5906
Ind. Eng. Chem. Res. 2008, 47, 5906–5912
Modeling of the Fully Developed Zone in the Riser of Circulating Fluidized Beds Yashar Khalighi, Rahmat Sotudeh-Gharebagh,* and Navid Mostoufi Oil and Gas Centre of Excellence, Department of Chemical Engineering, UniVersity of Tehran, P. O. Box 11155/4563, Tehran, Iran
Cluster formation in the risers of circulating fluidized beds has been verified by experimental works of different investigators. In this work, a model has been developed based on the force and mass balances over the riser and core cross section in order to characterize the core-annulus structure of the fully developed zone by the cluster-based approach. The clusters are considered to be spherical with constant diameter. Radial profiles of cluster velocity, solid mass flux, and solids holdup have been obtained by this model as the intermediate variable. The comparison is made between the results of the model and the experimental data and modeling reported in the literature. The model is able to explain the experimental data satisfactorily, and the core radius which is the key parameter in the core-annulus flow structure has been estimated. The results of this model could be used to understand the flow structure in the riser of a circulating fluidized bed largely adapted for industrial applications. 1. Introduction In order to improve various industrial technologies, such as coal combustion, gasification, and drying, circulating fluidized beds (CFBs) have been used in the past decades. High gas-solids contact efficiency, high gas and solids throughput, and reduced axial dispersion of both gas and solid phases are some of the advantages of CFBs compared to conventional systems. Modeling of the CFBs is rather complex. Therefore, it seems necessary to develop sophisticated modeling approaches which can describe the gas-solid flow structure with a sufficient accuracy. As seen in the literature, Harris and Davidson1 classified hydrodynamic models in three groups: • type I: models that predict only axial profile of the solids holdup or simple axial solids distribution models (e.g., Smolders et al.2) • type II: models that predict both axial and radial variation of the solids holdup by assuming a core-annulus structure (e.g., Hyre et al.3) • type III: models that employ fundamental equations of fluid dynamics to predict two-phase gas/solid flow Type III models are based on gas and solid phase continuity, momentum, and energy equations (CFD models) (e.g., Mathiesen et al.,4 Ibsen et al.5). The agreement of type I models with experimental data is rather acceptable. Based on experimental evidence, the core-annulus models of type II are good approximations of the time-averaged radial flow structure in the CFB riser.6 Type III models are the most rigorous, but require simplifying assumptions when balanced against their mathematical complexity. From another point of view on solid hydrodynamics, two different approaches could be used for modeling the hydrodynamics of CFBs. One common approach is the particle-based approach (PBA), which models the hydrodynamic phenomena based on moving single particles within the riser. Experimental investigations on the hydrodynamics of circulating fluidized beds by various researchers proposed the existence of discrete solid clusters instead of individual particles in fluidized beds.7–11 Therefore, it could be suggested that the hydrodynamics of circulating fluidized beds could be modeled through the cluster* To whom correspondence should be addressed. Tel.: (98-21) 66967797. Fax: (98-21) 6646-1024. E-mail:
[email protected].
based approach (CBA), which assumes that the solids move as clusters rather than as single particles. Developing mechanistic models based on the CBA, rather than the PBA, may provide better results in predicting the properties of the riser. For instance, by using an effective cluster terminal velocity, Avidan and Yerushalmi7 have modeled the bed expansion. Sabaghan et al.10 used the CBA for successful modeling of the acceleration zone in the riser of circulating fluidized beds. Characteristics of clusters were implemented to estimate the axial height of the acceleration zone. Karimipour et al.11 used the same approach and developed a model based on the force balance over a cluster moving along the axial length of the downer where satisfactory prediction was reported for hydrodynamic characteristics of the downer. In this work, the CBA is used for modeling the radial distribution of solids in the fully developed
Figure 1. Schematics of the core-annulus structure. Table 1. Correlation of Xu et al.12 dcl / dp ) A( Fp / Fcl ) A ) (3333Udg - M2)(1 - εmf)(Fp - Fg) / (Q - 2M2)Fp Q1 ) (Fp - Fg)g / Fp U0 + Udεmf / (1 - εmf) + Utεmf4.7 / 4 M2 ) (Umf + Udεmf / (1 - εmf) )g
[
10.1021/ie8002717 CCC: $40.75 2008 American Chemical Society Published on Web 07/17/2008
]
Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 5907
Figure 2. Flowchart of the method of solution of the hydrodynamic model presented in this work. Table 2. List of Experimental Cases That Includes Particle Properties and Operational Conditions case
particle
1 2 3 4 5 6 7
FCC FCC FCC FCC FCC FCC FCC
dp (µm) 70 70 70 70 54 54 54
Gs (kg/m2 s) 325 250 250 250 91.7 44.2 21.7
region of the riser for the aim of characterizing the core-annulus structure. Clusters were characterized by using the operating conditions. 2. Modeling Based on the core-annulus flow structure, two flows coexist in the riser. The gas flow conveys the clusters upward in the core region of the riser where solids flow downward adjacent to the wall. Thus the presence of a point with zero solid velocity and solid mass flux is evident. In fact, the positive sign of the cluster velocity in the core region turns negative in the annulus at this point. Figure 1 illustrates such a structure schematically. The following assumptions were considered in developing the model of the present work:
U0 (m/s)
Fp (kg/m3)
Dt (m)
reference
4 5 7 8 4.33 4.33 4.33
1600 1700 1700 1700 1545 1545 1545
0.1 0.2 0.2 0.2 0.14 0.14 0.14
Liu18 Kirbas19 Kirbas19 Kirbas19 Yang et al.14 Yang et al.14 Yang et al.14
• The core-annulus structure is dominant in the riser. • The gas passing through the core of the riser is assumed to be plug flow while the gas in the annulus is almost stagnant. • The solids move up as clusters in the core and move downward as a combination of clusters and particles adjacent to the wall. • The clusters are rigid spheres. The equivalent hydrodynamic diameter of the clusters was estimated from the correlation of Xu and Kato.12 The formulas required for calculating this diameter are listed in Table 1. In the present work, the fully developed zone of the riser was modeled from two points of view. First, the velocity profile of the clusters in the core was estimated and mass balance equations were developed. Second, the momentum balance
5908 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008
Figure 3. (a) Model prediction of local solids mass flux at a solid mass flux of 250 kg · m-2 · s-1 and gas superficial velocity of 5 m · s-1 (data of Kirbas; 19 Dt ) 0.2 m, dp ) 70 µm, Fs ) 1700 kg · m-3). (b) Model prediction of local solids mass flux at a solid mass flux of 250 kg · m-2 · s-1 and gas superficial velocity of 7 m · s-1 (data of Kirbas;19 Dt ) 0.2 m, dp ) 70 µm, Fs ) 1700 kg · m-3). (c) Model prediction of local solids mass flux at a solid mass flux of 250 kg · m-2 · s-1 and gas superficial velocity of 8 m · s-1 (data of Kirbas;19 Dt ) 0.2 m, dp ) 70 µm, Fs ) 1700 kg · m-3). (d) Model prediction of local solids mass flux at a solid mass flux of 325 kg · m-2 · s-1 and gas superficial velocity of 4 m · s-1 (data of Liu;18 Dt ) 0.1 m, dp ) 70 µm, Fs ) 1600 kg · m-3).
equations over the riser cross section were used to calculate the solids velocity in the core. These allow the calculation of a mean velocity of solids in the annulus. Solving the equations requires an iterative procedure in which the annulus thickness would be changed until the convergence was reached on the solids velocity calculated by both methods. 2.1. Mass Balance Based Annular Cluster Velocity. In order to model the fully developed region of the riser, the following expression proposed by Rhodes et al.13 was used in this work to calculate the radial solids mass flux profile:
[ ( )]
Gr r )a 1Gs Rs
m
+1-
ma m+2
(1)
where m ) 5 was found to best fit the experimental data. Since at r ) rc the local solids mass flux becomes zero, the constant a could be evaluated from a)
1
() ( ) rc R
m
2 m+2
(2)
Knowing the cluster velocity profile and solids mass flux profile, the radial voidage profile could be obtained from εr ) 1 -
Gr FclVcl,r
(3)
For calculating radial cluster velocity, a parabolic profile was assumed:6 r2 + Vcl,r)0 (4) 4P Evaluation of the constants of eq 4 requires two data points on the solids velocity profile. In accordance to the core-annulus flow structure, the core radius, rc, may be considered as the radial location at which the velocity of solids is zero: Vcl,r )
Vclr)rc ) 0
(5)
Moreover, at the centerline of the riser, the solids, which are assumed to be moving as clusters, move at their terminal velocity:
Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008 5909
Vclr)0 ) Ug - Ut
(6)
According to Yang et al., gas velocity at the center of the riser is 1.5-2.0 times the superficial gas velocity; thus Vclr)0 ) 1.5
Ug,c - Ut εc
Ug,c )
U0 φcl
(8)
where φcl is the ratio of the core to the riser cross-sectional area and U0 is the gas superficial velocity. Consequently, the average voidage and solids mass flux in the core could be calculated as follows: 1 εc ) πrc2
∫
rc
0
2πrcεr dr
(9)
∫
rc
0
2πrcGr dr
(10)
The mass flux in the annular layer could be determined by the following equation:
(7)
where Ug,c is the gas superficial velocity in the core region. The annular layer tightens the cross section for passing the gas; therefore, Ug,c could be calculated by the following equation, assuming that all the gas passes through the core:
1 πrc2
Gc )
14
Ga ) Gs - Gc
(11)
where, considering that the solids move only as clusters, solids velocity could be obtained from Gs ) FclVcl(1 - ε)
(12)
where the cluster density, Fcl, is Fcl ) (1 - εcl)Fp + εclFg
(13)
εcl is considered εmf in this work. Using the empirical correlation of Patience et al.15 for predicting the value of slip factor: ψ)
U0 5.6 + 0.47Frt0.41 )1+ εVp Fr
(14)
the average solids holdup in the fully developed region could be predicted by combining eqs 12 and 14: ε)
U0Fcl Gsψ + U0Fcl
(15)
In order to estimate the gas voidage in the annulus, the material balance for clusters was written over the cross section of the riser: εa )
φclεc - ε φcl - 1
(16)
Thus, the cluster velocity could be calculated from Vcl,a )
Ga Fcl(1 - εa)(1 - φcl)
(17)
2.2. Momentum Balance Based Annular Cluster Velocity. The momentum balances for the mixture in the core, considering the pressure drop in the gas phase only, would be
Figure 4. (a) Model prediction of local cluster velocity at a solid mass flux of 325 kg · m-2 · s-1 and gas superficial velocity of 4 m · s-1 (data of Liu;18 Dt ) 0.1 m, dp ) 70 µm, Fs ) 1600 kg · m-3). (b) Model prediction of local cluster velocity at gas superficial velocity of 4 m · s1 (data of Yang;14 Dt ) 0.14 m, dp ) 54 µm, Fs ) 1545 kg · m-3).
Figure 5. Model prediction of local solid fraction profile at a solid mass flux of 325 kg · m-2 · s-1 and gas superficial velocity of 4 m · s-1 (data of Liu;18 Dt ) 0.1 m, dp ) 70 µm, Fs ) 1600 kg · m-3).
5910 Ind. Eng. Chem. Res., Vol. 47, No. 16, 2008
4τw dp + Fgg + (1 - φcl)(Fcl - Fg)g )0 dz Dt
(18)
4τi dp + Fgg + )0 dz D φ 1⁄2
(19)
procedure. This iterative procedure of solving the model equations is outlined in Figure 2. When the solution is converged, all the hydrodynamic properties of the riser would be known, including the annulus thickness and profiles of voidage, solids velocity, and solids mass flux.
i cl
The shear stresses τw and τi could be calculated using the wall and interfacial friction factors as fw )
2τw(1 - φcl)2
3. Result and Discussion The profiles of solids mass flux, voidage and cluster velocity were estimated from eqs 1, 3, and 4. Results of the proposed model of this work were compared with the experimental data of Liu,18 Kirbas,19 and Yang et al.14 The operating conditions and solids and gas properties are listed in Table 2. 3.1. Radial Profiles. Figure 3 presents the model prediction of the radial solids mass flux profiles for different operating conditions. The experimental data of Kirbas19 and Liu18 are also shown in these figures. The dashed lines illustrate the result of the PBA model proposed by Pugsley and Berutti.6 These figures illustrate that the CBA model developed in this work accurately predicts the upward mass flux in the core compared with the PBA model for the data used in this study. The Pearson correlation coefficient in these figures is in the range 0.95-0.98 for the CBA model while it is in the range 0.92-0.95 for the PBA model. It can be seen in Figure 3 that, near the wall, the PBA and CBA models coincide. It can be concluded that since the clusters move slower near the wall, they become rather packed; thus, the same behavior could be expected from the particles and clusters. It is worth mentioning that Bhusarapu et al.,20 who studied the solids flow mapping in a gas-solid riser, also concluded that the clustering phenomenon exists through the riser cross section (mostly close to the wall) along with the particle exchange between core and annulus. Radial profiles of the cluster velocity are illustrated in Figure 4 against the experimental data of Liu18 and Yang et al.14 This figure confirms that the CBA model developed in this work is in close agreement with the experimental data. The PBA model prediction is far from the experimental data, especially in the core region. Profiles of the local particle voidage predicted by the CBA and PBA and the experimental data points of Liu18 are shown in Figure 5. In this case, the prediction of the CBA model is considerably closer than that of the PBA model. The Pearson correlation coefficient for the CBA model was evaluated to be 0.80 while this value is 0.75 for the PBA model. 3.2. Thickness of Annulus. The thickness of the annulus is defined as the radius from the wall to the position where the time-average net solids flow direction changes from being downward to upward. Bi et al.21 showed that the thickness of the annulus may depend on the measurement method. If the thickness is measured based on the location where the timemean particle velocity becomes zero, the thickness would be smaller than if the time-mean local solids mass flux is zero.
(20)
Fcl(1 - εa)Vcl,a2
Combination of eqs 18, 19, and 20 leads to the following equation: 2RfiUg,c2 φcl2.5
+
2fwβVcl,a2 (1 - φcl)2
) 1 - φcl
(21)
where R)
Fgεc (Fcl - Fg)gDt
(22)
β)
Fcl(1 - εa) (Fcl - FggDt)
(23)
and
The interfacial friction factor could be evaluated from Blasius:16 0.3164 (24) Re1⁄4 in which Reynolds number is defined based on the core radius as fi )
Re )
2rcUg,cFg µg
(25)
The wall friction factor could be evaluated from Konno:17 fw ) 0.0025V-1
[
(26)
]
Thus the annular cluster velocity could be determined as
Vcl,a )
(1 - φcl) -
2RfiUg,c2 φcl2.5
2fwβ/(1 - φcl)2
0.5
(27)
2.3. Method of Solution. Given the operating conditions, materials properties, and riser diameter, the average annular cluster velocity could be evaluated both from the mass balance method through eq 17 and from the momentum balance method through eq 27. The only unknown parameter in these equations is the core radius, which could be determined by an iterative Table 3. Correlations for Estimating Annular Thickness Layera reference Bi et al.
21
Bai et al.22 Patience and Chaouki15 Kim et al.
23
Zhang et al.24 Pugsley and Berruti6 this work a
correlation/model
AAD (%)
δ / D ) 0.5[1 - √1.34-1.30εs0.2+εs1.4] δ / D ) 0.403εs0.7
[
√
(
21.53 0.083FrD
)
]
δ / D ) 0.5 1 - 1⁄ 1+1.1FrD Gs / FpUg 0.16 δ / D0.85(Ug2/g)0.15 ) 1.73(V/)0.21(Frdp)-0.97( Fpεs / Fgε ) ; V*
