Hydrodynamic Modeling of GasâSolid Bubbling Fluidization Based on

Article pubs.acs.org/IECR

Hydrodynamic Modeling of Gas−Solid Bubbling Fluidization Based on Energy-Minimization Multiscale (EMMS) Theory Xinhua Liu,*,† Yuefang Jiang,†,‡ Cenfan Liu,† Wei Wang,† and Jinghai Li*,† †

State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, P.R. China ‡ State Key Laboratory of Heavy Oil Processing, China University of Petroleum, Qingdao 266555, P.R. China ABSTRACT: Hydrodynamic modeling of gas−solid bubbling fluidization is of significance to the development of gas−solid bubbling reactors since it still remains at the stage of experimental and empirical science. As is the role of particle clusters in gas− solid fast fluidization, gas bubbles characterize the structural heterogeneity of gas−solid bubbling fluidization, and their evolution is mainly subject to the constraints of the stability and boundary conditions of the system. By considering the expansion work of gas bubbles against the normal pressure stress in the emulsion phase, an improved necessary stability condition is proposed to close a gas−solid bubbling model. Applying the upgraded gas−solid bubbling model at the scale of vessels, the steady-state hydrodynamics of gas−solid bubbling fluidization can be reproduced without introducing bubble-specific empirical correlations such as for diameter and/or acceleration. The unified modeling of the entire gas−solid fluidization regime from bubbling to fast fluidization is performed by integrating the upgraded gas−solid bubbling model with the original energy-minimization multiscale (EMMS) model. Incorporating the upgraded gas−solid bubbling model into commercial computational fluid dynamics (CFD) software at the scale of computational cells, the unsteady-state simulation of gas−solid bubbling fluidization is realized with a higher accuracy than that based on homogeneous drag models.

1. INTRODUCTION Gas−solid bubbling fluidization has found wide application in the fields of energy utilization, chemical synthesis, mineral processing, etc. However, this technology today may still be considered as an experimental and empirical science because of the involved complex multiscale gas−solid interactions. For instance, mean voidage of gas−solid bubbling fluidization is often assumed to be the minimum fluidization voidage (εmf) in many mathematical models, though this assumption is obviously in conflict with the experimental observation.1,2 Accordingly, the design and scale-up of highly efficient gas− solid bubbling reactors calls for a theoretical description of hydrodynamics of gas−solid bubbling fluidization. As a typical nonlinear and nonequilibrium multiscale complex system, gas−solid bubbling fluidization is featured with the mesoscale heterogeneous structure called a gas bubble. As is the role of particle clusters in gas−solid fast fluidization, this heterogeneous structure has a significant effect on the heat and mass transfer in gas−solid bubbling reactors.3−5 By considering mesoscale gas−solid interactions in gas−solid bubbling fluidization as in energy-minimization multiscale (EMMS) theory,6,7 Shi et al.8 developed a so-called EMMS/ bubbling model and coupled it with computational fluid dynamics (CFD) simulation successfully. However, they have to introduce some bubble-specific empirical correlations such as for diameter and acceleration in order to obtain numerical solution of the model. These bubble-specific empirical correlations, similar to the cluster diameter correlation in the original EMMS model, make it difficult to extend and generalize the EMMS theory.9,10 The authors have been devoting themselves to addressing the problems related to cluster or bubble diameter in the EMMS © 2014 American Chemical Society

theory. For example, taking cluster number density as a much more comprehensive indicator to the axial heterogeneity in gas−solid fast fluidization than cluster diameter only, Hu et al.11 successfully established an axial EMMS model to reproduce the axial S-shaped voidage profile in circulating fluidized bed risers, in which particle cluster diameter is not determined empirically but defined by the stability conditions of the system. Illuminated by this successful attempt, the authors recognized that bubble growth must not be unconstrained even if no boundary condition exists in gas−solid bubbling fluidization, because the formation of a large gas bubble must induce greater energy consumption than the production of many small gas bubbles at the same bubble volume fraction. With this in mind, through gaining an insight into the interphase energy consumption related to the expansion of gas bubbles, the authors aim to propose an improved stability condition for gas−solid bubbling fluidization, in order to realize the hydrodynamic modeling of gas−solid bubbling fluidization without introducing bubble-specific empirical correlations such as for diameter and/or acceleration. This research can be expected to not only provide quantitative references for the development of gas−solid bubbling reactors but also lay a solid basis for the total-system steady-state modeling of complex gas−solid reactors with multiple flow regimes.12 Received: Revised: Accepted: Published: 2800

September 5, 2013 December 23, 2013 January 27, 2014 January 27, 2014 dx.doi.org/10.1021/ie4029335 | Ind. Eng. Chem. Res. 2014, 53, 2800−2810

Industrial & Engineering Chemistry Research

Article

ρ 3 C Di e fb Usi 2 = fb (ρe − ρg )(g + ab) 4 db

2. ENERGY CONSUMPTION IN GAS−SOLID BUBBLING FLUIDIZATION Following the multiscale analysis method in the EMMS theory, a gas−solid bubbling system can be resolved into three subsystems including the dense emulsion phase, the dilute bubble phase, and the interphase in between. Correspondingly, the total mass-specific suspension and transporting energy consumption rate (Nst) in the system can be expressed as the sum of following three parts: Nst,e for the emulsion phase, Nst,b for the bubble phase, and Nst,i for the interphase. However, the interphase interaction between the dense emulsion and gas bubbles differs much from that between the dilute emulsion and particle clusters, though both gas bubbles and particle clusters originate from the compromise between the fluid and the particles.11 In gas−solid bubbling fluidization, the normal pressure stress instead of the shear drag force may become to be significant under some situation and, thereby, cannot be neglected in gas−solid bubbling mathematical models.13 That is, unlike in the gas−solid fast fluidization where the interphase energy consumption mainly results from the shear drag force between the emulsion and cluster phases, in gas−solid bubbling fluidization the expansion work done by gas bubbles against the emulsion pressure stress may also contribute much to the interphase energy consumption between the emulsion and bubble phases. In order to differentiate from the traditional interphase suspension and transporting energy consumption rate (Nst,i) defined by the EMMS theory, the energy consumption rate resulting from the expansion of gas bubbles is denoted as Nex,i in this research, though the work can actually be taken as a part of the interphase suspension and transporting energy consumption. It is worth noting that the emulsion pressure stress was found to increase with bubble size and to scale with both particle density and bubble size in gas−solid fluidization.14 This implies that the maximum stable diameter of gas bubbles in gas−solid bubbling fluidization must be constrained by total energy consumption rate including Nex,i under the given operating conditions before it increases to be comparable with vessel size, since Nex,i is dependent on bubble diameter at a fixed volume fraction of gas bubbles. Therefore, taking into account the expansion work of gas bubbles against the emulsion pressure stress, it is possible to establish an improved stability condition for the gas−solid bubbling model to avoid introducing bubble-specific empirical correlations such as for diameter and/or acceleration.

(1)

• Force balance for the emulsion phase in unit bed volume: the sum of the drag force of the gas in both the emulsion and bubble phases is equal to the effective weight of particles in the emulsion phase in unit bed volume ρg ρ 3 3 C De (1 − fb )(1 − εe)Use 2 + C Di e fb Usi 2 4 dp 4 db = (1 − fb )(1 − εe)(ρp − ρg )(g + a p)

(2)

In the above formulas, ρe = ρp(1 − εe) + ρgεe is average density of the emulsion phase; nonzero Use and Usi are gas−solid slip velocities in the emulsion and interphases respectively, Use = Uge −

εeUpe 1 − εe

≥0

(3)

Usi = [Ub(1 − εe) − Upe](1 − fb ) ≥ 0

(4)

where Usi originates from the drag of the particles in the dense emulsion phase on gas bubbles in gas−solid bubbling fluidization,15 corresponding to that of the gas in the dilute emulsion phase on particle clusters in gas−solid fast fluidization; two drag coefficients, CDe and CDi, for calculating the gas−solid interactions in the emulsion and interphases respectively can be correlated as shown in Table 1. The detailed derivation processes of eqs 1 and 2 can be found in Appendix A. Table 1. Definitions of the Two Drag Coefficients for Multiscale Interactions parameter Reynolds number CD for a single particle or bubble CD for fluidized particles or multiple bubbles

emulsion phase16,17 Ree = Used pρg /μg

C De0 =

24 3.6 + 0.313 Ree Ree

C De = C De0εe−4.7

interphase18,19

Re i = Usidbρe /μe −1.5 ⎧ 0 < Re i ≤ 1.8 ⎪ 38Re i C Di0 = ⎨ ⎪ ⎩ 24/ Re i Re i > 1.8

C Di = C Di0(1 − fb )−0.5

Both the gas and the solids should meet the continuity condition for mass flow rate at any cross-section of the bed. • Mass balance for the gas: the net mass flow rate of the gas through the whole cross-section of the bed should be equal to the sum of the gas flow rate through both the emulsion and bubble phases

3. MATHEMATICAL FORMULATION AND STABILITY CONDITION Assuming no particles in gas bubbles, a gas−solid bubbling fluidization system can thus be described by eight independent parameters including εe, ap, ab, f b, Uge, Upe, Ub, and db, among which εe, ap, Uge, and Upe refer to the emulsion phase, and ab, f b, Ub, and db, to the bubble phase. Similar to the original EMMS model, force balance equation in unit bed volume can be built for the dilute and dense phases, respectively. • Force balance for the bubble phase in unit bed volume: the drag force of the emulsion phase exerting on gas bubbles is equal to the effective buoyancy of gas bubbles in unit bed volume

Ug = Uge(1 − fb ) + Ubfb

(5)

• Mass balance for the particles: the net mass flow rate of the solids through the whole cross-section of the bed should be equal to the solid mass flow rate through the emulsion phase since there exist no particles in the bubble phase Up = Gs /ρp = Upe(1 − fb )

(6)

According to the definition of the mass-specific suspension and transporting energy consumption rate in the EMMS theory, 2801

dx.doi.org/10.1021/ie4029335 | Ind. Eng. Chem. Res. 2014, 53, 2800−2810


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4. STEADY-STATE MODELING OF GAS-SOLID BUBBLING FLUIDIZATION The steady-state modeling of gas−solid bubbling fluidization is realized by applying the upgraded gas−solid bubbling model at the scale of vessels. At the specified Ug and Gs, both the particle and bubble acceleration (ap and ab) are negligible, and thus, the remaining six independent variables (εe, f b, Uge, Upe, Ub, and db) can be determined by utilizing the two-dimensional optimization method according to following computation scheme: (1) Traverse εe within the range of [εmin, εmax), where εmin is the minimum voidage of the emulsion phase under the given operating condition and εmax is the maximum voidage for occurrence of the dense emulsion phase for the specified gas−solid system. (2) Traverse f b within the range of (0, 0.5) at a specified εe. (3) Calculate Use at the given εe and f b according to eqs 1 and 2. (4) Calculate Upe, Uge, Ub, and Usi from eqs 3−6. (5) Calculate db according to eq 2. (6) Calculate Nst,e, Nst,i, Nex,i, and Nst/NT according to eqs 7 and 8 and 11−13. (7) Find optimal f b through Nst/NT = min among all possible roots at a constant εe. (8) Find optimal εe and f b through Nst/NT = min among all above optimized roots. (9) Determine the remaining parameters at the optimal εe and f b. The two-dimensional optimization process of the parameters f b and εe is exemplified in Figure 1. It can be seen from Figure 1a that there always exists a minimum value of Nst/NT for three tested gas velocities during the variation of f b, indicating that an

Nst,e and Nst,i in the emulsion (dense) and interphases are calculated respectively as ⎞ πd 3 ρ 1 − εe ⎛ 1 ⎜C De p g Use 2⎟Uge(1 − f ) ⎟ b (1 − ε)ρp πd p3/6 ⎜⎝ 4 2 ⎠ ρp − ρg [(1 − f )(g + a p) − f (g + ab)] b b Uge = 1 − fb ρp (7)

Nst,e =

Nst,i = =

fb ⎛ πdb3 ρe 2⎞ 1 ⎜C Di Usi ⎟Ubfb (1 − ε)ρp πdb3/6 ⎝ 4 2 ⎠ ρp − ρg

fb 2

ρp

1 − fb

(g + ab)Ub (8)

And Nst,b in the bubble (dilute) phase equals zero due to the assumption of no particles inside gas bubbles. The pressure stress in the emulsion phase can be correlated with particle density and bubble diameter as follows:14 Pe = Kρp gdb

(9)

where K is generally taken as a constant (0.1). Consequently, the expansion work done by a single gas bubble against the emulsion pressure stress is determined as Wex0 =

∫

Pe dVb =

∫

⎛ πd 3 ⎞ π Kρp gdb d⎜ b ⎟ = Kgρp db 4 8 ⎝ 6 ⎠

(10)

leading to the energy consumption rate (Nex,i) resulting from the expansion of gas bubbles with respect to unit mass of particles Nex,i = =

nb Wex0 Ms fb UgA t /Vb Ms

Wex0

fb 3 KgdbUg 4 Hmf (1 − εmf ) fb 3 KgdbUg = 4 Hs(1 − εs) =

(11)

where nb is the generation rate of gas bubbles in the bed. So, a new necessary stability condition for gas−solid bubbling fluidization can be expressed as Nst /NT = (Nst,e + Nst,i + Nst,b + Nex,i)/NT = min

(12)

where NT is total suspension and transporting energy consumption rate with respect to unit mass of particles under the condition of zero wall friction NT =

ρp − ρg ρp

(g + a p)Ug (13)

Therefore, the upgraded gas−solid bubbling model can be summarized as a set of equations consisting of eqs 1 and 2 and 5 and 6 under the constraint of eq 12.

Figure 1. Two-dimensional optimization of (a) f b and (b) εe with respect to Nst/NT = min at various superficial gas velocities. 2802



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optimal f b can be determined according to the optimization with respect to Nst/NT = min. Moreover, the optimal f b, as expected, increases with increasing superficial gas velocity under the tested situation. On the other hand, as shown in Figure 1b, Nst/NT monotonously increases with the increase of εe at lower superficial gas velocities (Ug = 10, 30Umf) at the optimal f b, indicating that εe always tends to be εmin because the gas tends to form bubbles instead of flowing through the emulsion phase under these conditions. However, at a higher superficial gas velocity (Ug = 50Umf), the minimum value of Nst/NT at the optimal f b occurs at a higher εe than εmin because the high superficial gas velocity results in the expansion of the emulsion phase. These calculation results are in good agreement with experimental observation qualitatively,20,21 showing the feasibility of the numerical scheme for solving the upgraded gas− solid bubbling model. According to eq 11, the expansion work of gas bubbles is mainly dependent on f b, db, and Ug for a specified gas−solid bubbling system. Figure 2 shows that Nex,i varies inversely with

superficial gas velocities. Compared with the case for Geldart A particles, this phenomenon obviously becomes much more significant for Geldart B particles, as large gas bubbles are likely to form under this situation. These calculation results consolidate that the expansion work of gas bubbles should not be neglected in the steady-state modeling of gas−solid bubbling fluidization, especially for gas bubbling fluidization of large particles at relatively high gas velocities. In order to further validate the prediction of the upgraded gas−solid bubbling model, the calculation results of the model are compared with the experimental data from various airfluidizing systems. It should be noted that in real gas−solid bubbling reactors the maximum stable bubble diameter is constrained by both the stability and boundary conditions. However, the expansion of gas bubbles is thought to be mainly constrained by the stability condition until they grow up to be comparable with vessel size. So, if the calculated db is greater than 2/3Dt, db is set to be a constant and equal 2/3Dt, since gas bubbles with a diameter larger than 2/3 of the column diameter no longer grow further and are generally regarded as slugs.22,23 Under this situation, a determinate f b corresponding to the specified Dt can be analytically calculated without involving the optimization with respect to Nst/NT = min. As shown in Figures 4a and b, the calculated f b and ε agree with the average data based on the whole reactor in a wide range of operating conditions, regardless of the types of particles (e.g., Geldart A or B). However, at superficial gas velocities lower than about 2Umf, the upgraded gas−solid bubbling model has no numerical solution, which is consistent with the fact that the upgraded gas−solid bubbling model as a typical two-phase model is not suitable to be applied into the situation near the minimum fluidization. The blue dashed line for Geldart B particles in Figure 4a is just a natural extension of the trend for the variation of f b with decreasing superficial gas velocity, which is however in good agreement with the measurement data. Figure 4c presents the calculated and experimental db as well as the mean bubble size using the literature correlation of Horio and Nonaka30 as a comparison. It is clear that the predicted variation trend of db is in agreement with the experimental and correlated results, qualitatively. However, the calculated db seems to increase much more sharply with superficial gas velocity than the experimental data and becomes constrained by the boundary conditions much more quickly than expected. This may be mainly attributed to the idea that the upgraded gas−solid bubbling model does not take into consideration the gas flow inside gas bubbles31,32 and assumes that all gas outside the emulsion phase flows through the reactor in the form of bubbles.

Figure 2. Variation of Nex,i with f b and db at a given superficial gas velocity.

f b but directly with db at a fixed superficial gas velocity. This phenomenon implies that a maximum stable bubble diameter may be defined by the improved stability condition (Nst/NT = min), thus avoiding the introduction of an empirical correlation for db. Figure 3 plots the variation of Nex,i/Nst with Ug for typical Geldart A and B particles. It can be seen that Nex,i/Nst increases with increasing superficial gas velocity for the both types of particles, and even up to 15% at relatively high

5. UNIFIED MODELING OF THE ENTIRE GAS-SOLID FLUIDIZATION REGIME The upgraded gas−solid bubbling model is proved to have a good predictability for the steady-state hydrodynamics of gas− solid bubbling fluidization at Up = 0. In fact, the model can also be applied into the situation of Up > 0 in theory, which provides a possible way for the unified modeling of gas−solid fluidization across various flow regimes by incorporating with the original EMMS model. With increasing Ug from zero gradually, a typical gas−solid system may successively experience five major operating regimes called as fixed, bubbling, turbulent, fast, and transport beds corresponding to four transition velocities (Umf, Uc, Utr, and Upt) respectively. It is out of question that the fixed bed can

Figure 3. Variation of Nex,i/Nst with Ug for typical Geldart A and B particles. 2803



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Figure 5. Unified modeling of gas−solid fluidization: (black line) bubbling model; (red line) EMMS model.

with Ug increasing from Umf to Ut for all tested particles. At Ug = Ut, the calculated voidage under the condition of Up > 0 is smaller than that in case of Up = 0, but this phenomenon gradually disappears with increasing particle diameter. This may be explained by the fact that the motion of small particles has more degree of freedom at Up > 0 than that at Up = 0 and, thereby, facilitates a full minimization of voidage in the emulsion phase. Beyond Utr, as plotted by the red line in Figure 5, the original EMMS model reasonably predicts the slight increase of voidage with an increase of Ug. When Ug ranges from Ut to Utr, for small particles (e.g., Geldart A), the two curves calculated from the original EMMS model and the upgraded gas−solid bubbling model generally cut at some specified gas velocity close to Uc. With increasing particle diameter, the calculation results from both models, however, tend to be consistent with each other. At Ug = Utr, the original EMMS model successfully captures the coexistence of the dilute and dense phases in the system, while the upgraded gas−solid bubbling model seems to provide a global hydrodynamic description since voidage calculated from this model generally falls between the dilute and dense phase voidage. Considering the good predictability of the upgraded gas− solid bubbling model at low superficial gas velocities, it is suggested that for Geldart A particles the voidage at Ut ≤ Ug ≤ Uc should be calculated from the upgraded gas−solid bubbling model, while the voidage at Uc ≤ Ug ≤ Utr, from the original EMMS model. For Geldart A/B boundary particles, the upgraded gas−solid bubbling model enables the calculation of voidage in both bubbling and turbulent fluidization regimes. And for Geldart B particles the voidage at Ut ≤ Ug ≤ Utr can be approximately determined from either the original EMMS model or the upgraded gas−solid bubbling model since the

Figure 4. Comparison between the calculation results and the experimental and correlated data: (a) f b; (b) ε; (c) db.

be modeled by the Ergun equation, while the transport bed is approximated as a uniform distribution. During the regime transition from bubbling to turbulent fluidization, the continuous dense emulsion phase begins to break up into discrete particle clusters immersed in a newly formed continuous dilute emulsion phase, and Uc is generally thought to be the critical velocity corresponding to this phase inversion. So, the original EMMS model based on particle clusters may be applicable to both the turbulent and fast fluidization in theory, while the gas−solid bubbling model grounded on gas bubbles is much more suitable to enable the hydrodynamic modeling of bubbling fluidization. However, considering the regime transition from bubbling to turbulent fluidization is generally a gradual, not an abrupt, process, the gas−solid bubbling and original EMMS models are applied into the both regimes at Up > 0 in this research, in order to clarify the application ranges of the two models. Solids circulation rate (Gs) is set to be zero at Ug < Ut, while to be saturation carrying capacity (Gs*) corresponding to Utr at Ug ≥ Ut. As indicated by the black line in Figure 5, the calculated voidage from the upgraded gas−solid bubbling model increases 2804



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and βe is measured with the heterogeneity index Hd33

discrepancy between the calculation results from both models is very small. According to the preceding modeling scheme, the unified steady-state modeling of the entire gas−solid fluidization regime can thus be realized by incorporating the upgraded gas−solid bubbling model with the original EMMS model, which facilitates the total-system hydrodynamic modeling of complex gas−solid reactors with multiple flow regimes.12

Hd = βe /βw where βw is Wen and Yu βw =

correlation with

3 1−ε ρ Uslipε−2.65 C D0 4 dp g

(17)

Two lab scale gas−solid bubbling fluidized beds are simulated, as shown in Figure 7 (hereinafter referred as bed I and bed II, respectively). Geldart A particles are used as fluidized material in bed I, and Geldart B particles in bed II. The main part of bed I has an inner diameter of 0.267 m and a height of 2.464 m, whose enlarged freeboard region has an inner diameter of 0.667 m and expands to a height of 4.2 m to avoid excessive particle entrainment.35 Bed II is a quasi-2D Plexiglas column of 1.0 m height and 0.025 m thickness.27 In both beds, the injected gas flows through a velocity inlet at the bottom and exits through a pressure outlet at the top. A no-slip boundary condition is specified at the wall for the gas, while a partial slip boundary condition for the wall. The physical properties of particles and gas in the two beds are summarized in Table 2. According to the computation scheme shown in Figure 6, we calculated corresponding eight structural parameters and further fitting functions for Hd at Ug = 0.2 and 0.4 m/s for bed I and at Ug = 0.38 and 0.46 m/s for bed II as summarized in eqs 18, 19, 20, and 21, respectively:

6. UNSTEADY-STATE SIMULATION OF GAS-SOLID BUBBLING FLUIDIZATION Taking both ab and ap into consideration, the upgraded gas− solid bubbling model also enables the unsteady-state simulation of gas−solid bubbling fluidization by incorporating it into commercial CFD software at the scale of computational cells. Under this situation, voidage can be provided from CFD results and correlated with εe and f b as follows: ε = (1 − fb )εe + fb

(16) 34

(14)

The acceleration of particles in the emulsion phase (ap) is assumed to be zero in gas−solid bubbling fluidization, so the remaining seven independent variables (εe, ab, f b, Uge, Upe, Ub, and db) in the set of eqs 1−4 and 14 can be similarly optimized under the constraint of eq 12, as shown in Figure 6, where amax means the maximum ab to ensure Use ≥ 0.

Bed I (Ug = 0.2 m/s) ⎧ 5.56 − 12.16ε εmf < ε < 0.45 ⎪ ⎪ ε − 23.69ε 2 0.45 < ε < 0.935 Hd = ⎨−2.66 + 13.74 3 ⎪ + 14.11ε ⎪ ⎩1.0 0.935 < ε < 1.0 (18)

Bed I (Ug = 0.4 m/s) ⎧ 5.55 − 12.29ε εmf < ε < 0.45 ⎪ 2 ⎪ ⎪− 4.87 + 39.17ε − 124.03ε 0.45 < ε < 0.995 Hd = ⎨ + 193.42ε 3 − 148.30ε 4 ⎪ 5 ⎪ + 45.17ε ⎪ ⎩− 93.44 + 94.44ε 0.995 < ε < 1.0

(19)

Bed II (Ug = 0.38 m/s) ⎧ 23.24 − 84.71ε + 79.74ε 2 ε < ε ≤ 0.588 mf Hd = ⎨ ⎩1.0 0.588 < ε ≤ 1.0 ⎪

⎪

Figure 6. Computation scheme for solving the gas−solid unsteadystate bubbling model.

(20)

With the eight structural parameters in the model to be determined, so-called structure-dependent drag coefficient (βe) can be expressed as

Bed II (Ug = 0.46 m/s) ⎧ 4.09 − 15.27ε + 16.62ε 2 ε < ε ≤ 0.617 mf Hd = ⎨ ⎩1.0 0.617 < ε ≤ 1.0 ⎪

⎪

2

βe =

ε (ρ − ρg )(1 − ε)[(1 − fb )(g + a p) Uslip p + fb (g + ab)]

(21)

Equations 18−21 are approximately extended to computational cells to modify drag coefficient for the sake of simplicity, as in

(15) 2805



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Figure 7. Schematic drawings of two lab scale gas−solid bubbling fluidized beds.

Table 2. Summary of the Physical Properties of Particles and Gas bed I

Table 3. Simulation Settings in Fluent space time viscous pressure−velocity coupling momentum discretization volume fraction discretization granular temperature granular viscosity granular bulk viscosity frictional viscosity angle of internal friction solids pressure frictional pressure radial distribution specularity coefficient close packing density time step

bed II

items

gas

particles

gas

particles

density (kg/m3) viscosity (Pa·s) diameter (μm) bulk voidage (−)

1.16997 1.91 × 10−5

1780

1.225 1.82 × 10−5

2500

65.0 0.37

275.0 0.40

the case of Yang et al.36 This simplification is adopted because the mesoscale structural parameters are mainly defined by macro-scale or globally hydrodynamic conditions according to the EMMS theory. Hexahedron meshes are used in the simulation of bed I, whose grid size is generated with Gambit at the scale of 1 cm as the grid independence has been validated with this size for the same geometry.37 For bed II, a uniform, 2D mesh with grid size of 0.467 cm × 0.417 cm is used because it is a quasi-2D experimental facility. The kinetic theory of granular flows (KTGF) is used to close the solid pressure and viscosities.38 The phase-coupled SIMPLE algorithm is selected for the pressure-velocity coupling. The statistic results between 20 and 30 s in the simulation are used to compare with the experimental data. The computational settings in Fluent are listed in Table 3. For comparison, simulation results with the Gidaspow drag are also provided for bed I in this subsection.41 The snapshots of solid concentration calculated at different superficial gas velocities in bed I are shown in Figure 8. It can be seen from Figure 8a and c that the simulation based on our drag model captures typical flow structures with bubbles surrounded by continuous emulsion. However, as indicated by Figure 8b and d, even if for the lower gas velocity case (0.2 m/ s), using the Gidaspow drag model at the current grid resolution predicts a flow pattern similar to turbulent fluidization, in which the discrete phase is hard to distinguish

max iterations per time step static bed height

bed I 3D; bed II 2D unsteady, second-order implicit laminar phase-coupled SIMPLE second-order upwind quick discretization second-order upwind Gidaspow38 Lun et al.39 Schaeffer40 30° Lun et al.39 based KTGF Lun et al.39 0.6 bed I 0.63; bed II 0.60 bed I 0.001 s and 0.0005 s at Ug = 0.2 and 0.4 m/s, respectively; bed II 0.0005 s 30 bed I 0.9 m; bed II 0.4 m

from the continuous one. In general, the Gidaspow drag model overpredicts the drag force with the grid resolution specified in this case, resulting in a lower solid concentration but a higher expansion height of more than 2 times the static bed height. The simulation results for bed I are further compared with experimental data in Figure 9. Generally, the axial pressure drop profiles predicted by using the bubbling model are in better 2806



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Figure 8. Instantaneous solid concentration in the gas fluidized bed of Geldart A particles.

Figure 9. Comparison between the simulation and experimental data for bed I.

drag model predicts a bed expansion ratio much closer to the experimental data of about 1.47 at Ug = 0.38 m/s. Figure 11 compares the simulation and experimental data in terms of radial solid concentration profiles at z = 0.2 m under the two gas velocities. Our calculated results of solid concentration are generally lower than the corresponding experimental data at both gas velocities. Similar results have been reported in Taghipour et al.,27 even using the drag models for homogeneous systems. In fact, the heterogeneity index (HD) for Geldart B particles (e.g., eqs 20 and 21) gives little correction to the homogeneous drag only at the range of voidage near close packing state. Compared with the correction for Geldart A particles (e.g., eqs 18 and 19), we can see that the need of drag modification, or the effects of mesoscale structure, lessens with the increase of particle size or the Archimedes number. That coincides with the conclusion of Lu et al.43

agreement with experiments than using the Gidaspow model at both gas velocities. And the comparatively larger discrepancy in the axial profiles at z = 0.8 m may be partly explained by the uncertainty of measurements at the bed surface. Correspondingly, the radial solid concentration profiles calculated from using our drag model are also in better agreement with the experimental data. The comparatively larger discrepancy near the bed wall may be attributed to the uniform grid scheme used in computation and the lack of wall corrections,42 which have not been accounted for in current model. Much effort is needed in these aspects. The snapshots of transient solid concentration simulated under different superficial gas velocities in bed II are shown in Figure 10. It can be seen that the simulation based on our drag model captures the increase of bubble size with increase of height at both gas velocities. Compared with what was reported by Taghipour et al. with various drag models,27 our bubbling 2807



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bubbling model into commercial computational fluid dynamics (CFD) software at the scale of computational cells, the unsteady-state simulation of gas bubbling fluidization is realized with a higher accuracy than the one based on homogeneous drag models. The prediction of the gas−solid bubbling model can be expected to be improved further by taking into consideration the gas flow inside gas bubbles. The proposed stability condition for gas−solid bubbling fluidization needs to be further demonstrated experimentally or theoretically.

■

APPENDIX A

• Force Balance for the Bubble Phase in Unit Bed Volume

The drag of the emulsion phase exerting on a single gas bubble can be calculated as π 1 FDi0 = C Di db 2 ρe Usi 2 4 2

leading to the drag of the emulsion phase exerting on all gas bubbles in unit bed volume ρ ⎞ ⎛ π ⎞⎛ π 1 3 FDi = ⎜fb / db3⎟⎜C Di db 2 ρe Usi 2⎟ = C Di e fb Usi 2 ⎝ 6 ⎠⎝ 4 ⎠ 2 4 db

Figure 10. Instantaneous solid concentration in the gas fluidized bed of Geldart B particles.

The effective buoyancy of all gas bubbles in unit bed volume can be determined as Fef = fb (ρe − ρg )(g + ab)

Thus, according to the force balance for the bubble phase in unit bed volume FDi = FeF, we get ρ 3 C Di e fb Usi 2 = fb (ρe − ρg )(g + ab) 4 db • Force Balance for the Emulsion Phase in Unit Bed Volume

In the emulsion phase, the drag force of the gas acting on a single particle can be written as

Figure 11. Comparison between the simulation and experimental radial distribution of solid volume fraction at z = 0.2 m for bed II.

π 1 FDe0 = C De d p2 ρg Use 2 4 2

leading to the drag of the emulsion gas acting on all particles in the emulsion phase in unit bed volume

7. CONCLUSIONS Following the multiscale analysis method in the energyminimization multiscale (EMMS) theory, this research gains an insight into the interfacial interactions between the emulsion and bubble phases in gas−solid bubbling fluidization and clarifies that both the shear drag force and the normal pressure stress in the emulsion phase affect the dynamical evolution of gas bubbles. The energy consumption rate related to the expansion of gas bubbles against the emulsion pressure stress is proved to have an effect on the maximum stable diameter of gas bubbles before it increases to be comparable with vessel size. An upgraded gas−solid bubbling model is developed on the basis of a new necessary stability condition accounting for bubble expansion, and the model predicts well the steady-state hydrodynamics of gas−solid bubbling fluidization without introducing bubble-specific empirical correlations such as for diameter and/or acceleration. The steady-state hydrodynamics of the entire gas−solid fluidization regime ranging from bubbling to fast fluidization can also be reproduced by integrating the upgraded gas−solid bubbling model with the original EMMS model. Incorporating the upgraded gas−solid

⎞ ⎡ 1 π ⎤⎛ π FDe = ⎢(1 − fb )(1 − εe)/ d p3⎥⎜C De d p2 ρg Use 2⎟ ⎣ ⎠ 6 ⎦⎝ 4 2 ρ 3 g = C De (1 − fb )(1 − εe)Use 2 4 dp

In unit bed volume, the drag of gas bubbles exerting on all particles in the emulsion phase should be equal to the drag of the emulsion phase exerting on all gas bubbles ρ 3 FDi = C Di e fb Usi 2 4 db In unit bed volume, the effective weight of all particles in the emulsion phase can be expressed as Feg = (1 − fb )(1 − εe)(ρp − ρg )(g + a p)

Therefore, according to the force balance for particles in the emulsion phase in unit bed volume FDe + FDi = Feg, we get 2808



Article

ρg ρ 3 3 C De (1 − fb )(1 − εe)Use 2 + C Di e fb Usi 2 4 dp 4 db

Ug = superficial gas velocity (m/s) Uge, Upe = superficial gas and particle velocities in the emulsion phases (m/s) Umf = minimum fluidization velocity (m/s) Up = superficial particle velocity (m/s) Upt = transition velocity from fast fluidization to dilute transport (m/s) Use, Usi = superficial slip velocities in the emulsion and inter phases (m/s) Uslip = superficial gas−solid slip velocity (m/s) Ut = terminal velocity of particles (m/s) Utr = transition velocity from turbulent to fast fluidization (m/s) Vb = bubble volume (m3) Wex0 = expansion work with respect to a gas bubble (kg·m2/ s2) z = axial location (m)

= (1 − fb )(1 − εe)(ρp − ρg )(g + a p)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Tel.: +86-10-82544940. Fax: +86-10-62558065. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The pertinent comments of the reviewer on this research are acknowledged. We would like to thank Prof. Wei Ge and Dr. Nan Zhang for their valuable discussion with the authors. We are also grateful to Prof. Chaohe Yang and Dr. Hui Zhao for recommending graduate students to the EMMS group. This research was financially supported by the International Science & Technology Cooperation and Exchange Program of Ministry of Science and Technology (No. 2011DFA60390), the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDA07080400), and the Natural Science Foundation of China (No. 21376244 and No. 91334107).

Greek Letters

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NOMENCLATURE ap, ab = acceleration of emulsion particles and gas bubbles (m/s2) At = cross-sectional area of the bed (m2) CDe, CDi = effective drag coefficients in the emulsion and interphases db = bubble diameter (m) dp = particle diameter (m) Dt = bed diameter (m) f b = volume fraction of bubble phase g = gravity acceleration (m/s2) Gs = solids circulation rate (kg/(m2 s)) Gs* = saturation carrying capacity (kg/(m2 s)) H = bed height (m) Hd = heterogeneity index Hmf = bed height at the minimum fluidization (m) Hs = static bed height (m) K = constant Ms = total mass of particles (kg) Nex,i = energy consumption rate due to the expansion of gas bubbles (m2/s3) Nst = suspension and transporting energy consumption rate (m2/s3) Nst,b = energy consumption rate in the bubble phase (m2/s3) Nst,e = energy consumption rate in the emulsion phase (m2/ s3) Nst,i = energy consumption rate in the inter phase (m2/s3) NT = total suspension and transporting energy consumption rate (m2/s3) P = pressure (Pa) Pe = pressure stress in the emulsion phase (Pa) Re = Reynolds number Ub = superficial velocity of gas bubbles (m/s) Uc = transition velocity from bubbling to turbulent fluidization (m/s)

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βe = structure-dependent drag coefficient βw = homogeneous drag coefficient ε = voidage εe = voidage in the emulsion phase εmax = maximum voidage for clustering εmf = voidage at the minimum fluidization εmin = minimum voidage in the emulsion phase εs = bulk voidage of particles μe = shear viscosity of the emulsion phase (Pa·s) μg = gas shear viscosity (Pa·s) ρe = average density of the emulsion phase (kg/m3) ρg = gas density (kg/m3) ρp = particle density (kg/m3)

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