids flux are close to the experimental data for all three sets of grid resolution. That again verifies our statement on the grid dependency of the EMMS-based drag model for simulation of CFB.
Our previous works on the cluster-based EMMS model was restricted to simulation of high-velocity CFB with Gs40. In this article, we extended the application of the bubble-based EMMS model to more flow regimes ranging from bubbling to fast fluidization. This work aims to unify two types of structural descriptions, i.e., bubble and cluster, under the umbrella of the SFM. More quantitative validation has been discussed in our previous efforts, involving such as the radial distribution of solids concentration and gas/solids velocities. Thus, we will not extend validation to more cases.
7. Conclusions
This is follow-up to our previous efforts on the EMMS/bubbling model (Shi et al., 2011a) and the structure-dependent multi-fluid model (SFM) (Hong et al., 2012). To be consistent with structures, the SFM was proposed based on the dilute-dense two-phase characterization, and it can be reduced to the conventional TFM as well as the bubble-based and cluster-based EMMS models. Thus, these models can be unified under the umbrella of the SFM. A new version of the bubble-based EMMS model was presented thereby and evaluated with comparison to experimental data. This new bubble-based EMMS model was verified suitable for wide flow regimes ranging from bubbling to fast fluidization with satisfactory accuracy and acceptable computational cost.
Fig. 6. Snapshot of predicted solids concentration for (a): bubbling fluidized bed (Zhu et al., 2008), (b): turbulent fluidized bed (Venderbosch, 1998) (The numbers at the bottom represent the grid resolution for the main section of the bed and similar grid size is adopted for the expanded section.) and (c): circulating fluidized bed (Li and Kwauk, 1994).
Fig. 7. predicted axial profiles of solids concentration for (a): bubbling fluidized bed (Zhu et al., 2008), (b): turbulent fluidized bed (Venderbosch, 1998) (The numbers at the bottom represent the grid resolution for the main section of the bed and similar grid size is adopted for the expanded section.) and (c): circulating fluidized bed (Li and Kwauk, 1994).
Table 7 The predicted solids flux and its standard deviation for CFB case of Li and Kwauk (1994).
Grid resolution Experimental solids flux [kg/(m2s)]
Predicted solids flux [kg/(m2s)]
Standard deviation [kg/(m2s)]
40300 14.3 16.77 5.74 60450 16.52 5.37 80600 15.31 5.30
K. Hong et al. / Chemical Engineering Science 99 (2013) 191–202200
Nomenclature
a inertial term, m/s2
Cd effective drag coefficient for a particle or a bubble Cd0 standard drag coefficient for a particle or a bubble db bubble diameter, m dcl cluster diameter, m dp particle diameter, m
Dt diameter of fluidized bed, m es particle–particle restitution coefficient ew particle–wall restitution coefficient f volume fraction of dense phase F/F drag force, N g/g gravitational acceleration, m/s2
Gs solids flux, kg/(m2s) h riser height, m Hd heterogeneity index
Table A.1 The summary of parameters and definitions in the bubble-based EMMS model (adapted from Shi et al. (2011a)).
Dilute phase Dense phase Inter-phase
Characteristic diameter dp dp db Voidage εgf εgc 1−f
Uslip;f ¼Ugf − Uscεgf εsf
Uslip;c ¼Ugc− Uscεgc εsc
Uslip;i ¼ f ðUf −UcÞ Characteristic Reynolds number Ref ¼ ρgdpUslip;f
μg Rec ¼ ρgdpUslip;c
μg Rei ¼ ρcdbUslip;i
þ 3:6 Re0:313
2:7þ 24 Rei
Rei41:8
8< :
Effective drag coefficient Cdf ¼ Cdf0ε −4:65 gf Cdc ¼ Cdc0ε
−4:65 gc Cdb ¼ Cdb0f
−0:5
mc ¼ 1−εgc πd3p=6
mi ¼ 1−f πd3b=6
Drag force on each particle or bubble Ff ¼ Cdf πd2p 4
ρg 2 U
πd2p 4
ρc 2 U
2 slip;i
Drag force in unit volume Fdf ¼mf Ff Fdc ¼mcFc Fdi ¼miFi
Where the averaged densities of the dilute and dense phases are ρb¼εgfρg+εsfρs, ρc¼εgcρg+εscρs, respectively. The dense-phase viscosity is μc¼μg[1+2.5εsc+ 10.05εsc2+0.00273exp(16.6εsc)](Thomas, 1965). The averaged velocities of the dilute and dense phases are defined by Uf¼(ρgUgf+ρsUsf)/ρb, Uc¼(ρgUgc+ρsUsc)/ρc, respectively.
K. Hong et al. / Chemical Engineering Science 99 (2013) 191–202 201
k ratio m number density Nst mass-specific energy consumption for suspending and
transporting particles, W/kg NT total mass-specific energy consumption, W/kg p pressure, Pa Re Reynolds number u real velocity, m/s ui interface velocity between the dilute and dense
phases, m/s U superficial velocity, m/s Umf superficial gas velocity at minimum fluidization, m/s ut terminal velocity of a single particle, m/s
Greek letters
β drag coefficient, kg/(m3s) βe effective drag coefficient, kg/(m3s) βWY Wen and Yu drag coefficient, kg/(m3s) Γ mass exchange, kg/(m3s) Δt time step, s Δx grid size, m εg void fraction εgc voidage of dense phase εgf voidage of dilute phase εmf incipient voidage εsc solids concentration in the dense phase εsf solids concentration in the dilute phase εmax maximum voidage for particle aggregation εs,max close packing density μ viscosity, (Pa s) ρ density, kg/m3
τ stress tensor, (Pa) φ specularity coefficient
Subscripts
b bubble (or void) c dense phase cl particle cluster f dilute phase g gas phase gc dense-phase gas gf dilute-phase gas i meso-scale interphase
mb minimum bubbling mf minimum fluidization p particles pt value for choking point s solid phase sc dense-phase solid sf dilute-phase solid slip slip velocity
(Bold characters are for vectors or tensors.)
Acknowledgment
The first author gratefully thanks Dr. Bona Lu, Junwu Wang and Mr. Atta Ullah for valuable discussions and suggestions. This work is financially supported by the National Basic Research Program of China under Grant no. 2012CB215003, the National Natural Science Foundation of China under Grant nos. 21176240, 21106157 and the “Strategic Priority Research Program” of the Chinese Academy of Sciences, Grant no. XDA07080000.
Appendix
In this appendix, the relevant definitions and parameters in the bubble-based EMMS model are summarized in Table A.1.
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Introduction
Cluster-based Fdi
Bubble-based Fdi
Restoration to the cluster-based EMMS model
Restoration to the bubble-based EMMS model
Closure of EMMS: Stability condition
Reduction to the TFM
Validation of the bubble-based EMMS model
Simplified solution of SFM: Coupling of TFM and bubble-based EMMS drag
Model results
Conclusions
Nomenclature
Acknowledgment
Appendix
References