Ind. Eng. Chem. Res. 1999, 38, 81-89
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KINETICS, CATALYSIS, AND REACTION ENGINEERING Radial Hydrodynamics in Risers Larin Godfroy,† Gregory S. Patience,*,‡ and Jamal Chaouki† Department of Chemical Engineering, E Ä cole Polytechnique de Montre´ al, C.P. 6079, Succursale Centre-Ville, Montre´ al, Que´ bec, Canada H3C 3A7, and E.I. du Pont de Nemours & Company, Wilmington, Delaware 19880-0262
On the basis of the benchmark modeling exercise at Fluidization VIII, predicting riser hydrodynamics continues to be more of an art than a science. Ten different hydrodynamic models were compared with a set of experimental data that covered a wide range of operating conditions and showed reasonable to poor overall agreement. Herein, we describe the model that gave the best overall agreement with the experimental data. Density is calculated by a correlation based on slip factor, and the radial voidage profile depends solely on the cross-sectional average void fraction. Both the gas and velocity profile follows a power law type expression; the gas velocity at the wall is zero. The model predictions agree well with experiments conducted with sand but not as well as those conducted with fluidized catalytic cracking catalyst. 1. Introduction Circulating fluid bed (CFB) hydrodynamic models are useful for understanding gas-solids mixing, scale-up, plant optimization, and control. Hydrodynamics impact reactor performance: conversion, selectivity, and heat transfer. Furthermore, CFB riser operating conditions affect the efficiency of downstream equipment such as cyclones, filters, standpipes, and so forth. Hydrodynamic modeling is useful for understanding and optimizing plant conditions, but they do not offer the level of confidence required to design, a priori, a new commercial plant. Rather, new commercial facilities are designed on the basis of extensive piloting and conservative extrapolations of pilot-plant basic data. Pilot plants, at a sufficiently large scale, minimize the risk of projecting performance to commercial scale and provide the means with which to test alternative designs rapidly and economically. Three examples of this approach include the NUCLA power generation facility (1), Mobil’s short contact time catalytic cracker (2), and DuPont’s butane to maleic anhydride process (3). In the last two decades, significant advances have been made in experimental measurements of riser hydrodynamics and a number of models have emerged to characterize these data. However, most models are developed on the basis of a limited data set and their extrapolation to conditions outside the range is not welldocumented. For this reason J. Chen proposed a “benchmark modelling exercise” to compare model predictions against unpublished experimental data that cover a wide range of operating conditions. T. Knowlton prepared the exercise and invited modelers to predict the axial pressure drop, radial void fraction, and mass flux in two different risers. He disclosed the CFB geometry, * To whom correspondence may be addressed. † E Ä cole Polytechnique de Montre´al. ‡ E.I. du Pont de Nemours & Co.
particle characteristic, and operating conditions. Ten teams accepted this challenge and T. Knowlton, D. Geldart, and J. Matsen presented the results of the exercise at Fluidization VIII. In this paper, we discuss the benchmark modeling database and describe in detail the model proposed by Chaouki, Godfroy, and Patience. Throughout this discussion, we highlight some difficulties in measuring experimental data and the strengths and weaknesses of our model. 2. Design Considerations Operational flexibility is of particular importance in many CFB applications. In both combustion and fluid catalytic cracking (FCC), operators often require the ability to treat a variety of feedstocks. Flexibility is an advantage of CFB technology but, at the design stage, this flexibility often translates into uncertainty. The largest uncertainty relates to predicting the solid volumetric fractionssolid holdup or inventorysas a function of geometry and operating conditions. Holdup increases with an increasing solid circulation rate and decreases with an increasing gas velocity. The solid holdup not only affects the riser pressure drop but may also affect reactor performance: for example, in FCC units, higher solid holdup, resulting from increasing the solid circulation rate, may alter the temperature profile and, thus, the hydrocarbon product distribution. Together with an increasing temperature, an increased inventory affects the specific reaction rates. Figure 1 is a simplified schematic of the principle interactions between reaction kinetics and hydrodynamics at the design stage. To meet economic objectives, the process equipment size should be minimized and process yields maximized (conversion, X, and selectivity, S). Process equipment sizing depends on both the overall catalyst inventory and gas volumetric flow rates. Therefore, process design is an exercise in minimizing the
10.1021/ie960784i CCC: $18.00 © 1999 American Chemical Society Published on Web 12/03/1998
82 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999
G
Figure 1. Design flow sequence to determine the geometry and operating conditions.
catalyst inventory. Generally, the design begins with specifying the production rate at a desired level of conversion and selectivity (which depends on temperature, T). Together with the reaction kinetics (k), volumetric flow rate, and inventory, we can now begin to estimate the reactor size and concentrate on hydrodynamics. The two principle operating parameters are the gas velocity (Ug) and solid mass flux (Gs); the two design parameters are the reactor diameter (D) and height (H). The relationship between these four factors are complicated, and to arrive at the final design may require several iterations, as shown in Figure 1. The bounds of the riser diameter are set on the basis of gas velocity considerations: high gas velocities may lead to excessive catalyst attrition and low velocities may result in poor solids circulation stabilityschoking. Solid mass flux is the next parameter to consider; it affects solid suspension density and may also impact conversion and selectivity. Higher mass flux results in higher suspension densities at a constant gas velocity and riser diameter. Therefore, to satisfy the solid inventory requirements, by specifying mass fluxsand thus suspension densitysthe riser height may be calculated. The riser height is constrained by pressure buildup considerations in the recycle loop; therefore, suspension densities in the riser must be sufficiently low so that the pressure buildup in the recycle loop (∆Precirculation) is greater than the riser pressure drop (∆Priser). High mass flux is generally preferred, but it may result in too high a pressure dropsresulting in lower than design solid circulation ratesand too much backmixing, which may reduce both the conversion and selectivity, as shown by the dotted line. In the case of low solid mass flux, the suspension density becomes low, which results in a very tall reactor. Generally, suspension densities in the recycle loop are near the bulk density; therefore, catalyst inventory in tall reactors is considerably greater than that in shorter vessels. To satisfy the economic objectives, the ratio of the riser inventory to the recycle loop inventory (Wriser/Wrecirculation) should be minimized. Independent parameters that impact riser hydrodynamics include geometrysentrance and exit configuration as well as bed diametersgas velocity, circulation rate, and particle characteristics. These parameters form the basis of hydrodynamic models and are used to predict the overall riser pressure drop, and radial density and velocity profile.
literature to predict the relationship between solid holdup, operating conditions, and riser geometry. Harris and Davidson (1994) classified the models into three broad categories: (I) those that predict the axial solid suspension density profile, but not the radial profile; (II) those that predict the radial profile by assuming two or more regions, such as core-annulus or clustering annular flow models; (III) those that employ the fundamental equations of fluid dynamics to predict twophase gas-solid flow. Type III models, because of their generality, are suitable for predicting the effects of a complex geometry. However, the constituent equations for two-phase gas-solids flow is not well-developed and the numerical complexity often discourages their use. Proponents of the type I and II models cite ease of understanding and usage along with generally very good agreement with experimental data as the main advantages. Detractors argue that the assumptions of the flow structure associated with such models oversimplify the complex flow pattern. One important consideration in selecting a modeling approach is its intended application: type I and II models may best be employed as a design tool to investigate the effects of operating conditions and riser dimensions on the riser flow structure as well as for control models. In addition, they may be easily coupled with reaction kinetic models to simulate the performance of CFB reactors (Pugsley et al. (1992), Patience and Mills (1994), Bolkan-Kenny et al. (1994)). Type III models, however, are well-suited to investigate the riser local flow structure and the impact of geometry, such as corner effects in CFB combustors, or a unique inlet configuration such as those studied by Pita and Sundaresan (1993). They could also be useful for control development and testing. However, they are too complex to use directly in control algorithms. The three models may also be characterized by the mathematical approach taken. This characterization is obvious for type III models in which a system of equations contains the continuity, momentum, pseudo-thermal energy balance, and constitutive relations are solved. However, types I and II may involve the use of correlations based on experimental data, referred to as the “lumped approach” or a combination of correlations and fundamental relationships. More detail of each class of model is provided in a review of hydrodynamics by Berruti et al. (1995).
3. Hydrodynamic Models
Over the past several years, significant advances have been made in both the experimental measurements of riser hydrodynamics and characterization. To highlight recent advances and define future directions, a bench-
Several modeling efforts employing very different mathematical formulations have appeared in the recent
4. CFB Workshop
Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 83 Table 1. Test Matrix for Hydrodynamic Predictions FCC, dp ) 76 µm, Fp ) 1712 49 kg/m2‚s
Ug, Gs 5.2 m/s 7.6 m/s 11 m/s
kg/m3,
D ) 0.2 m
196 kg/m2‚s
489 kg/m2‚s
782 kg/m2‚s
X
X X X
X
X
Sand, dp ) 175 µm, Fp ) 2145
kg/m3,
Table 2. Suspension Densities in Different Riser Diameters study
D (m)
Ug (m/s)
Gs (kg/m2‚s)
F (kg/m3)
Avidan (1980) Avidan (1980) Rhodes et al. (1992) Rhodes et al. (1992)
0.076 0.152 0.152 0.305
4 4 4 4
60 60 60 60
39 20 56 29
D ) 0.2 m
Ug, Gs
15 kg/m2‚s
29 kg/m2‚s
51 kg/m2‚s
2.4 m/s 4 m/s 5.8 m/s
X
X X X
X
Sand, dp ) 120 µm, Fp ) 2600 kg/m3, D ) 0.4 m Ug, Gs
50 kg/m2‚s
4.2 m/s
X
mark modeling exercise was proposed at Fluidization VII. The test matrix is shown in Table 1. Knolwton (1995) measured the axial pressure drop, radial void fraction profile, and the radial profile of the axial solid mass flux in two tall risers. One riser was 14.2 m tall and 0.2 m in diameter and the other was 9 m tall and 0.4 m in diameter. Both risers were equipped with round smooth elbows at the top and solids were fed at the bottom through L-valves. Sand and FCC were the test solids and air was the fluidizing medium. Most of the data were collected in the taller, narrower reactor. Only one test was conducted in the larger diameter reactor. With FCC powder, Knowlton (1995) varied the gas velocity from 5.2 to 11 m/s and the solid mass flux from 49 to 782 kg/m2‚s. (There is some error associated with the data at the highest mass flux because the integrated cross-sectional value is only 640 kg/m2‚s.) Experimental conditions with sand were at lower gas velocities and solid mass flux that more closely approximate CFB combustor operation: 2.4 < Ug < 5.8 m/s and 15 < Gs < 51 kg/m2‚s. In Jan 1995, Knowlton invited modelers to predict the riser hydrodynamics of each experiment, but he only provided the riser operating conditions, reactor geometry, and particle characteristics. In May 1995, the model predictions and experimental data were compared in a public forum in the CFB Workshop at Fluidization VIII. Ten research groups accepted the challenge to predict the hydrodynamics: Bernard; Sundaresan; Arastoopour and Kim; Gidaspow and Sun; Johnsson; Chaouki, Godfroy and Patience; Pugsley and Berruti; Rhodes and Wang; O’Brien and Syamlal; Chen. Comparison between predictions and the experimental data were presented by Knowlton, Matsen, and Geldart. Two plots were shown for each participating groupsone which represented the best agreement between experimental data and model predictions and the other in which the agreement was considered the worst. Most of the groups did not attempt to model all the test conditions; nonetheless, the exercise provided a good perspective of the state of the art in CFB modeling. Only types II and III modelers participated. The conclusions of the exercise were that (a) no single model exists that can adequately predict all the conditions and trends in the data; (b) type II shows better agreement with the data compared to type III; (c) some models are good over a limited range of conditions; (d) most models fail to properly represent radial density and the solid mass flux profile of an FCC catalyst at the highest mass flux; (e) no model adequately predicts
the increase in suspension density at the top of the riser; (f) Bernard’s model predicts FCC data quite well; (g) the models of Chaouki, Godfroy, and Patience and Pugsley and Berruti show the best overall agreement with the experimental data; (h) the best type III model is that of Gidaspow and Sun, which matched some significant trends in the radial mass flux profiles. During subsequent discussion, the necessity of including additional criteria in the assessment of the different models was pointed out. One of the most crucial requirements is user-friendliness and speed in generating results. The benchmark modeling test provided a fair representation of the accuracy and applicability of present hydrodynamic models and it indicated direction for future developments. 5. Hydrodynamic Model Most experimental data show strong radial velocity and solid volume fraction gradients at the wall. To characterize this phenomena, we approximate the radial profiles of particle velocity and suspensions density with continuous functions. The input parameters are the superficial gas velocity (Ug), the solid circulation rate (Gs), the riser diameter (D), and the particles density and diameter. The model predicts (1) a fully developed average void fraction, (2) a radial void fraction profile, (3) a radial profile of the axial gas velocity, (4) a radial profile of the axial particle velocity, and (5) a radial profile of the axial solid mass flux. The state of riser modeling is handicapped by the inconsistency in the published experimental data. Controversies abound in the open literature. A clear example is the effect of diameter on void fraction. As shown in Table 2, Avidan (1980) reports lower density in a larger diameter column at the same operating conditions: 39 kg/m3 in a 7.6 cm column versus 20 kg/ m3 in a 15.2 cm diameter column. These trends are also reported by Rhodes et al. (1992). They show that the density increases by a factor of 2 in the smaller diameter riser and attibuted the increase to solid friction and a wall phenomenon effect. Gas velocities are near zero at the wall and solid residence times are higher there, which leads to higher local densities. Therefore, proportionately more solids will tend to agglomerate at the wall in small diameter risers compared to large diameter risers because of their higher surface-to-volume ratio. However, although the trends in the two studies agree, the absolute values differ by a factor of almost 3: the density reported by Rhodes et al. (1992) is 56 kg/m3 in a 15.2 cm column compared to only 20 kg/m3 in Avidan’s test. That difference is greater than the diameter effect. Discrepancies such as these are not uncommon in the literature and contribute to the lack of convergence of a unique interpretation of hydrodynamic trends. The wall phenomenon effect may not be the dominant factor relating diameter and suspension densities; it does not, for example, account for experimental findings
84 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999
Figure 2. Comparison between predicted and experimental pressure drops for sand (closed symbols, D ) 0.4 m; open symbols, D ) 0.2 m).
of Contractor et al. (1992). They reported lower densities after introducing surface area in the form of a vertical heat-transfer tube. The reason for the discrepancy may be related to the difficulty in accurately measuring solid mass flux and gas velocities. For example, pressure gradients in tall reactors are significant and the pressure at the bottom of the riser may as high as 0.1-0.2 barg. Thus, the true superficial velocity will be 10-20% lower than that reported. 5.1. Fully Developed Average Void Fraction. In industrial-sized FCC risers, Matsen (1976) reported that the ratio of the gas velocity to solid velocitysslip factor, ψswas about 2. Patience et al. (1992) correlated data from a number of experimental risers using this concept and developed a nondimensional expression to account for the effects of particle characteristics, riser diameter, and gas velocity.
ψ ) 1 + 5.6/Fr + 0.47Frt0.41 ) (Ug/)/Vp
(1)
where Fr is the Froude number and Frt is the terminal Froude number. The three main variables that affect ψ are the gas velocity, particle characteristics, and riser diameter. The slip factor decreases with an increasing gas velocity. Equation 1 predicts that the void fraction increases with an increasing diameter. This trend is opposite to the results of Rhodes et al. (1992) and Avidan (1980). The void fraction increases with an increasing gas velocity, decreasing solid circulation rate, and decreasing slip factor.
)
1 1 + Gsψ/(UgFp)
(2)
Figure 2 compares the calculated pressure gradient with experimental data for the 0.2 and 0.4 m diameter riser with sand. In the fully developed region, it shows good agreement between the model and the experimental
Figure 3. Comparison between predicted and experimental pressure drops for FCC (Gs ) 489 kg/m2‚s).
data. In the 0.2 m diameter riser, two experiments were reported at a constant mass flux of 29 kg/m2‚s and gas velocities of 4 and 5.8 m/s. We predict the correct trend inasmuch as the pressure gradientssolid holdups decreases with an increasing gas velocity. In the 0.4 m diameter riser, the gas velocity was 4.2 m/s and the mass flux was 50 kg/m2‚s; a gas velocity similar to that in the experiment in the 0.2 m unit was almost twice the mass flux. Again, we show good agreement in the fully developed region, which suggests that the slip factor correlation adequately predicts the influence of the riser diameter for this data set. FCC data are compared with model predictions in Figure 3. In the fully developed region, we consistently overpredict the pressure gradient by about 30%. However, the model predicts the relative change in the suspension density with the change in the gas superficial velocity. For example, at a height of 8.2 m, the experimental pressure drop decreases from 870 Pa/m at 7.6 m/s to 490 Pa/m at 5.2 m/s. A difference of 380 Pa/m. For the same conditions, the model predicts a drop of 415 Pa/m. The slip factor correlation is based on a number of studies with sand and FCC powders. The database for FCC at high gas velocities and mass flux was taken from van Swaaij et al. (1970). They report slip factors in the range of 1.6-2.2. More recently, Contractor et al. (1994) published data for FCC in the same operating range in a 15 cm riser and their results agree well with those of van Swaaij et al. As shown in Table 3, slip factors reported by Knowlton (1995) for FCC are much lower than those found in the data in the earlier studies. Viitanen (1993) measured both solid and gas velocities using radioactive tracers in a commercial FCC riser 1.3 m in diameter. He reported solid and gas velocities from 12 to 14 m/s and solid velocities between 6 and 8 m/s, which gives a slip factor near 2. This value is significantly higher than that reported by Knowlton (1995).
Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 85 Table 3. Slip Factors for FCC Powder Ug (m/s)
Gs (kg/m2‚s)
D (m)
ψa
11 7.6 5.2 11 7.6 5.7 8.9 10.1 5.6 7 9
500 500 500 500 300 300 500 500 550 550 550
0.2 0.2 0.2 0.18 0.18 0.18 0.18 0.18 0.15 0.15 0.15
1.16 1.45 1.74 1.9
ψb
ψc
ψd
2.5 2.2 1.9
1.9 2.3 2.7 1.6 2.2 2.5 2.1 2 2.5 2.2 2
2.3 2.3 2 2.3
a Knowlton (1995). b van Swaaij et al. (1970). c Contractor et al. (1994). d Equation 1.
Furthermore, it is inconsistent with Rhodes et al. (1992) and Avidan (1980) experimental data, which suggests that the slip factor decreases with an increasing diameter. Equally controversial as the effect of diameter on the slip factor is the variation of the suspension density along the riser axis, particulary at the entrance and exit. Generally, when solids enter the riser horizontally from an L-valve, the suspension density decays exponentially to an asymptote. We refer to the region where the solid density is essentially constant as that in the fully developed region. The length of the acceleration region depends on the gas injection, riser geometry, and operating conditions; it increases with suspension density; it is higher at lower gas velocities and higher solid mass flux. Many experimental risers demonstrate a sigmoidal distribution with a V-valve solid feed device: density is highest at the base and constant up to a certain height at which point it transitions exponentially to a lower value. Considerable industrial research is devoted to distributing the gas and solids evenly across the riser and thus minimizing the entrance region. Entrance and exit region effects in experimental risers require further investigation. Measuring accurately the solid circulation rate is a difficult task, and for this reason the published literature is inconsistent. Perhaps the first benchmark exercise should have been to compare solid circulation rate measurements as well as density measurements. 5.2. Radial Void Fraction Profile. Zhang et al. (1991) reported radial void fraction profiles for four different powders in three different risers up to 0.3 m in diameter. They found that the normalized radial void fraction was a unique function of radial distance. Data reported by Herb et al. (1989), Mineo (1989), and Tung et al. (1989) also support this observation. Zhang et al. (1991) correlated their data and found that the centerline void fraction was best approximated by cl ) 0.191. However, this expression appears to underestimate the actual void fraction inasmuch as it poorly accounts for the observed trends in solid mass flux. Solid mass flux is the product of the local density and local particle velocity. The local particle velocity is the highest at the center, as is often the case for mass flux. However, Zhang’s correlation predicts a local minimum for centerline mass flux for most experimental risers. For this reason, Patience and Chaouki (1995) recorrelated Zhang’s data with mass flux data and developed the following relationship:
0.4 - ξ 0.4 -
) 4ξ6
(3)
Figure 4. Normalized void fraction as a function of the radial position for sand.
The only input parameter is sthe average void fraction. The numerator in eq 3 is the difference between the centerline and average void fraction while the denominator is the difference between the centerline void fraction and the void fraction at a nondimensional distance, ξ. The void fraction at the centerline, 0.4, is much higher than the value proposed by Zhang et al. (1991). In the turbulent and bubbling fluidization regimes, the centerline void fraction is higher yet. Data published by Abed (1983) suggest that the range is from 0.5 to 1.0; the exponent increases with a decreasing gas velocity. As shown in Figure 4, overall agreement between the experimental void fraction data with sand and that predicted by eq 3 is good. The curves show the same characteristic shape, which is consistent with Zhang et al.’s postulate that the reduced radial profile is a unique function of the radial distance. In Figure 5, we show the experimental FCC data compared with eq 3 in its reduced form. Under dilute operating conditions, the reduced radial void fraction profile appears to depend on radial distance only; agreement between model predictions and experimental data is generally good at a high gas velocity. However, at a low gas velocity and high mass fluxshigh suspension densitiessthe shape of the curve changes. For example, at 7.6 m/s and 489 kg/m2‚s, the reduced void fraction is about 2 for ξ > 0.75. The solid density at the wall appears to reach a limiting value; with further increases in the suspension density (either by an increasing solid circulation rate at a constant gas velocity or by a decreasing gas velocity at a constant circulation rate) solids at the wall do not compact further; rather they migrate toward the center, thus increasing the average local density until it reaches the limiting value. Presumably, the limiting void fraction at the wall is largely a function of the gas velocity. 5.3. Radial Profile of Axial Gas Velocity. van Zoonen (1962) reported that the local solid holdup was greater near the wall compared to that near the cen-
86 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999
solid velocity datasto identify parameters that affect the velocity profile. The data set included both sand and FCC powders and covered a large range of operating conditions and riser diameters up to 0.3 m. They found that the profile depends most strongly on gas velocity and to a lesser extent on mass flux and riser diameter,
R ) 1 + 0.036Fr(UgFp/Gs)0.5
(5)
With an increasing gas velocity, the solid suspension density decreases and the velocity profile approaches turbulent. A decreasing mass flux has the same effect on suspension density; the velocity profile approaches turbulent at a lower mass flux. The correlation predicts an interesting affect with respect to the riser diameter: at a constant gas velocity and mass flux, the velocity profile tends toward a turbulent profile with a decreasing diameter. This trend is consistent with the slip factor model, which predicts that the solid holdup increases with an increasing diameter. However, as already mentioned, the effect of the diameter on suspension density is not well-documented and it is less well-understood for gas hydrodynamics. The value of γ in eq 4 is calculated from a mass balance on the gas: Figure 5. Normalized void fraction as a function of the radial position for FCC.
terline and that the radial particle velocity was nearly parabolic. On the basis of these observations, Rowe (1962) suggested that the gas might have a similar radial velocity distribution. He reasoned that the presence of solids would dampen the turbulence and cause the gas to be in streamline flow. Data reported from the NUCLA (1) power-plant experimental program support this hypothesis. They sampled the gas stream over the radius at two heights 7 m apart and found similar radial concentration profiles. If radial gas mixing was significant, then we would expect that the concentration profile might become flat. However, steady-state injection of helium gas in experimental risers show that gas backmixing declines with an increasing gas velocity (Cankurt and Yerushalmi, 1978) but that radial dispersion can be an important effect (Yang et al., 1983). Other important issues remaining concerning gasphase hydrodynamics relate to the wall boundary condition and entrance effects. Some data suggest that gas is carried down by the solids along the wall, while other data indicate that the no slip boundary condition is an appropriate approximation. Under certain conditions of high mass flux and suspension density, solids have been reported to ascend at the wall, and this effect has not yet been well-documented. In our model, we assume that the radial profile of (axial) gas velocity is continuous and can be approximated by a power law type expression, similar to that proposed by Martin al. (1992). We assume the no slip boundary condition at the wall and that the velocity is maximum at the center:
Vg,ξ ) Ug/γ(1 - ξR)
(4)
The value of R varies between 1 and 7, where 1 approximates a triangular profile, 2 is parabolic, and 7 approximates a turbulent profile. Patience et al. (1996) correlated experimental datasboth gas velocity and
1 1 Ug,ξ2ξ dξ ) ∫ξ)0ξVg,ξ2ξ dξ ) Ug ∫ξ)0
(6)
where Ug,ξ is the local superficial velocity. Solving for γ gives
γ)-
8(0.4 - ) 20.4 + R+2 R+8
(7)
5.4. Radial Profile of Axial Solid Velocity. On the basis of van Zoonen’s (1962) experimental data, Rowe (1962) suggested that both gas and solids might have a similar parabolic velocity profile. He further postulated that the slip velocity between the two phases would also be similar at all radii. We assume that the slip velocity at the center equals the particle terminal velocity:
Vp,cl ) Vg,cl - Vt
(8)
Solid holdup is low and the particles are well-dispersed. For the radial solid velocity profile, we use a power law expression with the same exponent, R, used in the radial gas velocity profile, eq 4: R Vp,ξ ) Vp,cl(1 - (ξ/φ1/2 s ) )
(9)
where φs is the fraction of the cross-sectional area in which particles ascend. Gas and solid radial velocity profiles are similar but the slip velocities across the radius are not equal, as Rowe (1962) postulated. At the wall, particle velocities may be negative and considerably greater than the particle terminal velocity while the gas velocity is zero. The expression for the radial velocity profile has the same form as that for the gas velocity profile except that the velocity is negative for ξ > ξp,0. φs is calculated based on an overall mass flux balance 1 Fp(1 - ξ)Vp,ξ2ξ dξ ≡ Gs ∫ξ)0
(10)
The solid mass flux is the integral of the product of the local solid holdup (eq 3) and the local solid velocity (eq
Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999 87
Figure 6. Selected figure showing good agreement (radial profile of axial solids mass flux; sand: D ) 0.4 m; Ug ) 4.2 m/s; Gs ) 50 kg/m2‚s).
9). Integrating this product and solving for φs gives
(
)
2(1 - 0.4) (0.4 - ) +8 R+2 R+8 φs ) (1 - ) - Gs/(FpVp,cl)
2/R
(11)
which simplifies to With an increasing gas velocity at a
φs )
(
)
γ - + 2/(R + 2) (1 - )(1 - Vp/Vp,cl)
2/R
(12)
fixed mass flux, the value of φs will also increase. It is essentially constant with mass flux over 2 orders of magnitude. With a larger riser diameter, eq 12 predicts that the cross-sectional area in which solids descend along the wall is greater. This trend is consistent with the slip factor model and eq 4 relating the gas velocity profile with operating conditions: in larger diameter risers, suspension densities are higher, the velocity gradient is steeper, and more solids descend along the wall. 5.5. Radial Profile of Axial Solid Mass. Solid mass flux is a product of the local velocity and local suspension density:
Gs,ξ ) VpFp(1 - ξ)
(13)
where the local solid velocity is calculated on the basis of eq 9 and the local void fraction is from eq 3. During the Fluidization VIII Workshop, J. Matsen was charged with the task of presenting a graph from each participating research group in which agreement between experimental data and model predictions was the closest. D. Geldart had the misfortune to compare a graph that showed the least agreement between experimental data and model predictions. Matsen selected Figure 6 as an example where our model agreed well with the
Figure 7. Selected figure showing poor agreement (radial profile of axial solids mass flux; FCC: D ) 0.2 m; Ug ) 11 m/s).
data. In the 0.4 m diameter riser, mass flux at the centerline is almost 2 times the average. Wall mass flux is negative at the wall and about 4 times the average. The transition from upflow to downflow occurs at φ0.5 s ) 0.9. All of these characteristics were captured by the model remarkably well. Geldart chose Figure 7 as the example where the model predictions fit the experimental data poorly. He pointed out that we overpredict the centerline mass flux and show negative values at the wall (although the experiments do not include data near the wall). He did add that the model captured some unique characteristics of the data and in fact “was not that bad”. For example, the centerline velocity is at a local minimum and the absolute maximum is nearer to the wall. The model fits the experiments conducted at a mass flux below 200 kg/m2‚s very well. The agreement is poor at the highest mass flux. It appears that the difference is, in part, due to an error in the experimentally reported value. Integrating the experimental local flux gives 640 kg/m2‚s compared to their reported value of 782 kg/m2‚s. At an average mass flux of 640 kg/m2‚s, our model predicts 875 kg/m2‚s at the center, whereas we calculated 1100 kg/m2‚s with 782 kg/m2‚s. Regardless of the error in the reported mass flux, we still overpredict Gs at the center. A contributing factor to the difference may be due to the slip factor model that overpredicts the average density: higher densities in the center will result in higher local mass flux. 6. Conclusions We described a simple two-dimensional hydrodynamic model to approximate CFB riser hydrodynamics in the fully developed region. The model predicts solid holdup, radial void fraction, and the radial profile of axial gas and solid velocity and mass flux. At the Fluidization VIII CFB Workshop it was cited as providing the best
88 Ind. Eng. Chem. Res., Vol. 38, No. 1, 1999
overall agreement with experimental data. The essential features of the hydrodynamic model are as follows: (1) The slip factor increases with the riser diameter and decreases with the gas velocity (eq 1). (2) The normalized void fraction profile is a unique function of the radial distance (eq 3). (3) The centerline void fraction is equal to the average raised to the power of 0.4 (eq 3). (4) The gas velocity at the wall is zero. (5) The radial profile of axial gas velocity follows a power law (eqs 4 and 5). (6) At the center, the slip velocity equals the singleparticle terminal velocity (eq 8). (7) The radial profile of axial particle velocity follows the same power law as that of the gas (eqs 5 and 9). (8) The solid wall velocities are calculated on the basis of a solid flux balance (eq 12). The model predicts an axial pressure gradient for experiments conducted with sand quite well. It overpredicts the pressure gradient for the FCC data; however, it correctly predicts the trend and the fractional change in the pressure drop. We show that the normalized void fraction is a unique function of the radial distance and that the centerline void fraction is equal to the cross-sectional average void fraction raised to the power of 0.4. Again, as with the pressure gradient, the agreement is better for experiments with sand compared to those with FCC. Model predictions are good at low suspension densities with FCC, but the shape of the curve changes at high suspension densities. We calculate analytically the radius in which the solids flow transitions from upflow to downflow and the predictions agree very well with the experiments conducted with sand. Acknowledgment This research was funded by the Natural Science and Engineering Research Council of Canada. Nomenclature D ) riser diameter, m dp ) particle diameter, m k ) kinetic constant Fr ) Froude number, Ug/x(gD) Frt ) terminal Froude number, Vt/x(gD) g ) gravitational constant, m/s2 Gs ) solid mass flux, kg/m2‚s Gs,ξ ) solid mass flux at radius ξ, kg/m2‚s H ) riser height, m Q ) gas flow rate, m3/s r ) radial coordinate, m R ) riser radius, m S ) selectivity T ) temperature, K Ug ) gas superficial velocity, m/s Vg ) actual gas velocity, m/s Vg,ξ ) local gas velocity at radius ξ, m/s Vp ) particle velocity, m/s Vp,ξ ) local particle velocity at radius ξ, m/s Vt ) terminal velocity, m/s W ) catalyst inventory, kg X ) conversion z ) axial coordinate, m R ) exponent in the gas and particle velocity profile (eqs 4, 5, and 9) ∆P ) pressure drop, Pa ) mean void fraction
cl ) centerline void fraction z ) mean void fraction at height z ξ ) void fraction at radius ξ φs ) fraction of the cross-sectional area in which particles ascend ψ ) slip factor γ ) parameter defined by eq 7 F ) average suspension density, kg/m3 Fp ) particle density, kg/m3 ξ ) nondimensional radial coordinate, r/R ξp,0 ) nondimensional radius where axial particle velocity equals zero
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Resubmitted for review August 2, 1998 Accepted October 19, 1998 IE960784I
