Hydrodynamic simulation of gas–solids downflow reactors

Chemical Engineering Science 63 (2008) 5107 -- 5119

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Hydrodynamic simulation of gas–solids downflow reactors S. Vaishali a , Shantanu Roy a,∗ , Patrick L. Mills b a b

Department of Chemical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India Department of Chemical and Natural Gas Engineering, Texas A&M University—Kingsville, 700 University Boulevard, MSC 188, Kingsville, TX 78363-8202, USA

A R T I C L E

I N F O

Article history: Received 6 June 2007 Received in revised form 5 June 2008 Accepted 23 June 2008 Available online 25 June 2008 Keywords: CFB downer CFD Fluidization Granular temperature Hydrodynamics Mathematical modeling Momentum transfer

A B S T R A C T

In this work, we investigate the radial flow structure in gas–solids downer using Euler–Euler computational fluid dynamics (CFD) models. Solids are modeled as pseudo-fluid using kinetic theory of granular flow. In addition to the mass and momentum conservation equations, transport equation for fluctuating kinetic energy of the solids (modeled as granular temperature) is solved. The main focus of this work is the systematic investigation of the most suitable closures for the various force interactions in the system of interest. Results are presented for mean solids velocity, volume fraction, granular temperature and slip velocities for various closure forms. Sensitivity of the predicted results to the choice of closure forms is presented. Finally we emphasize the idea of matching slip velocities and the trends thereof with solids fraction as the key to developing a robust CFD model which has predictive capability over a wide variety of flow conditions. © 2008 Published by Elsevier Ltd.

1. Introduction One of the major goals of chemical reaction engineering in recent times has been to make the manufacturing process of chemicals more efficient and environmentally friendly. In addition to the conventional practice of designing reactors for various catalysts and chemistries (Krishna and Sie, 1995; Dudukovic et al., 1999), chemical reaction engineering is also now aiming to improve the `atom economy' and `atom efficiencies' (Trost, 1991) of chemical processes in much more tailored and directed fashion. Indeed this is being brought about by the parallel innovations in catalyst science, leading to the development of rapid acting and hyper-selective catalysts (e.g. Abul-Hamayel et al., 2005; Contractor et al., 1987). Thus, the goal of the reaction engineer is turning towards innovating flow patterns and contacting mechanisms that would be able to `exploit' the `better' catalysts in an `optimal' manner. One such innovation which has been developed and successfully commercialized since the 1970s is the circulating fluidized bed (CFB) reactor. These involve fluidization at velocities high enough (3–15 m/s) for the catalyst particles to be elutriated away with the gas, and thus, it has to be recycled back in order to ensure steady flow. The original thinking that brought about the use of CFB (see for example, Grace, 1990) were to have higher throughputs, and arguably less backmixing of solids as compared to conventional batch/bubbling fluidized beds. The use of fine

∗

Corresponding author. Tel.: +91 11 26596021; fax: +91 11 26581120. E-mail address: [email protected] (S. Roy).

0009-2509/$ - see front matter © 2008 Published by Elsevier Ltd. doi:10.1016/j.ces.2008.06.014

particles which could overcome internal and external mass and heat transfer resistances were an added bonus. Not withstanding their relatively common use in the process industry at present, their flow behavior is far from what may be considered optimal from a reactor engineering perspective. It is found that as one moves to risers of large diameter or those that operate at high solids fluxes, these contactors exhibit core-annular flow regime (Sun and Koves, 1998), wherein the solids tend to segregate in an annulus at the wall while the gas bypasses the center (core) of the column. This causes poor gas–solids contact and solids backmixing resulting in lower yields. Additionally make-up catalyst needs to replenish the catalyst lost due to solids attrition, and this gets even more critical because to begin with the net solids inventory in these systems is quite low. The `CFB downer' configuration (Gartside, 1989) has been proposed as an alternative to the riser for more efficient gas–solids contacting. This involves cocurrent downflow of gas and solids, which in turn reverses the role of `drag' and hence high solids concentration can be associated with regions of high solids velocity (something that is impossible to achieve in a riser). This is thought to result in more uniform extent of reaction across the vessel cross section. The `downer' is claimed to offer several other advantages over riser reactor, such as more uniform gas–solids contact, narrow residence time distribution (Zhu et al., 1995) and forward mixing (Bolkan et al., 1994). Therefore, downer could be potentially used in fast and highly selective reactions. Examples of such reaction systems include coal liquefaction by flash hydropyrolysis (Oberg and Falk, 1980), deep catalytic cracking (Deng et al., 2004), fluid catalytic cracking

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Table 1 Key experimental studies in `circulating fluidized bed downer' (CFBD) system Reference

Shimizu et al. (1978) Kim and Seader (1983) Wang et al. (1992) Wei et al. (1994) Wei and Zhu (1996) Herbert et al. (1998) Tuzla et al. (1998) and Schiewe et al. (1999) Johnston et al. (1999) Lehner and Wirth (1999) Zhang et al. (1999) and Zhang and Zhu (2000) Cao et al. (2000) Krol et al. (2000) Zhang et al. (2003) Nova et al. (2004)

Downer dimensions

Particle properties

ID (mm)

L/D ratio

dp (m)

 (kg/m3 )

Ut (m/s)

Gas superficial velocity (m/s)

Solid circulating rate (kg/m2 s)

28 13 140 140 50,140 50 150 100 150 100 127 25.4 418 26.3

136 516 41 54 51 92 57.3 93 57 93 46 120 16 70

45–170 329 59 54 54,90 75 125 66 60,130 67 82 65 77 76

8978 2480 1545 1710 1710 1400 2480 1500 2500 1500 1480 1722 1398 1722

3.2 2.6 0.34 0.33 0.226,0.42 0.4 0.99 0.37 0.47,1.03 0.26 0.46 0.40 0.34 0.47

5–24.3 11.5–35 2–8 2.3–9.0 2.6–8 0.4–10 0–6.6 5.1–9.6 0–7 3.7–10.2 2.9–5.1 0.4–3 1.8–10 1–3

50,000–5 (Solid loading ratio) 385 30–180 5–60 8–80 49–120 30–100 44–185 30–100 49–205 50–236 3–7 19–180 2–12

(Bolkan et al., 1994; Talman et al., 1999), residual oil catalytic cracking (Murphy, 1992), ultra pyrolysis of biomass (Freel et al., 1987) and partial oxidation of n-butane (Emig et al., 2002). Pilot plant studies have already shown encouraging results for the use of downer for FCC (Abul-Hamayel, 2004). Hence, clearly there is a need for more rigorous analysis of hydrodynamics of downer in order to realize its full potential in various applications. The research in downers was mainly started in late 1970s (Shimizu et al., 1978; Kim and Seader, 1983), and thus far significant amount of experimental studies have been carried out. Table 1 enlists some of the key experimental studies with the system configuration and operating parameters. Most of the studies thus far have concluded that the downer is characterized with more uniform gas and solids flow as compared to the riser. However, there was no universal agreement on the nature of radial profile of solids phase in the so-called `fully developed' zone of downer. The FLOTU group (Bai et al., 1991; Wang et al., 1992) has reported that solids concentration exhibits a peak near the wall region whereas Zhang et al. (1999) and Zhang and Zhu (2000) suggest that this peak vanishes for higher L/D ratio. In contrast, Cao and Weinstein (2000) claimed that downer exhibits maximum solids concentration at wall. In spite of experimental studies, there have been very few attempts to model gas–solids dispersed flows in a downer configuration. Table 2 reports the previous modeling attempts in gas–solids downer system along with some remarks. As can be seen, modeling the onedimensional axial development of downer flow has been more substantial (Bolkan et al., 1994, 2003; Deng et al., 2004) as compared to that of radial flow structure. However, radial flow structure is equally crucial as it plays a role in dispersion and mixing behavior of the reactor. In recent decades, `computational fluid dynamics (CFD)' has emerged as a promising option to solve multiphase flows using advanced numerical techniques (Ranade, 2000). However, much of the work is as yet at a point that one cannot use those tools for effective design and scale-up, and therefore they have limited applicability. Two-phase CFD models are often subject to tuning of several parameters in the `closure' equations which account for various interphase interactions in the system. Hence, even if one were to obtain good agreement with experimental data under certain limited set of conditions, strong scientific basis is not always available for the choice of tuning parameters and it becomes quite impossible to extend the simulations to other conditions or configurations of interest. In one of the significant efforts at using CFD to model downer flow, Cheng et al. (1999) used kinetic theory of granular flow (KTGF) to model solids as a continuum phase in addition to gas phase turbulence.

Operating conditions

This CFD model predicted excessive densification of solids at wall and also showed significant parametric sensitivity to the choice of the restitution coefficient. Cheng et al. (2001) subsequently extended their earlier work to find the effect of inlet geometry of downer on the flow development. The downer inlet region was found to be characterized with very high (around 0.3 m2 /s2 ) granular temperature, signifying the dominant role of particle–particle collisions in the flow dynamics. Jian and Ocone (2003) introduced a counter-diffusive solids concentration term in the solids stress tensor in order to account the inter-particle cohesive forces for Geldart group A particles. This model predicted excessive segregation (about five times the experimental data) of solids near wall. Ropelato et al. (2005) modeled solid phase as hypothetical ideal fluid (fluid with no viscosity) and studied hydrodynamic features for various inlet geometries. In the past there have been few attempts (Du et al., 2006; Gomez and Milioli, 2003; Johansson et al., 2006) in gas–solid flows to assess role of various interactions on the behavior of the systems such as spouted bed, riser and fluidized bed. However, as listed in Table 2, none of the work published so far makes a systematic analysis of the various interactions in the downer, viz, gas phase interactions (turbulence), particle phase interactions (particle–particle collisions) and interphase interactions via drag (Fig. 1). Further, a quantitative analysis of the various closures and their relative impact on the global hydrodynamics of gas–solid flows has not been reported. In this work, we have attempted to address these issues.

2. Modeling Downer represents a special category of dispersed flows in which solids volume fraction is relatively low (1–5%), and the solids are moving with very high velocities (∼ 3–10 m/s), and it is `aided' by gravity (which distinguishes it clearly from a riser, in which the solids flow against gravity). In the present work, a pilot-plant scale downer system has been considered for the study, which involves a large number of particles in the solids inventory. Hence, a `Euler–Euler' approach (Anderson and Jackson, 1968; Harlow and Amsden, 1975) has been used for CFD modeling, treating the solids phase as a pseudocontinuum. Both phases, namely, the gas phase and the pseudocontinuous solids phase, are allowed to exist at the same point and at the same time forming an interpenetrating continuum. More details of the philosophical and practical basis of this approach can be found elsewhere (Drew, 1983; Drew et al., 1998).

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Table 2 Key modeling efforts in `circulating fluidized bed downer' (CFBD) system Sr. no.

Reference

Model features

Result/conclusions

Remark

1

Bolkan et al. (1994)

1. 1D, unsteady, pseudo homogeneous approach

Model slightly over predicted solids fraction in the inlet acceleration zone, whereas slightly under predicted in the constant velocity section

The only empirical parameter used was gas–solid drag coefficient

Wall effect plays an important role in the flow dynamics of downer

1. One of the initial models to describe downer dynamics with one adjustable parameter 2. Slip velocity was found to be sensitive to this empirical parameter

1. Radial profile of solids fraction is a strong function of downer diameter 2. Inlet and outlet region is of critical importance

1. Model is very sensitive to the restitution coefficient Peak in the solids fraction near the wall region might be due to the assumption of parabolic profile of gas velocity at wall

Collisions between particles and solid phase turbulence play an important role in the flow dynamics of downer

Modeling of inlet geometry makes possible to follow, not only the radial, but also the axial development of the flow

The modified drag coefficient yields in an improved prediction of velocity profiles as compared to that of Bolkan et al. (1994)

Two adjustable parameters  and  were used in drag coefficient. The best fitted values are  = 1 and  = 1

1. Cluster diameter was found to be 8 to 12 times mean particle size 2. Unlike earlier model, solids fraction was slightly under predicted in the inlet zone, whereas good fitting was obtained in the constant velocity zone 1. Gas phase turbulence does not contribute significantly in flow dynamics of downer

Cluster formation was inherently assumed in the model

2

Kimm et al. (1996)

2. Model is based on the experimental observation that downer consists of three axial zones, first acceleration section, second acceleration section, and constant velocity section 1. 2D steady state, pseudohomogeneous approach 2. No slip condition for solids at wall

3

Cheng et al. (1999)

1. Two fluid approach with KTGF 2. 2D, axisymmetric, steady state

4

Cheng et al. (2001)

3. Turbulent energy equation of particle phase has also been incorporated 4. Restitution coefficient is assumed to be function of radial position 5. Gidaspow's correlation was used for interphase momentum transfer via drag Similar to Cheng et al. (1999)

5

Emig et al. (2002)

1. 1D, steady

Bolkan et al. (2003)

2. It was assumed that increase in slip velocity increases the rate of breakage of clusters and increase in solids concentration increases formation of clusters 1. 1D, unsteady

6

2. In order to account clustering effect, `equivalent agglomerate diameter' was incorporated in the downer model 7

Jian and Ocone (2003)

1. 2D, axisymmetric, steady state

Radial profile of granular temperature showed constant value in the central region and decreasing trend with minima at the wall. Thus the region of maximum solid velocity and solids fraction showed minimum granular temperature

2. Non dimensional form of two fluid equations was used 3. In order to account particle nature inter-particle cohesive force was introduced 4. Eight empirical parameters 5. Prandtl's mixing length turbulence model was used 6. Wen and Yu's drag closure

2. Solids volume fraction was over predicted 3. Model predicts five times high lateral segregation of solids at the wall as compared to that of experiment

EMMS model with one adjustable parameter showed good agreement with experimental profiles of solids fraction and velocity in the fully developed region

In the core region slip velocity is of the order of terminal velocity of particle, whereas it increases sharply in the near wall region (r/R = 0.8–0.9) and again drops at the wall to the same value as that of core region

1. For constant volume case, voidage was slightly over predicted in the entrance region

1. This hydrodynamic model is useful for the reaction in which change in volume is taking place

8

Li et al. (2004)

1. 2D steady state

9

Deng et al. (2004)

1. 1D steady state

2. EMMS approach

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Tabele 2 Continued Sr. no.

10

Reference

Ropelato et al. (2005)

Model features

Result/conclusions

Remark

2. Model was applied at the varying superficial gas velocity

2. In case of changing superficial gas velocity, there is no constant velocity section but the solids are accelerating throughout the length of the reactor depending on the amount of change in volume and reactor length 3. Drag force is highest in the entrance region and then suddenly decreases and remains constant along the length of the downer 1. Downer with transversal inlet gave the best match with experimental data 2. Upwind scheme and higher upwind scheme showed the similar results

2. No experimental validation

1. 3D unsteady

2. Two fluid model

1. Only solids fraction profile was considered for validation of the model 2. Higher L/D ratio was recommended for good solid distribution

3. Solid phase is assumed to behave like a hypothetical fluid without shear stress 4. Three different inlet designs were tried to find the best suited design

Total volume conservation:

s + f = 1

(4)

2.2. Closures for solids fluctuations

Fig. 1. Types of interactions in gas–solid dispersed flow.

In this work, the ensemble averaged formulation of the twofluid equations has been adopted and the simulations have been performed using the FLUENT library of codes, specifically in Fluent v.6.2.16 (Fluent. Inc., New Hampshire, USA). 2.1. Conservation equations The conservation equations that are solved are as follows: Continuity (kth phase): n  j k) = ˙ pk (k k ) +  · (k k u m jt p=1

k=f k=s

for fluid for solids

(1)

(Right-hand side is zero in the present case since there is no mass transfer between the two phases.) Momentum (fluid phase):

j f )  ) +  · ( f  f u f ⊗ u (  u jt f f f s − u  f ) + Ff = −f p +  · ¯¯ f + f f g + Ksf (u

(2)

Momentum (solids phase):

j s)  ) +  · (s s u s ⊗ u (  u jt s s s f − u  s ) + Fs = −s p − ps +  · ¯¯ s + s s g + Ksf (u

(3)

As shown schematically in Fig. 1, modeling of dilute gas–solid suspension needs to take into account interactions within and between mean flow field as well as the fluctuating flow fields of each phase. As such, the fluctuations are at multiple scales and hence there could be, in principle, correlations between the components of velocities of either of these phases at the spectrum of scales. In the approach adopted, all the scales of fluctuations are `lumped' into a single fluctuating velocity field (or kinetic energy field), and one hopes to resolve the correlations between fluctuating fields of each phase, as within the phase, through suitable `turbulence' modeling. In this approach, in spite of the sophistication in the modeling approach and the numerical schemes involved, at the end one has to pick out the `best suited' closures for a given flow system. These closures are derived either from experiments or from fine scale simulations, and very often there is no consensus in their choice. We feel that there is a need to systematically benchmark them (through the full CFD calculations), against the same set of reliable experimental data before we can recommend the use of any particular closure or sets of closures for a certain application. In this work, we have tried to find the `best suited' set of closures for gas–solids dispersed flow in downer from the closures available. For the present work, interaction between fluctuating fields of gas phase and solids phase has not been taken into account as it is expected be a correlation of lower order as compared to other three interactions (Fig. 1). Interaction between random flow field and mean flow field of particulate phase generates stresses in the particle assembly and gives rise to solids phase pressure and viscosity. KTGF, which is a theory inspired by the kinetic theory of dense gases (Chapman and Cowling, 1961), is used to model the solids phase interactions (Jenkins and Savage, 1983; Lun et al., 1984). In this approach, the fluctuating kinetic energy of solids is represented by the `granular temperature' (s ), by analogy with the description of translational kinetic energy of gas molecules by the thermodynamic temperature in the kinetic theory of gases. Other solid phase transport properties such as solid phase pressure, solids stresses, etc., are expressed in terms of granular temperature. The quantity granular temperature s , (representing the kinetic energy of fluctuations or random motions in the solids phase) is transported as per the

S. Vaishali et al. / Chemical Engineering Science 63 (2008) 5107 -- 5119

following equation (Chapman and Cowling, 1961):   3 j  s s ) (s s s ) +  · (s s u 2 jt   s +  · (k s ) −  + fs = (−psI + s ) : u s s

(5)

2.3. Closures for gas–solids drag Finally, the interaction of prime importance in downer flows is that the mean flow field of gas phase and mean flow field of solids phase is modeled here via the `drag' concept. Owing to the detailed sensitivity analysis with this closure force, we present here some discussion of the various drag closures and their evolution, and how that has influenced the systematic analysis of the closures presented in this work. Clearly the drag force depends on the slip velocity between the phases (absolute difference between the mean velocity of gas phase and solids phase), and for a multi-particle system such as a downer it also depends on the local volume fraction of the dispersed phase. In general, drag closures are determined from either bed expansion experiments (for example, Richardson and Zaki, 1954; Wen and Yu, 1966) or bed pressure drop experiments (for example, Ergun, 1952; Gibilaro et al., 1985). More recently, techniques like direct numerical simulation have been employed to obtain the drag closure from appropriate fine scale simulations and subsequently coarse-graining the results to yield ad hoc expressions (Li et al., 2001; McLaughlin, 1994). As per Richardson and Zaki (1954), the drag force in multi-particle system is related to the slip velocity to the single-particle terminal velocity through a voidage function: Ug = ng Ut

(6)

where n = 4.65 + 20  n = 4.4 + 18  n = 4.4 + 18

dp for Ret < 0.2 D  dp Re−0.03 for 0.2 < Ret < 1 t D  dp Re−0.01 for 1 < Ret < 200 t D

n = 4.4Re−0.01 t n = 2.4

for 200 < Ret < 500

for 200 < Ret < 500

This drag closure was derived from the sedimentation experiments and hence strictly applicable for homogeneous, quiescent systems. The more commonly used drag closure in fluidization literature is that of Wen and Yu (1966). It postulates that the ratio between the drag force Fdrag on a particle in a fluidized bed and the drag force Fdrag,s on a single sphere is affected by the bed voidage and is given as Fdrag Fdrag,s

= −n g

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In case of dilute gas–solids dispersed flow with Geldart group A particles having average volume fraction 0.1–5%, due to externally imposed hydrodynamic effects (Horio and Clift, 1992) particles show a tendency to move in a group. This motion is characterized by high degree of correlation of velocities of particles in such a group, and this entity is referred to as a `cluster' (Wilhelm and Kwauk, 1948; Horio and Kuroki, 1994). Thus, effectively bigger particles (Nieuwland et al., 1994) are moving with higher velocity (about 20–30 times, as per Yerushalmi et al., 1976), thereby causing higher slip velocity. This feature must be reflected by a drag closure to be used for dilute dispersed flows. In the past there have been numerous drag closures (Matsen, 1982; Nieuwland et al., 1994) for dilute dispersed flows. In case of gas–solids downer wherein the flow is along gravity, it is speculated that clustering will be highly unstable (Zhu et al., 1995; Krol et al., 2000) resulting in smaller clusters (cluster diameters in the range of 2–6 dp ), and with smaller lifetime (few milliseconds). Observed slip velocities in the case of downer (Roques et al., 1994; Cao et al., 1994; Herbert et al., 1998) were about 5–10 times singleparticle terminal velocity. Therefore, it becomes highly imperative to investigate how drag gets modified in gas–solid downer flow. In order to find the `best suited' drag closure from those available in open literature in addition to the commonly used Wen and Yu (1966) drag closure, four other drag closures as enlisted below were tried for prediction of the flow in the downer. • Matsen (1982) proposed the empirical expression for slip velocity of a particle in multi-particle system based on bed expansion experiments similar to Richardson and Zaki's (1954) drag closure. The distinct feature of this drag closure is that, it proposed minimum possible slip velocity of a particle in multi-particle system as a single-particle terminal velocity which increases as non-linear function of solid concentration. This drag closure is applicable to the system with uniform particle size and negligible wall and inertial effect. • Di Felice et al. (1994) proposed a general drag correlation for a wide variety of gas–solid systems ranging from fixed bed to dilute flow suspensions from bed pressure drop data. It is assumed that, the voidage function which accounts for the presence of other particles may be expressed as − , where  is a function of particle Reynolds number and is not dependent on any other system variables. • Nieuwland et al. (1994) attributed higher slip velocity to the `cluster formation' of particles. Therefore, a non-linear voidage function is expressed as a ratio of single-particle terminal velocity to the slip velocity which in turn is expressed as the empirical function of solids concentration. • Yang et al. (2004) proposed drag closure based on `energy minimization multiscale (EMMS)' approach. In this model solid phase is resolved at two scales, cluster scale and particle scale, and subsequently the interactions of cluster and particle with the gas phase are taken into account for the determination of momentum exchange via drag. In addition to the mass and momentum conservation, system is subjected to the minimization of energy consumption.

(7)

where n = 4.7. Wen and Yu (1966) carried out number of experiments in liquid–solid systems, and under a wide range of operating conditions. The force on single sphere (Fdrag,s ) was found out from Schiller and Naumann (1935) correlation; the force Fdrag was evaluated from the apparent weight of particle and then subsequently exponent n of void fraction was evaluated. Gidaspow (1994) proposed that Wen and Yu's drag correlation is applicable for the region having voidage greater than 0.8 and for region having voidage lesser than 0.8 (that is, solids fraction of 20% or more), the drag is governed by Ergun's equation.

Interphase momentum exchange factors appearing in momentum conservation equation for the corresponding closure are listed in Table 3. In summary, Wen and Yu's (1966) drag closure suggests a modification in the single-particle drag force by accounting for the presence of other particles. This is done by raising the fluid volume fraction to some constant power n. Other investigators offer minor improvements, by expressing the power of fluid volume fraction (n) as a function of other system variables. In clear contrast, Matsen's (1982) drag closure attempts to modify the multi-particle drag force for dilute systems as a function of solids phase volume fraction and

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Table 3 Interphase momentum exchange factors for various drag closures Reference

Momentum exchange coefficient

Wen and Yu (1966)

s g g |vp − vg | −2.65 3 CD g 4 g

Matsen (1982)

=

Di Felice (1994)

 = 0.006475CD

s g g |vp − vg | g −0.586  g mix s

(0.0003 < s < s,choking ) 3  = CD s 2g g |vp − vg |−g 4 where   2 1 1 CD = 0.63 + 4.8 g Rep

2 [1.5 − log(Rep ) ] 2 3 s g g |vp − vg |  = CD g() 4 g 1 ,  < 0.1276 g() = 0.997 + 442.35s − 1733.422s g() = 0.034,   0.1276 

Nieuwland et al. (1994)

Yang et al. (2003)

= 3.7 − 0.65 exp −

3 s g g |vp − vg | CD

(g ) (g > 0.74) 4 g (1 − g )g (1 − g )g |vp − vg |  = 150 + 1.75 (g > 0.74) g d2p dp where 0.0214

(g ) = −0.5760 + (0.74 < g  0.82) 2 4(g − 0.7463) + 0.0044 0.0038 (0.82 < g  0.97) = − 0.0101 + 2 4(g − 0.7789) + 0.0040 = − 31.8295 + 32.8295g (g > 0.97) 24 0.687 CD = [1 + 0.5(g Rep ) ] g Rep g dp |p − g | Rep =

=

g

it also premultiplies it by a constant. Thus, as compared to other drag closures Matsen's expression offers significantly lower interphase momentum exchange factor (Eqs. (2) and (3)). 3. Results and discussion As discussed earlier, in the current work, CFD has been used to study gas–solid interactions with Euler–Euler approach. k– model is used to take into account gas phase turbulence, whereas KTGF has been used to take into account solid phase interactions. Initially a `base case' CFD model is established with the parameters listed in Table 4. Validation of the model is done by the comparison of radial flow behavior of solids predicted by the model with that obtained from experiments in fully developed zone. Zhang et al. (1999) and Zhang and Zhu (2000) have published experimental data across a wide range of operating conditions, including solids velocity and solids concentration in pilot plant scale gas–solids downer system at the same location for various operating conditions. Therefore, our system specifications were chosen to be same as that of Zhang et al. (1999) and Zhang and Zhu (2000), and the developed CFD model is evaluated against experimental data. As appropriate, the most suitable amongst the available closures have been modified to improve the predictability of the experimental data. 3.1. Base case simulation For base case simulation, low mass flux operating condition Gs = 49 kg/m2 s with Ug = 3.7 m/s is chosen. Figs. 2a and b show the

comparison of radial profile of time averaged solids fraction and solids velocity for various grid sizes, viz, 15 × 1860, 21 × 2604 and 30 × 3720 at axial location Z = 4.398 m. As can be seen, the numerical predictions for various grid sizes are within acceptable range. Therefore, for further simulations grid size of 15 × 1860 is chosen. As far as time step is concerned, the time step was varied from 10−4 to 10−5 s, so that the numerical simulations satisfy the convergence criterion which is set at 10−3 (dimensionless, deviation normalized by the converged value of variables like velocity and pressure) for both continuity and momentum equations. Figs. 3a and b show the comparison of radial profile of mean solids fraction and mean solids velocity with the experimental data at axial locations Z = 1.198, 2.112, 4.398, 6.227 and 8.056 m (distance from inlet). Since the current simulations are 2D axisymmetric, they can predict the flow behavior in fully developed region and cannot account for the entry and exit effects with fidelity. For further analysis, two axial locations (Z = 4.398 and 6.227 m), which represent fully developed region in the current downer reactor, are considered. Though the CFD simulation predicts similar trend of solids velocity as that of experimental measurements qualitatively, the values are under-predicted by the simulations and correspondingly, solids volume fraction are over-predicted (in order to satisfy the overall solids mass balance or continuity). For the particles used in this case (67 m diameter with particle density 1500 kg/m3 ), single-particle terminal velocity is 40 cm/s, where as the slip velocity predicted by Wen and Yu's (1966) drag closure is 18 cm/s. The significant underprediction of slip velocity demands modification in the drag closure employed. Therefore, CFD simulations are attempted with four different drag closures, as discussed earlier, and then their performance is evaluated against experimental data.

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Table 4 Numerical parameters used for `base case CFD simulation' of downer Downer specification (Zhang et al., 1999, 2000)

Length Diameter

9.1 m 0.1 m

Solid properties

Particle density

1500 kg/m3

Sauter mean diameter Solver Multiphase Turbulence

67 m 2D, axi-symmetric, unsteady Eulerian–Eulerian Standard K- model

Radial (uniform) Axial (uniform) Granular temperature Restitution coefficient (particle) Restitution coefficient (wall) Specularity coefficient

15 1860 1e–5 m2 /s2 0.95 0.90 0.5

Wall boundary condition

Gas phase Solid phase

No-slip condition Johnson Jackson

Inlet boundary condition

Gas and solid phase

Uniform velocity profile

Model

Grid (base case) KTGF parameters

Outlet boundary condition

Gas and solid phase

Outflow (satisfying overall mass balance)

Numerical method

Pressure–velocity coupling Discretization

Phase coupled SIMPLE Second order UPWIND

Under-relaxation parameters

Pressure Density Body forces Momentum Volume fraction Granular temperature Turbulent kinetic energy Turbulent dissipation rate Turbulent viscosity

0.3 1.0 1.0 0.7 0.2 0.2 0.8 0.8 1.0

Unsteady iterations

Time step

0.0001 s

Initialization

Gas phase axial velocity Solids fraction

1 m/s 0.0

3.2. Selection of `best suited' drag closure Figs. 4a and b show the comparison of radial profile of mean solids fraction and solids velocity at Z = 4.398 m. Most of the drag closures predict mean solid velocity similar to that of Wen and Yu's (1966) drag closure, whereas significant improvement is offered by Matsen's (1982) drag closure. When the exponent of solids volume fraction in the Matsen's (1966) interphase momentum exchange expression is changed from 0.586 to 0.01, significantly improved agreement is obtained with experimental data. Fig. 5 shows the radial profile of slip velocity in fully developed region predicted by the different drag closures. Di Felice's (1994) drag closure, Nieuwland et al.'s (1994) drag closure, as well as Yang et al.'s (2004) drag closure predicted the slip velocity practically similar to single-particle terminal velocity. Matsen's (1982) drag closure predicts slip velocity around five times that of the single-particle terminal velocity. Cao et al. (1994) have reported slip velocity two to six times that of the single-particle terminal velocity for FCC particles, for similar operating conditions in downers. Thus, Matsen's (1982) drag closure not only predicts the average solids velocity and solids volume fraction reasonably well, but also predicts the `highly probable' slip velocity in downers. In order to understand the effect of various drag closures on interphase momentum exchange, interphase momentum exchange factor (Eqs. (2) and (3)) predicted by these drag closures are compared as a function of slip velocity with the assumption of constant solid volume fraction of 1% (typical value in the downer). Fig. 6 shows that, interphase momentum exchange coefficient predicted by Yang et al.'s (2004) drag closure as well as that by Di Felice's (1994) drag closure is of similar magnitude as that of Wen and Yu's (1966) drag closure. Nieuwland et al.'s (1994) drag closure predicts interphase momentum exchange factor one magnitude lower than that of Wen and Yu's (1966) drag closure whereas Matsen's (1982) drag closure

predicts three magnitudes lower interphase momentum exchange coefficient. Modified Matsen's drag closure (this work) predicts interphase momentum exchange factor one magnitude lower than that of the original Matsen's drag closure. Fig. 7 shows the comparison of interphase momentum exchange factor predicted by `single-particle drag' and other multi-particle drag closures. For very dilute suspensions Wen and Yu's (1966) drag closure as well as Di Felice's (1994) drag closure predicts the system to be `free settling', which is distinctly different from the experimental observation in downers, and therefore, it is not possible to explain the high slip velocity in the system using these drag closures. In contrast to all the other drag closures, Matsen's (1982) drag closure (and modified Matsen's drag closure) predicts a reduction in the interphase momentum with an increase in the solids concentration. For gas–solids suspension in downers for the investigated operating conditions, considerable amount of `drag reduction' takes place within the cluster place resulting in higher slip velocities. Helland et al. (2007) have shown that for very dilute suspension (having solid volume fraction less than 3–5%), `diluted cluster' formation takes place causing considerable `drag reduction' in the system. At this point, more experimental investigations are required on the origin and nature of the clusters in the downer.

3.3. Solids phase kinetic energy Gas–solid dispersed flow is a multiscale phenomenon involving interactions at various scales, and the energy of the incoming flow (gas in this case) is cascaded through various scales from the mean flow of both phases and eventually dissipated through gas phase stresses or solids phase fluctuations. Therefore, it is important to consider second moment quantities which quantifies fluctuating com-

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Fig. 2. Radial profile of (a) mean solid velocity and (b) mean solids fraction at H = 4.398 m, for various grid sizes (Ug = 3.7 m/s and Gs = 49 kg/m2 s). Fig. 3. Radial profile of (a) mean solids fraction and (b) mean solids velocity, `base case simulation' at various axial positions (Ug = 3.7 m/s and Gs = 49 kg/m2 s).

ponent of momentum in the system. When `KTGF' is used to model solid phase interactions, the concept of granular temperature is introduced to signify the fluctuation kinetic energy of solids in the system. Typically for dispersed flows (Gidaspow, 1994), the granular temperature is assumed to be of the order of 10−5 m2 /s2 , and therefore, this value is specified at the inlet boundary for our CFD simulations. Fig. 8 shows the comparison of radial profile of granular temperature in fully developed region for various drag closures. Matsen's (1982) drag closure predicts granular temperature of the order 0.01 m2 /s2 , whereas other drag closures predict it in the range of 10−5 m2 /s2 (essentially they advect the value specified at the inlet boundary). Cheng et al. (2001) have reported that downer inlet region having FCC particles of mean diameter 54 m operated with Ug = 2.8 m/s and Gs = 47 kg/m2 s is characterized with higher granular temperature 0.3 m2 /s2 . In our simulations, we found that even when the inlet value was specified as 0.03 m2 /s2 , simulations with Wen and Yu's (1966) drag closure converged to a granular temperature in the range of 1e–5 m2 /s2 , whereas Matsen's (1982) drag closure converged to granular temperature in the range of 0.03 m2 /s2 in the developed zone. In the past there have been number of studies

(Cody et al., 1996; Gidaspow and Huilin, 1996; Jiradilok et al., 2006; Tartan and Gidaspow, 2004), which measured granular temperature for gas–solid dispersed flows (in riser) at superficial gas velocity in the range of 1–10 m. All these studies predicted granular temperature in the range of 0.01–10 m2 /s2 . Therefore, it may be reasonable to expect that downer would also be characterized with the granular temperature of the similar magnitude. As compared to risers, downer is a more dilute system with higher velocities (and correspondingly, fluctuations), which is perhaps the reason for the higher value of granular temperature exhibited by them. Also, one concludes from the above discussion that, in addition to the mean flow quantities, Matsen's (1982) drag closure seems to predict `highly probable' value of granular temperature, at least going by the limited experimental information in this regard that is available in the open literature (Cheng et al., 2001). Experimental work using radioactive particle tracking (RPT) is being planned in downer systems in our group for establishing this claim beyond doubt, something which we will be reporting in a future publication.

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Fig. 6. Interphase momentum exchange factor (Ksf in Eqs. (2) and (3)) vs. slip velocity at constant solids fraction of 1%.

Fig. 4. Radial profile of (a) mean solids fraction and (b) mean solids velocity at H = 4.398 m for various drag closures (Ug = 3.7 m/s and Gs = 49 kg/m2 s).

Fig. 7. Interphase momentum exchange factor (Ksf in Eqs. (2) and (3)) vs. mean solids fraction at slip velocity equal to single-particle terminal velocity (40 cm/s).

Fig. 5. Radial profile of slip velocity at H = 4.398 m for various drag closures (Ug = 3.7 m/s and Gs = 49 kg/m2 s).

Table 5 lists the KTGF closures used in the base case simulations which are decided from the previous CFD simulations reported for riser in the literature (Gidaspow, 1994). Since the average volume fraction of solids involved is too low (1–2%), CFD simulations are

Fig. 8. Radial profile of granular temperature at H =4.398 m for various drag closures (Ug = 3.7 m/s and Gs = 49 kg/m2 s).

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found not to be very sensitive to our choice of KTGF closures and the parameters contained therein. It is expected that in high density downers (downers whose solids distributor has been suitably modified to ensure high volume fraction of solids in the vessel) KTGF has a much more serious role to play.

regressed values of parameters, functional forms of the kind of Matsen's drag closure are the only ones that even stand a chance for predicting the solids fraction and velocity trends correctly. Clearly, the others are not even potential candidates for gas–solids downer flow.

3.4. Effect of gas phase turbulence In order to see the effect of gas phase turbulence on the overall hydrodynamics, simulations were done with laminar flow assumption and then subsequently compared with turbulent flow simulation (i.e., base case simulation). In case of laminar flow simulation, the average velocity of solids was found to be little smaller than that of base case simulation. Similar kind of result has earlier been reported by Ranade (1999) for the case of gas–solids risers. Since our findings in this are not very unique and the results were somewhat expected, in the interest of brevity, detailed discussion is not presented here. 3.5. Effect of operating conditions Downers are potential candidates for a wide variety of applications such as FCC, combustion, pyrolysis and synthesis reactions such as partial oxidation of n-butane. Operating conditions of downer reactor need to be chosen according to the application of interest and hence it is necessary to study the behavior of a downer under different operating conditions. Base case simulations with Matsen's (1982) drag closure are extended to higher superficial gas velocity and high solids mass flux conditions. Three sets of conditions are chosen, viz, Set I: Ug =3.7 m/s and Gs =101 kg/m2 s, Set II: Ug =7.2 m/s and Gs = 101 kg/m2 s, and Set III: Ug = 7.2 m/s and Gs = 208 kg/m2 s. Modified Matsen's drag closure was used for the Set I operating conditions, whereas Matsen's drag closure was used for Sets II and III. Figs. 9a and b shows the validation of the simulation results with a distinct set of experimental data Zhang et al. (1999) and Zhang and Zhu (2000). CFD simulations predictions of solids fraction and solids velocity are in satisfactory agreement with the experimental data. The point to be noted here is that the exponent of solids fraction in the Matsen's drag closure needs the modification depending on the operating conditions. There are two points with regards to Matsen's drag that we wished to highlight in this paper. The first is that it is the only functional form that seems to predict both velocity and holdup profiles together with any degree of accuracy. All the other's drag forms are successful with only one of the two. This has to do with the relationship between apparent slip velocity and solids fraction (voidage), which has been discussed above and in the earlier sections. In other words, what we are indicating is that, no matter what the chosen or

Fig. 9. Radial profile of (a) mean solids fraction and (b) mean solids velocity at H = 4.398 m for various operating conditions.

Table 5 KTGF closures used in CFD simulation KTGF parameter Granular conductivity

Granular viscosity Radial distribution

KTGF closure used in the CFD model

  12 2 15ds s s s  1+ ks = (4 − 3)s go,ss 4(41 − 33 ) 5 where 1 = (1 + ess ) 2

   2 4 s 1/2 4 10s ds s  1 + go,ss s (1 + ess ) s = s s ds go,ss (1 + ess ) + 5  96s (1 + ess )go,ss 5 go,ss =

 1−

where s max = 0.6 Solids pressure

1

s s max

1/3

Ps = s s s + 2s (1 + ess )2s go,ss s

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Fig. 10. Radial profile of slip velocity at H = 4.398 m for various operating conditions.

The second point is about the exponent of (1 − g ) (Table 3). Indeed, in order to `fit' the results quantitatively, one can `play' with this exponent, and a value of 0.01 seems to fit the reported experimental data under some of the conditions. However, even if the traditional value reported by Matsen (1982), of 0.586, is taken, the trends are still acceptable (Figs. 4a and b). They can only be further improved by regressing the exponent. One may choose not to do so and restrict the simulations to Matsen's (1982) suggested exponent of 0.586, and still one will get close to the experimental results. It may also be noted that in Matsen's original paper (1982), the value of 0.586 is not cited as a `universal' constant, but merely as a `fit' to the experimental data reported therein. Therefore, it is not surprising that when applied in a different system (downer), different geometry, different particle properties, and different flow conditions (CFD vs. correlation fitting), the new fit will be needed. In addition to the radial mean flow profiles, slip velocity, and hence, particle agglomeration (Bolkan et al., 2003) may get affected by operating conditions. Fig. 10 shows that at constant solid mass flux of 101 kg/m2 s, increase in superficial gas velocity from 3.7 to 7.2 m/s (Sets I and II) results in a decrease in slip velocity in the system, which is a typical characteristic of downer system (Roques et al., 1994). As stated earlier, in dilute gas–solid systems, higher slip velocity is observed due to cluster formation. As superficial gas velocity increases, suspension density decreases i.e., probability of direct interaction between the individual particles reduces, hence the extent of clustering reduces resulting into lower slip velocity. Also it can be seen that at constant superficial gas velocity of 7.2 m/s, doubling of the solids mass flux from 101 to 208 kg/m2 s showed negligible effect on the slip velocity. Similarly, there is very small effect of operating conditions on granular temperature (Fig. 11) in the range of investigated operating conditions, which we believe is due to low solid volume fraction. Since gas phase is the source of momentum in the system, it is important to investigate the effect of drag closures on gas velocity profile. Since there is no experimental data available for gas velocity in downer in the open literature for the investigated operating conditions, a correlation proposed by Kimm et al. (1996) is used for comparison with the gas velocity profile predicted by our model. Fig. 12 compares the gas velocity profile predicted by CFD simulation with Kimm et al.'s (1996) correlation for gas velocity profile. Clearly, it seems that our model works satisfactorily even in predicting gas

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Fig. 11. Radial profile of mean granular temperature at H = 4.398 m for various operating conditions.

Fig. 12. Comparison of radial profile of mean gas velocity predicted by CFD simulation with Kimm et al.'s (1996) correlation for various operating conditions.

phase profiles as well (note that Kimm et al., 1996 correlation was obtained through a wide range of experimental data). 4. Summary and conclusions Gas–solid dispersed flow in circulating fluidized bed downer is complex involving multiple modes of momentum transfer and should be modeled rigorously. In the present work, we have presented a CFD model of gas–solids flow in the fully developed region of downer. Effects of various interactions on the overall hydrodynamics of downer have been studied in a systematic manner. Among the various interactions involved, `gas–solids drag' is found to be the most dominating one, owing to the high slip velocities involved and the relatively less importance of direct solid–solid collisional interactions. The conventionally used Wen and Yu's (1966) drag closure, very popular in other gas–solids flow literature, is not found very suitable for predicting the gas–solids flow in the case of downers. For the investigated operating conditions in downers, Matsen's (1982) drag closure (with a suitably modi-

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fied exponent depending upon the operating conditions) is able to predict the average flow well and also predicts higher slip velocity as compared to other investigated drag closures (which is in line with the meager experimental evidence reported in the literature). This is because this drag formulation inherently incorporates the physical observation that the slip velocity increases with increase in solid concentration. CFD simulations also indicate that, downer is characterized by higher granular temperature (∼ 0.02 to 0.06 m2 /s2 ) signifying highly fluctuating nature of the flow. Using the modified Matsen's closure, our predictions are close to the granular temperature values predicted by CFD simulation, by Cheng et al. (2001). While this work indicates promising results for the use of the developed CFD model for design and scale-up of downers, clearly more experimental validation is necessary. This is particularly with reference the higher moments of the velocity fields, namely the solids phase fluctuations typified by the granular temperature. Only higher order validations, in addition to the `matching' of radial profiles of velocity and holdup, would be acceptable for rigorous use of such a model. In addition, one needs detailed simulations for the solids motions with numerical techniques such as discrete-particle methods, to be able to capture the fluctuations with greater fidelity. Work on some of these experimental and theoretical aspects is underway in our group and would be communicated in a future publication. Notation

Cd,s d D e Fdrag Fdrag,s G go,ss ks Ks,f p Re t U u z

single-particle drag coefficient diameter of the single particle, m diameter of the column, m restitution coefficient drag force on multiple particles (N) drag force on single particles (N) solids mass flux, kg/m2 s radial Distribution function diffusion coefficient, kg/ms interphase momentum exchange factor, kg/m3 s pressure, N/m2 Reynolds number time, s superficial velocity, m/s velocity, m/s axial distance, m

Greek letters

     

collisional dissipation rate of solids fluctuating kinetic energy, J/m3 s volume fraction granular temperature, m2 /s2 granular viscosity, kg/m s viscosity, kg/m s shear stress, N/m2 energy exchange rate per unit volume between two phases, J/m3 s

Subscripts and superscripts g s T

gas phase solid, granular terminal

Acknowledgments SR would like to acknowledge the financial support for this work from DuPont Center for Collaborative Education and Research,

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